Math. Nachr. (1999),

Local Spectra and Index of Singular Integral Operators with Piecewise Continuous Coefficients on Composed Curves

By C.J. Bishop of Stony Brook, A. Böttcher of Chemnitz, Yu.I. Karlovich of Chemnitz, I. Spitkovsky of Williamsburg

“The spectrum gives an operator a personality.”

L.N. Trefethen [60]

(Received January 21, 1997)

Abstract. We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space L^{p}(Γ, w) where 1 < p < ∞, Γ is a composed Carleson curve, and w is a Muckenhoupt weight in the class A_{p}(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate “N projections theorem”, and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.

1. Introduction

In 1960, Harold Widom discovered that the spectrum of the Cauchy singular integral operator on the space L^{p}(0, ∞) is a circular arc joining the points −1 and 1 whose shape depends on p and which degenerates to the line segment [−1, 1] in case p = 2. About ten years later, I. Gohberg and N. Krupnik had accomplished their Fredholm theory of singular integral operators with piecewise continuous coefficients on L^{p} spaces over composed Lyapunov curves with power weights. Only in recent time, some phenomena caused by general Carleson curves and general Muckenhoupt

1991 Mathematics Subject Classification. Primary 47B35; Secondary 16R20, 30C20, 30D55, 30E20, 42A50, 45E05, 46J15, 47A53, 47C05, 47D30.

Keywords and phrases. Ahlfors-David curve, Carleson condition, Muckenhoupt condition, singular integral operator, spectral theory, index, idempotent, geometric function theory

weights were understood, and it became clear that Carleson curves and Muckenhoupt weights effect a surprising and beautiful metamorphosis of the circular arcs prevailing the Widom/Gohberg/Krupnik theory.

Some kind of symbol calculus for deciding whether a singular integral operator with piecewise continuous coefficients is Fredholm on L^{p}(Γ, w) (w ∈ A_{p}(Γ)) is presently available in two cases: if Γ is a composed Lyapunov curve [35] or if Γ is a Carleson Jordan curve [8]. Curves composed by arcs with oscillating tangents were considered in [48], [49], [11] under the assumption that w is a so-called power weight. In this paper we provide a symbol calculus for the case of an arbitrary composed Carleson curve. We remark that, in a sense, composed Carleson curves constitute the maximal class of curves for which a symbol calculus with matrices of finite order is possible.

The paper is organized as follows. Sections 2 and 3 are devoted to the preliminaries. In Section 2 we introduce composed Carleson curves and Muckenhoupt weights. In Section 3 we discuss some basic properties of singular integral operators and define the notion of the local spectrum of such operators. The main results are stated in Section 4. Several special cases of the main theorem as well as pieces of its history are illuminated in Section 5. The main results are proved in Sections 6 to 10.

In Section 6 we prove that Carleson stars can be extended to Carleson flowers and that Muckenhoupt weights on a Carleson star can be extended to Muckenhoupt weights on the corresponding Carleson flower. In Section 7 we apply an appropriate N projections theorem to local algebras of Carleson flowers, which reduces the invertibility problem in these algebras to the problem of studying the invertibility of certain matrix functions living on certain sets. The latter sets are called leaves. Some preliminary information about the structure of the leaves is obtained in Section 8. In Section 9 we connect the local algebras of an arbitrary composed Carleson curve with the local algebras of Carleson flowers. The precise shape of the leaves is then determined in Section 10.

Once the theorems of Section 4 are available, many results on singular integral operators can be extended to the most general setting by using standard techniques. In Section 11 we supplement the results of Section 4 by showing that neither the consideration of systems of singular integral equations nor relaxation of the question about Fredholmness to the one about semi-Fredholmness yield qualitatively new phenomena. In Section 12 we establish index formulas which do not only extend known index formulas to the case of Carleson curves with Muckenhoupt weights but also differ in their form from previous index formulas. Section 13 is dedicated to “Coburn’s lemma”, which says that certain operators, including the operator S_{Γ} − λI, are invertible if and only if they are Fredholm of index zero. The spectrum of the Cauchy singular integral operator S_{Γ} over an arbitrary composed Carleson curve with an arbitrary Muckenhoupt weight is finally determined in Section 14.

2. The Cauchy singular integral

An oriented rectifiable curve in the complex plane is called a simple arc if it is homeomorphic to a line segment. The union of finitely many simple arcs each pair of which have at most endpoints in common is referred to as a composed curve.

Let Γ be a composed curve and equip Γ with Lebesgue length measure |dτ|. A measurable function w : Γ → [0, ∞] is called a weight if w^{−1}(0) and w^{−1}(∞) have measure zero. We denote by L^{p}(Γ) (1 ≤ p ≤ ∞) the usual Lebesgue spaces of Γ and by L^{p}(Γ, w) (1 < p < ∞) the weighted Lebesgue spaces with the norms

∥f∥_{p,w} := ( ∫_{Γ} |f(τ)|^{p}w(τ)^{p} |dτ|)^{1/p}.

We write A_{p}(Γ) (1 < p < ∞) for the class of all weights w on Γ such that w ∈ L^{p}(Γ), w^{−1} ∈ L^{q}(Γ), and

(2.1)  sup_{t∈Γ} sup_{ε>0} \frac{1}{ε}( ∫_{Γ(t,ε)} w(τ)^{p} |dτ|)^{1/p}( ∫_{Γ(t,ε)} w(τ)^{−q} |dτ|)^{1/q} < ∞,

where here and in what follows

Γ(t, ε) := {τ ∈ Γ : |τ − t| < ε} and 1/p + 1/q = 1.

The weights in A_{p}(Γ) are usually called Muckenhoupt weights.

Not every composed curve Γ may carry a Muckenhoupt weight. Indeed, if (2.1) holds then Hölder’s inequality gives

(2.2)  sup_{t∈Γ} sup_{ε>0} \frac{1}{ε}|Γ(t, ε)| < ∞

where |Γ(t, ε)| is the Lebesgue length measure of the portion Γ(t, ε). On the other hand, if (2.2) is valid then constant weights obviously belong to A_{p}(Γ) for all p ∈ (1, ∞). Thus, A_{p}(Γ) ≠ ∅ if and only if (2.2) is true. Composed curves Γ satisfying (2.2) are usually referred to as Carleson curves (Ahlfors-David curves, Ahlfors regular curves, David regular curves, or simply as regular curves).

Now let 1 < p < ∞, let Γ be a composed curve, and let w be a weight on Γ. For f ∈ L^{1}(Γ) and t ∈ Γ, consider the limit

lim_{ε→0} \frac{1}{πi} ∫_{Γ\Γ(t,ε)} \frac{f(τ)}{τ − t} dτ. (2.3)

If the limit (2.3) exists, it is denoted by (S_{Γ}f)(t). Suppose

(i) for each f ∈ L^{1}(Γ) ∩ L^{p}(Γ, w) the limits exist a.e. on Γ and define a function S_{Γ}f ∈ L^{p}(Γ, w);

(ii) L^{1}(Γ) ∩ L^{p}(Γ, w) is dense in L^{p}(Γ, w);

(iii) there is a constant B = B(Γ, p, w) ∈ (0, ∞) such that ∥S_{Γ}f∥_{p,w} ≤ B∥f∥_{p,w} for all

f ∈ L^{1}(Γ) ∩ L^{p}(Γ, w).

In that case there is a unique extension of the map

L^{1}(Γ) ∩ L^{p}(Γ, w) → L^{p}(Γ, w), f ↦ S_{Γ}f

to a bounded linear operator on L^{p}(Γ, w). This operator is called the Cauchy singular integral operator on L^{p}(Γ, w) and is denoted by S_{Γ}.

Theorem 2.1. Let 1 < p < ∞, let Γ be a composed curve, and let w be a weight on Γ. Then (i), (ii), (iii) hold if and only if w ∈ A_{p}(Γ).

In other words, the Cauchy singular integral S_{Γ} is a well-defined and bounded operator on L^{p}(Γ, w) if and only if Γ is a Carleson curve and w is a Muckenhoupt weight in A_{p}(Γ). Note that if w ∈ A_{p}(Γ), then automatically L^{p}(Γ, w) ⊂ L^{1}(Γ) by Hölder’s inequality, so that (i) and (ii) hold with L^{1}(Γ) ∩ L^{p}(Γ, w) replaced by L^{p}(Γ, w) in this case.

Under various hypotheses concerning Γ and w, the boundedness of S_{Γ} on L^{p}(Γ, w) had been studied for a long time by many mathematicians, including Hardy, Littlewood, M. Riesz, Mikhlin, Babenko, Khvedelidze, Helson, Szegö, Widom, Forelli, Danilyuk, Shelepov, Calderón, and many others. Theorem 2.1 as it is quoted here was established by Hunt, Muckenhoupt, and Wheeden [38] in the case where Γ is the complex unit circle T. Paatashvili and Khuskivadze [45] showed that if S_{Γ} is bounded on L^{2}(Γ), then Γ must be a Carleson curve. Finally, David [19], [20] proved that for S_{Γ} to be bounded on L^{2}(Γ) it is necessary and sufficient that Γ be Carleson. After David’s result, a theorem like Theorem 2.1 was stated by many people (see, e.g., [23], [33], [35], [6]).

To compute S_{Γ}f one may employ the formula

(S_{Γ}f)(t) = \frac{1}{πi} ∫_{Γ} \frac{f(τ) − f(t)}{τ − t} dτ + lim_{ε→0} \frac{f(t)}{πi} ∫_{Γ\Γ(t,ε)} \frac{dτ}{τ − t}

in which the first integral does not contain a singularity provided f is sufficiently nice, say a rational function. In this connection the following result proves useful.

Theorem 2.2. Let Γ be a composed (rectifiable) curve, 1 < p < ∞, and w ∈ L^{p}(Γ). Then the set R(Γ) of all rational functions without poles on Γ is dense in L^{p}(Γ, w).

This theorem is well known, but we have not found an explicit reference. For the reader’s convenience, we will give a proof.

Proof. Let first η ⊂ Γ be a simple rectifiable arc. Without loss of generality assume |η| = 2π, put I := [0, 2π], and denote by μ : I → η any homeomorphism such that |μ′(x)| = 1 for almost all x ∈ I. Put ρ := (w|η) ∘ μ. Since

∫_{Γ} |f(τ)|^{p}w(τ)^{p}|dτ| = ∫_{I} |f(μ(x))|^{p}w(μ(x))^{p}dx = ∫_{I} |f(μ(x))|^{p}ρ(x)^{p}dx,

the map f ↦ f ∘ μ is an isometric isomorphism of L^{p}(η, w|η) onto L^{p}(I, ρ). In particular, if w|η ∈ L^{p}(η) then ρ ∈ L^{p}(I).

We claim that C(I) is dense in L^{p}(I, ρ). To see this, assume C(I) is not dense in L^{p}(I, ρ). Then there is a nonzero function h ∈ (L^{p}(I, ρ))^{∗} = L^{q}(I, ρ^{−1}) such that

∫_{I} h(x)φ(x)dx = 0 for all φ ∈ C(I). (2.4)

Hölder’s inequality gives

∫_{I} |h(x)|dx ≤ ( ∫_{I} |h(x)|^{q}ρ(x)^{−q}dx)^{1/q}( ∫_{I} ρ(x)^{p}dx)^{1/p},

and since h ∈ L^{q}(I, ρ^{−1}) and ρ ∈ L^{p}(I), it follows that h ∈ L^{1}(I). Taking φ(x) = e^{inx} with n ∈ Z in (2.4), we see that all Fourier coefficients of h vanish, which implies that h = 0 a.e. on I. This contradiction proves that C(I) is dense in L^{p}(I, ρ). Let χ_{ε} ∈ C(I) be any function satisfying

0 ≤ χ_{ε} ≤ 1, χ_{ε}(0) = χ_{ε}(2π) = 0, χ_{ε} = 1 on [ε, 2π − ε].

Then for φ ∈ C(I),

∫_{I} |φ(x) − χ_{ε}(x)φ(x)|^{p}ρ(x)^{p}dx ≤ ∥φ∥_{∞}^{p} ∫_{[0,ε]∪[2π−ε,2π]} ρ(x)^{p}dx = o(1) as ε → 0

because ρ^{p} ∈ L^{1}(I). We therefore arrive at the conclusion that C_{0}(I), the functions in C(I) vanishing at the endpoints of I, is also dense in L^{p}(I, ρ).

Thus, given f ∈ L^{p}(η, w|η) we can find functions φ_{n} ∈ C_{0}(I) such that f ∘μ−φ_{n} → 0 in L^{p}(I, ρ), implying that f − φ_{n} ∘ μ^{−1} → 0 in L^{p}(η, w|η). Consequently, the functions in C(η) vanishing at the endpoints of η are dense in L^{p}(η, w|η).

We may write Γ = η_{1} ∪ … ∪ η_{m} where η_{1}, …, η_{m} are simple arcs each pair of which have at most endpoints in common. Put

C_{0}(η_{j}) := {f ∈ C(Γ) : f|η_{j} ∈ C(η_{j}), f|Γ \ η_{j} = 0},
C_{0}(η_{1}) + … + C_{0}(η_{m}) := {f_{1} + … + f_{m} : f_{j} ∈ C_{0}(η_{j})}.

From what was shown in the preceding paragraph we infer that C_{0}(η_{1}) + … + C_{0}(η_{m}) is dense in L^{p}(Γ, w). Clearly, C_{0}(η_{1})+…+C_{0}(η_{m}) ⊂ C(Γ), and since, by Mergelyan’s theorem, R(Γ) is (uniformly) dense in C(Γ), it follows that R(Γ) is dense in the space L^{p}(Γ, w). □

3. Singular integral operators

Let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Obviously, for every function a ∈ L^{∞}(Γ) the multiplication operator f ↦ af is bounded on L^{p}(Γ, w). This operator is traditionally denoted by aI, and in case it follows another operator, say B, one abbreviates aIB to aB. Let B(Γ) := B(L^{p}(Γ, w)) denote the Banach algebra of all bounded linear operators on L^{p}(Γ, w). In the notation B(Γ) and in similar notations we will introduce in the following, we suppress the dependence on p and w; we rather think of Γ as an object which is a curve and carries a number p and a weight w. Given a closed subalgebra E of L^{∞}(Γ), we denote by alg (S_{Γ}, E) the smallest closed subalgebra of B(Γ) containing the operator S_{Γ} and all multiplication operators aI with a ∈ E. The operators in alg (S_{Γ}, E) are called singular integral operators with coefficients in E.

An operator A ∈ B(Γ) is said to be Fredholm if it is invertible modulo compact operators. Thus, if we denote by K(Γ) := K(L^{p}(Γ, w)) ⊂ B(Γ) the ideal of all compact operators, then an operator A ∈ B(Γ) is Fredholm if and only if the coset π(A) := A + K(Γ) is invertible in the quotient algebra B(Γ)/K(Γ). The essential spectrum sp_{ess} (A) of an operator A ∈ B(Γ) is the set

sp_{ess} (A) := {λ ∈ C : A − λI is not Fredholm on L^{p}(Γ, w)}.

Clearly, the algebra alg (S_{Γ}, E) is the more complicated the larger the coefficient algebra E is. One might expect that the simplest case is the case where E is the algebra C(Γ) of all continuous functions on Γ. However, the restriction to continuous functions is unnatural. In fact, the simplest and at the same time natural class of coefficients is the algebra P C(Γ) of all piecewise continuous functions on Γ. Here P C(Γ) stands for the smallest closed subalgebra of L^{∞}(Γ) containing all characteristic functions of simple subarcs of Γ. For example, if Γ = η_{1} ∪…∪η_{m} where η_{1}, …, η_{m} are simple arcs, if a ∈ L^{∞}(Γ) and if a|η_{1}, …, a|η_{m} are continuous functions, then a ∈ P C(Γ) (but, in general, a ∉ C(Γ)). That P C(Γ) is more natural than C(Γ) but not more complicated than C(Γ) can be seen as follows. Let Δ be a proper subset of Γ and suppose Δ itself is also a composed curve. Denote the characteristic function of Δ by χ_{Δ}. If a is any function in C(Δ), then the operator aS_{Δ} is Fredholm on L^{p}(Δ, w|Δ) if and only if the operator χ_{Δ}aS_{Γ}χ_{Δ}I + χ_{Γ\Δ}I is Fredholm on L^{p}(Γ, w). Thus, considering operators in alg (S_{Δ}, C(Δ)) naturally results in studying operators in alg (S_{Γ}, P C(Γ)).

Our purpose is to establish Fredholm criteria and thus to give descriptions of the essential spectrum for operators A in alg(S_{Γ}, P C(Γ)). The essential spectrum sp_{ess}(A) may be a rather intricate set, which leads to the problem of explaining why certain parts of the essential spectrum appear. This problem is tackled by identifying the local spectra of A.

The following theorem is the local principle of G.R. Allan [3] and R.G. Douglas [21] (see also [14] for a proof and for a discussion).

Theorem 3.1. Let B be a (complex) Banach algebra with identity element e and let Z be a closed subalgebra of B such that e ∈ Z and

(3.1) bz = zb for all z ∈ Z and all b ∈ B.

Denote the maximal ideal space of Z by Ω, and for ω ∈ Ω, let J_{ω} refer to the smallest closed two-sided ideal of B containing the ideal ω. Then an element b ∈ B is invertible in B if and only if b + J_{ω} is invertible in B/J_{ω} for every ω ∈ Ω.

A subalgebra Z of B satisfying (3.1) is usually said to be a central subalgebra. The algebra B/J_{ω} is referred to as the local algebra of B at ω ∈ Ω, and the spectrum of b + J_{ω} is called the local spectrum of b at ω ∈ Ω. We may write the conclusion of Theorem 3.1 also in the form

sp_{B} b = ⋃_{ω∈Ω} sp_{B/J_{ω}}(b + J_{ω}),

where sp_{A} a := {λ ∈ C : a − λe is not invertible in A}. Sometimes, if the algebra A is clear from the context, we will abbreviate sp_{A}a simply to sp a.

Now let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). We are interested in Fredholmness and thus in the invertibility of cosets π(A) := A + K(Γ) in the quotient algebra π(B(Γ)) := B(Γ)/K(Γ). Put

(3.2) Z(Γ) := {π(cI) : c ∈ C(Γ)},
Λ(Γ) := {A ∈ B(Γ) : π(A)π(cI) = π(cI)π(A) for all c ∈ C(Γ)}.

Thus, Λ(Γ) is the collection of all operators in B(Γ) which commute modulo compact operators with multiplications by continuous functions. Following [56], we refer to the operators in Λ(Γ) as operators of local type. It is easily seen that Λ(Γ) is a closed subalgebra of B(Γ) and that an operator A ∈ Λ(Γ) is Fredholm if and only if π(A) is invertible in the quotient algebra π(Λ(Γ)) := Λ(Γ)/K(Γ). Consequently, in order to decide whether operators in Λ(Γ) are Fredholm, we may have recourse to Theorem 3.1 with B = π(Λ(Γ)) and Z = Z(Γ) as in (3.2). The maximal ideal space of Z(Γ) may be identified with Γ: the maximal ideal corresponding to t ∈ Γ is the set

{π(cI) : c ∈ C(Γ), c(t) = 0}. (3.3)

For t ∈ Γ, let J_{t}(Γ) be the smallest closed two-sided ideal of π(Λ(Γ)) containing the set (3.3) and define

B_{t}(Γ) := π(Λ(Γ))/J_{t}(Γ), π_{t}(A) := π(A) + J_{t}(Γ).

Theorem 3.1 now gives the following.

Proposition 3.2. An operator A of local type is Fredholm on L^{p}(Γ, w) if and only if π_{t}(A) is invertible in B_{t}(Γ) for every t ∈ Γ. Equivalently, if A is of local type, then

(3.4) sp_{ess} (A) = ⋃_{t∈Γ} sp π_{t}(A). □

We call B_{t}(Γ) the local algebra of Γ (and of p and w) at t and sp π_{t}(A) the local spectrum of A at t. In (3.4), the essential spectrum of an operator is represented as the union of the local spectra of the operator. If φ ∈ C(Γ) and φ(t) = 1, then

π(φI) − π(I) = π((φ − 1)I) ∈ J_{t}(Γ)

and hence, π_{t}(φI) = π_{t}(I). It follows that

π_{t}(A) = π_{t}(φI)π_{t}(A)π_{t}(φI) = π_{t}(φAφI) (3.5)

for every operator A ∈ Λ(Γ). In particular, A and φAφI have the same local spectrum at a point t. Since φAφI does not depend on the properties of Γ and w outside the support supp φ of φ and because supp φ may be chosen to be contained in any open neighborhood of t, it is clear that spπ_{t}(A) depends in fact only on the “local properties” of Γ and w at t.

Proposition 3.3. The operators in alg (S_{Γ}, L^{∞}(Γ)) are of local type.

Proof. Multiplication operators are obviously of local type. Since Λ(Γ) is a closed algebra, it remains to show that S_{Γ} is of local type, i.e. that φS_{Γ} − S_{Γ}φI is compact on L^{p}(Γ, w) for every φ ∈ C(Γ). If φ ∈ R(Γ), then

((φS_{Γ} − S_{Γ}φI)f)(t) = \frac{1}{πi} ∫_{Γ} \frac{φ(t) − φ(τ)}{t − τ}f(τ) dτ (t ∈ Γ),

and on decomposing φ into partial fractions it is readily seen that φS_{Γ} − S_{Γ}φI is a finite-rank operator. Since R(Γ) is uniformly dense in C(Γ) by Mergelyan’s theorem, we arrive at the assertion. □

4. Main results

Let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). We first associate a certain compact subset L_{t}(Γ) := L_{t}(Γ, p, w) of the complex plane with each point t ∈ Γ, which is called the leaf of the triple p, Γ, w at t, and then we construct a Banach algebra homomorphism

Sym_{t,x} : alg (S_{Γ}, P C(Γ)) → C^{N(t)×N(t)}

for each x ∈ L_{t}(Γ) such that

sp π_{t}(A) = ⋃_{x∈L_{t}(Γ)} sp Sym_{t,x}(A)

for every A ∈ alg (S_{Γ}, P C(Γ)). Here C^{N(t)×N(t)} stands for the Banach algebra of all N(t) × N(t) matrices and sp Sym_{t,x}(A) is the spectrum (= set of eigenvalues) of the matrix Sym_{t,x}(A). We also emphasize that by a Banach algebra homomorphism we always understand a continuous (= bounded) algebra homomorphism.

Suppose first that η is a simple Carleson arc, 1 < p < ∞, and w ∈ A_{p}(η). Let t be an endpoint of η and denote by η^{+} the arc η with the orientation in which t is the starting point. Put S_{η}^{+} := S_{η^{+}} and P_{η}^{+} := (I + S_{η}^{+})/2. We define the leaf L_{t}(η) of the simple Carleson arc η at its endpoint t as the local spectrum of P_{η}^{+} at t:

L_{t}(η) := sp π_{t}(P_{η}^{+}). (4.1)

Notice that L_{t}(η) does not depend on the orientation of η. We remark that if both η and w are “nice” at the point t, then L_{t}(η) is a certain circular arc between the points 0 and 1. The points 0 and 1 always belong to L_{t}(η). More about the shape of leaves will be said in the next section.

Theorem 4.1. Let η_{1} and η_{2} be simple Carleson arcs which have a common endpoint t but no other common points. Let further 1 < p < ∞, w_{1} ∈ A_{p}(η_{1}), w_{2} ∈ A_{p}(η_{2}). If η_{1} ∪ η_{2} is a Carleson curve and the weight w given by w|η_{1} = w_{1} and w|η_{2} = w_{2} belongs to A_{p}(η_{1} ∪ η_{2}), then

L_{t}(η_{1}) = L_{t}(η_{2}).

In the case where both η_{1} and η_{2} are simple Lyapunov arcs, a result like the previous theorem is implicitly already in [35].

A composed Carleson curve γ comprised of a finite number N ≥ 2 of simple Carleson arcs η_{1}, …, η_{N} which have exactly one point in common is called a Carleson star and the common point is referred to as the center of the star. Let γ = η_{1} ∪ … ∪ η_{N} be a Carleson star with the center t, let 1 < p < ∞, and let w ∈ A_{p}(γ). Theorem 1.1 implies that w|η_{i} ∪ η_{j} ∈ A_{p}(η_{i} ∪ η_{j}). From Theorem 4.1 we so infer that

L_{t}(η_{1}) = … = L_{t}(η_{N}). (4.2)

We denote the set (4.2) by L_{t}(γ) := L_{t}(γ, p, w) and call it the leaf of the Carleson star γ at its center γ. Notice again that L_{t}(γ) is independent of the orientations of the arcs γ is comprised of.

Now let Γ be a general composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Pick a point t ∈ Γ and denote by Γ_{0}(t, ε) the connected component of the portion Γ(t, ε) containing t. If ε > 0 is sufficiently small, then Γ_{0}(t, ε) is a simple Carleson arc or a Carleson star and hence, we may define L_{t}(Γ_{0}(t, ε)). From (4.1), (3.5), and Proposition 3.3 we infer that L_{t}(Γ_{0}(t, ε)) does not depend on ε. We denote L_{t}(Γ_{0}(t, ε)) by L_{t}(Γ) := L_{t}(Γ, p, w) and refer to it as the leaf of the composed Carleson curve at the point t.

If ε > 0 is small enough, then the number of connected components of Γ_{0}(t, ε) \ {t} is independent of ε. This number is denoted by N(t) and called the multiplicity of t. Clearly, there is only a finite number of points t ∈ Γ with N(t) ≠ 2. Let

Γ_{0}(t, ε) = η_{1} ∪ … ∪ η_{N(t)} (4.3)

where η_{1}, …, η_{N(t)} are simple arcs. Each of these arcs has an orientation. We put ε_{j}(t) = 1 if η_{j} is an arc starting at t and we let ε_{j}(t) = −1 if η_{j} is an arc terminating at t. If a ∈ P C(Γ), then the limits

a_{j}(t) := lim ; \begin{matrix}τ ∈ η_{j}, \\ τ → t \end{matrix} a_{j}(τ) (j = 1, …, N(t)) (4.4)

exist and are finite.

We are now in a position to state our main result.

Theorem 4.2. Suppose Γ is a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). For t ∈ Γ, put E(t) := diag (ε_{1}(t), …, ε_{N(t)}(t)), and given (t, x) ∈ Γ × C, define the map

Sym_{t,x} : {S_{Γ}} ∪ {aI : a ∈ P C(Γ)} → C^{N(t)×N(t)}

as follows. For x ∉ {0, 1}, let

Sym_{t,x}(S_{Γ}) := E(t) ( \begin{matrix}2x − 1 & 2x − 2 & 2x − 2 & … & 2x − 2 \\ 2x & 2x − 1 & 2x − 2 & … & 2x − 2 \\ 2x & 2x & 2x − 1 & … & 2x − 2 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 2x & 2x & 2x & … & 2x − 1 \end{matrix} ) , (4.5)

for x ∈ {0, 1}, let Sym_{t,x}(S_{Γ}) := (2x − 1)E(t), (4.6)

and for all x, let Sym_{t,x}(aI) := diag (a_{1}(t), …, a_{N(t)}(t)). (4.7)

(a) For each x in the leaf L_{t}(Γ) the map Sym_{t,x} extends to a Banach algebra homomorphism

Sym_{t,x} : alg (S_{Γ}, P C(Γ)) → C^{N(t)×N(t)}.

(b) If A ∈ alg (S_{Γ}, P C(Γ)), then

sp π_{t}(A) = ⋃_{x∈L_{t}(Γ)} sp Sym_{t,x}(A).

(c) An operator A ∈ alg (S_{Γ}, P C(Γ)) is Fredholm on L^{p}(Γ, w) if and only if Sym_{t,x}(A) is an invertible matrix for all (t, x) in the leaf bundle

M := M(Γ, p, w) := ⋃_{t∈Γ} ({t} × L_{t}(Γ)).

In other terms,

sp_{ess} A = ⋃_{(t,x)∈M} sp Sym_{t,x}(A) = ⋃_{t∈Γ} ⋃_{x∈L_{t}(Γ)} sp Sym_{t,x}(A).

Clearly, part (c) of this theorem follows from part (b) and Propositions 3.2 and 3.3. Let us consider three simple examples.

We call a curve a Carleson Jordan curve if it is a Carleson curve homeomorphic to a circle and if it is oriented counter-clockwise. If Γ is a Carleson Jordan curve and t ∈ Γ, then Theorem 4.2. holds with

Sym_{t,x}(S_{Γ}) = ( \begin{matrix}2x − 1 & 2x − 2 \\ −2x & 1 − 2x \end{matrix} ) (x ∉ \{0, 1\}), (4.8)

(4.9) Sym_{t,0}(S_{Γ}) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , Sym_{t,1}(S_{Γ}) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,

Sym_{t,x}(aI) = ( \begin{matrix}a(t + 0) & 0 \\ 0 & a(t − 0) \end{matrix} ) . (4.10)

The eigenvalues of the matrices (4.8), (4.9) are −1 and 1. Hence spπ_{t}(S_{Γ}) = {−1, 1} for all t ∈ Γ and thus, sp_{ess} (S_{Γ}) = {−1, 1} and sp_{ess} (P_{Γ}) = {0, 1}, where P_{Γ} := (I + S_{Γ})/2.

Secondly, suppose Γ = η is a simple Carleson arc starting at t_{0} and terminating at t_{1}. We then have Theorem 4.2 with

Sym_{t_{0},x}(S_{Γ}) = 2x − 1, Sym_{t_{0},x}(aI) = a(t_{0} + 0),
Sym_{t_{1},x}(S_{Γ}) = 1 − 2x, Sym_{t_{1},x}(aI) = a(t_{1} − 0),

and with Sym_{t,x}(S_{Γ}) and Sym_{t,x}(aI) as in (4.8), (4.9), (4.10) for t ∈ η \ {t_{0}, t_{1}}. In particular,

sp π_{t_{0}}(S_{η}) = 2L_{t_{0}}(η) − 1, sp π_{t_{0}}(P_{η}) = L_{t_{0}}(η),
sp π_{t_{1}}(S_{η}) = 1 − 2L_{t_{1}}(η), sp π_{t_{1}}(P_{η}) = 1 − L_{t_{1}}(η).

Thirdly, assume Γ = η_{1} ∪ η_{2} is a Carleson star centered at t_{0} and η_{1} and η_{2} are outgoing arcs terminating at t_{1} and t_{2}, respectively. Then Theorem 4.2 is true with

Sym_{t_{0},x}(S_{Γ}) = ( \begin{matrix}2x − 1 & 2x − 2 \\ 2x & 2x − 1 \end{matrix} ) (x ∉ \{0, 1\}), (4.11)

Sym_{t_{0},0}(S_{Γ}) = diag (−1, −1), Sym_{t_{0},1}(S_{Γ}) = diag (1, 1),
Sym_{t_{1},x}(S_{Γ}) = 1 − 2x, Sym_{t_{2},x}(S_{Γ}) = 1 − 2x,

with Sym_{t,x}(S_{Γ}) as in (4.8), (4.9) for t ∈ Γ \ {t_{0}, t_{1}, t_{2}} and with Sym_{t,x}(aI) given by (4.10) for t ∈ Γ \ {t_{0}, t_{1}, t_{2}} and by

( \begin{matrix}lim_{t→t_{0},t∈η_{1}} a(t) & 0 \\ 0 & lim_{t→t_{0},t∈η_{2}} a(t) \end{matrix} ) , a(t_{1} − 0), a(t_{2} − 0)

for t = t_{0}, t = t_{1}, t = t_{2}, respectively. The eigenvalues of the matrix (4.11) are 2x − 1 ± 2i\sqrt{x(1 − x)}. Thus,

sp π_{t_{0}}(S_{Γ}) = \{2x − 1 ± 2i\sqrt{x(1 − x)} : x ∈ L_{t_{0}}(Γ)\}

where L_{t_{0}}(Γ) := L_{t_{0}}(η_{1}) = L_{t_{0}}(η_{2}).

Theorem 4.2 does everything provided we know the leaves L_{t}(Γ) (t ∈ Γ) or, equivalently, the local spectra of the operators P_{η}^{+} at the endpoints of all simple arcs η contained in Γ. We now proceed to the computation of the leaves.

Suppose first that Γ is a Carleson Jordan curve and fix a point t ∈ Γ. Given a weight ψ : Γ → [0, ∞] such that log ψ ∈ L^{1}(Γ), we construct a new function V_{t}^{0}ψ : (0, ∞) → [0, ∞], the V_{t}^{0} transform of ψ (this terminology proved useful in [8], [9]) as follows: for ε > 0, put

M_{t,ψ}(ε) := exp (\frac{1}{|Γ(t, ε)|} ∫_{Γ(t,ε)} log ψ(τ) |dτ|)

and then define

(V_{t}^{0}ψ)(ξ) := \{ \begin{matrix}lim sup_{R→0} M_{t,ψ}(ξR)/M_{t,ψ}(R) for ξ ∈ (0, 1], \\ lim sup_{R→0} M_{t,ψ}(R)/M_{t,ψ}(ξ^{−1}R) for ξ ∈ [1, ∞). \end{matrix}

We call

α(V_{t}^{0}ψ) := lim sup_{ξ→0} \frac{log(V_{t}^{0}ψ)(ξ)}{log ξ}, β(V_{t}^{0}ψ) := lim sup_{ξ→∞} \frac{log(V_{t}^{0}ψ)(ξ)}{log ξ}

the lower and upper indices of V_{t}^{0}ψ, respectively.

For τ ∈ Γ, we have τ − t = |τ − t|e^{i arg(τ−t)}, and arg(τ − t) may be chosen as a continuous function of τ ∈ Γ\{t}. Taking into account that Γ is a Carleson curve, one can show that the function τ ↦ arg(τ − t) belongs to L^{1}(Γ) (see [8]). It is also well known that log W ∈ L^{1}(Γ) whenever W ∈ A_{p}(Γ) (again see [8]). Hence, if we put

ϱ_{t}(τ) := e^{− arg(τ−t)} for τ ∈ Γ \ {t},

then log(ϱ_{t}^{x}W) = x log ϱ_{t} + log W ∈ L^{1}(Γ) for each x ∈ R and we may carry out the above construction with ψ := ϱ_{t}^{x}W to obtain

α_{t}(x) := α(V_{t}^{0}(ϱ_{t}^{x}W)), β_{t}(x) := β(V_{t}^{0}(ϱ_{t}^{x}W)). (4.12)

In [8], [9] it is shown that

−∞ < α_{t}(x) ≤ β_{t}(x) < +∞ for all x ∈ R, (4.13) −1/p < α_{t}(0) ≤ β_{t}(0) < 1/q, (4.14)

α_{t} is a concave function and β_{t} is a convex function.

(4.15)

The latter property implies that α_{t} and β_{t} are continuous functions on R. We call α_{t} and β_{t} the indicator functions of the pair Γ, W at t.

Given two continuous functions α, β : R → R such that α(x) ≤ β(x) for all x ∈ R, we put

Y (p; α, β) := \{z = x + iy ∈ C : \frac{1}{p} + α(x) ≤ y ≤ \frac{1}{p} + β(x)\} (4.16)

and

L^{0}(p; α, β) := \{\frac{e^{2πz}}{e^{2πz} − 1} : z ∈ Y (p; α, β)\} ∪ \{0, 1\}. (4.17)

We refer to L^{0}(p; α, β) as the leaf of the triple p, α, β. If α_{t}, β_{t} are the indicator functions of Γ, W at t, then L^{0}(p; α_{t}, β_{t}) is called the indicator leaf of the triple Γ, p, W at the point t. Sometimes, in case p and W are known from the context, we will simply speak of L^{0}(p; α_{t}, β_{t}) as the indicator leaf of the (Carleson and Jordan !) curve Γ at the point t.

Of course, the above construction is not easy. However, in the next section we will see that the indicator functions can be given by pretty nice formulas in many interesting cases. At the present moment we remark that solely (4.13) and the continuity of the indicator functions imply that L^{0}(p; α_{t}, β_{t}) is a compact connected set which does not separate the plane and which contains the points 0 and 1.

The following theorem identifies the leaves L_{t}(η) we are interested in in terms of indicator leaves.

Theorem 4.3. Let η be a simple Carleson arc, 1 < p < ∞, and w ∈ A_{p}(η).

(a) There exist a Carleson Jordan curve Γ and a weight W ∈ A_{p}(Γ) such that η ⊂ Γ and W |η = w.

(b) Let Γ be any Carleson Jordan curve and W ∈ A_{p}(Γ) be any weight such that η ⊂ Γ and W |η = w. For t ∈ Γ, let L^{0}(p; α_{t}, β_{t}) be the indicator leaf of Γ, p, W at t. Then L_{t}(η) = L^{0}(p; α_{t}, β_{t}) (4.18)

at each endpoint t of η.

Thus, in order to find L_{t}(η), we first have to extend η and w to a Carleson Jordan curve Γ and a weight W ∈ A_{p}(Γ) (which is possible by Theorem 4.3(a)) and then we have to compute the indicator leaf L^{0}(p; α_{t}, β_{t}) (which is independent of the extension due to Theorem 4.3(b)).

The following theorem describes the local spectrum of the Cauchy singular integral operator as the union of certain leaves. Suppose t has the multiplicity N := N(t) and that in (4.3) we have N_{o} outgoing and N_{i} incoming arcs. The difference ε := N_{o} − N_{i} is called the valency of t. Of course, ε = ε_{1}(t) + … + ε_{N}(t). For a continuous function f : R → R, define T_{n}f : R → R by (T_{n}f)(x) = f(nx).

Theorem 4.4. Let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Let t ∈ Γ and let α_{t}, β_{t} be any functions such that L_{t}(Γ) = L^{0}(p; α_{t}, β_{t}). Given integers n > 0 and k ≥ 0, put

L_{k,n}^{0} := L^{0}(p; \frac{1}{n}(\frac{1}{p} − \frac{n}{p} + k + T_{n}α_{t}), \frac{1}{n}(\frac{1}{p} − \frac{n}{p} + k + T_{n}β_{t})).

Then, sp π_{t}(S_{Γ}) equals

{−1, 1} if ε = 0, (4.19)

⋃_{k=0}^{ε−1}(2L_{k,ε}^{0} − 1) if ε > 0, (4.20)

⋃_{k=0}^{|ε|−1}(1 − 2L_{k,|ε|}^{0}) if ε < 0. (4.21)

We remark that, curiously, a result of this type was first explicitly stated only in the recent papers [11] and [50].

5. Comments and history

Theorem 4.2 stands at the end of a long chain of results. This theorem associates a family of matrix functions with every operator in alg (S_{Γ}, P C(Γ)) in terms of which one can decide whether the operator is Fredholm or not. These matrix functions have a special structure and live on certain sets, the leaves. The history of Theorem 4.2 is the history of understanding the shape of the leaves on the one hand and the history of finding the right structure of the matrix functions on the other.

5.1. Genesis of leaves

The theory of singular integral operators with continuous coefficients was essentially disposed of with I. Gohberg’s 1952 paper [28]. B.V. Khvedelidze was the first to investigate Fredholm properties of singular integral equations with piecewise continuous coefficients on L^{p} spaces with power weights [40] (also see his survey [41]). In particular, he established sufficient conditions for such operators to be Fredholm. In 1960, Shamir [54] and Widom [61] studied singular integral operators on the line and on the half-line. Widom’s paper [61] is the earliest reference we know in which the term “circular arc” explicitly appeared in connection with the spectral theory of singular integral operators. Also in the sixties, Simonenko created his local principle [56] and the method of generalized Wiener-Hopf factorization [55], [57] and thus laid the foundation of the today’s edifice of singular integral equations. In the late sixties, Gohberg and Krupnik turned to operators with discontinuous symbols and within only a few years elaborated a Fredholm theory for algebras of singular integral operators with piecewise continuous coefficients on composed Lyapunov curves with power weights; we here confine ourselves to refering to their book [33] and their three papers [30], [31], [32].

A simple arc η is called a Lyapunov arc if the derivative φ′ of its natural parametrization φ : [0, l] → C satisfies a Hölder condition,

|φ′(t) − φ′(τ)| ≤ C|t − τ|^{λ} for all t, τ ∈ η

with C > 0 and λ ∈ (0, 1). Two Lyapunov arcs η_{1} and η_{2} are said to be compatible if either η_{1} ∩ η_{2} = ∅ or η_{1} and η_{2} have exactly one endpoint in common, at which they do not form a cusp. A composed Lyapunov curve is a finite union of pairwise compatible Lyapunov arcs. It is easily seen that composed Lyapunov curves are composed Carleson curves. A power weight is a weight w of the form

w(τ) = ∏_{j=1}^{m} |τ − t_{j}|^{λ_{j}} (τ ∈ Γ) (5.1)

where t_{1}, …, t_{m} are distinct points on Γ and λ_{1}, …, λ_{m} are real numbers. It is well known that the weight (5.1) belongs to A_{p}(Γ) if and only if −1/p < λ_{j} < 1/q for all j; in the case of general Carleson curves this was perhaps first proved by Danilov [18] (his proof is reproduced in [6]).

Let now η be a simple Lyapunov arc, let w be the power weight (5.1) with Γ = η, and let t be an endpoint of η. Gohberg and Krupnik proved that then sp π_{t}(P_{η}^{+}) is a certain circular arc. In the language of Section 4, this means that the leaf L_{t}(η) is a circular arc. In terms of Theorem 4.3, this circular arc can be described as the indicator leaf corresponding to the indicator functions

α_{t}(x) = β_{t}(x) = λ_{t} (5.2)

where λ_{t} := 0 if t ∉ {t_{1} …, t_{m}} and λ_{t} := λ_{j} if t = t_{j}. Clearly, Y (p; α_{t}, β_{t}) is the horizontal line {z ∈ C : Im z = 1/p + λ_{t}}. The map z ↦ e^{2πz} maps Y (p; α_{t}, β_{t}) to a ray starting at the origin and making the angle 2π(1/p+λ_{t}) ∈ (0, 2π) with the positive real axis. Finally, the Möbius transform ζ ↦ ζ/(ζ − 1) maps this ray to a circular arc between 0 and 1 which degenerates to the segment [0, 1] in case 1/p + λ_{t} = 1/2. This circular arc is L_{t}(η).

In 1990, after a pause of about 15 years, the next step was done by one of the authors [58], who determined sp π_{t}(P_{η}^{+}) and thus the leaf L_{t}(η) in case η is a simple Lyapunov arc and w is an arbitrary weight in A_{p}(η). Using Theorem 4.3, the leaf L_{t}(η) can be identified as the indicator leaf induced by the indicator functions

α_{t}(x) = μ_{t} and β_{t}(x) = ν_{t},

where μ_{t} and ν_{t} are certain real numbers measuring the “powerlikeness” of the weight w at t. We always have −1/p < μ_{t} ≤ ν_{t} < 1/q. Thus, Y (p; α_{t}, β_{t}) is now the stripe {z ∈ C : 1/p + μ_{t} ≤ Im z ≤ 1/p + ν_{t}} and from what was said in the preceding paragraph we infer that L_{t}(η) is a horn, i.e. the closed set between two circular arcs joining 0 and 1.

In [1] (also see [2]), Aizenshtat, Litvinchuk, and one of the authors were led to the investigation of singular integral operators on Carleson curves very close to logarithmic spirals and stated a Fredholm criterion for such operators. Only recently, two of the authors turned attention to general Carleson curves and were able to describe spπ_{t}(P_{η}^{+}) for simple Carleson arcs η and weights w ∈ A_{p}(η) provided there is a Carleson Jordan curve Γ and a weight W ∈ A_{p}(Γ) such that η ⊂ Γ and W |η = w ( see [6], [7], [8], [9]). By virtue of Theorem 4.3(a), we therefore know sp π_{t}(P_{η}^{+}) and thus L_{t}(η) for arbitrary simple Carleson arcs η and arbitrary weights w ∈ A_{p}(η). We remark that the concept of indicator functions and the description of L_{t}(η) in terms of indicator leaves as in Theorem 4.3(b) were introduced only in [8].

We now examine some examples of leaves. Suppose η is a simple Carleson arc, t is an endpoint of η, and w ∈ A_{p}(η). Let Γ be a Carleson Jordan curve containing η and let W be a weight in A_{p}(Γ) extending w. Theorem 4.3(b) tells us that L_{t}(η) = L^{0}(p; α_{t}, β_{t}), where α_{t}, β_{t} are the indicator functions of Γ, W at t.

Recall first the definition (4.12) of the indicator functions. We call the two numbers

δ_{t}^{−} := α(V_{t}^{0}ϱ_{t}), δ_{t}^{+} := β(V_{t}^{0}ϱ_{t})

the spirality indices of the curve Γ at the point t. Obviously, δ_{t}^{−} and δ_{t}^{+} have nothing to do with a weight on Γ and are therefore “intrinsic” characteristics of the curve. The curve Γ is said to be spiralic at t if δ_{t}^{−} = δ_{t}^{+} =: δ_{t}. Let, for example,

η = {r exp (ih(log(− log r))(− log r)) : r ∈ (0, 1)} ∪ {0, 1}

where h is a twice continuously differentiable function and h, h′, h′′ are bounded on R. One can show that η is a Carleson arc and that the spirality indices at t = 0 of any extension Γ of η are

δ_{t}^{−} = lim inf_{x→+∞} (h(x) + h′(x)), δ_{t}^{+} = lim sup_{x→+∞} (h(x) + h′(x)).

If h(x) = δ + ε sin x, then

δ_{t}^{−} = δ − |ε|\sqrt{2} and δ_{t}^{+} = δ + |ε|\sqrt{2}.

In case h(x) = δ, the arc η is a logarithmic spiral and δ_{t}^{−} = δ_{t}^{+} = δ_{t} = δ. Piecewise C^{1} curves and, in particular, Lyapunov curves are always spiralic with δ_{t} = 0.

We need still another pair of indices. We put

μ_{t} := α_{t}(0) := α(V_{t}^{0}W), ν_{t} := β_{t}(0) := β(V_{t}^{0}W) (5.3)

and call μ_{t} and ν_{t} the indices of powerlikeness of the weight W at t. Note that in (5.3) the curve Γ does not explicitly appear, although, of course, the weight W lives on Γ, by virtue of which Γ enters (5.3) implicitly. We say that W is powerlike at t if μ_{t} = ν_{t}. To have a concrete example, suppose η := [0, 1] and

w(r) := exp (g(log(− log r)) log r) for r ∈ (0, 1)

where g is a twice continuously differentiable function on R for which g, g′, g′′ are bounded on R. One can show that w ∈ A_{p}(η) if and only if −1/p < μ_{t} ≤ ν_{t} < 1/q where

μ_{t} := lim inf_{x→+∞} (g(x) + g′(x)), ν_{t} := lim sup_{x→+∞} (g(x) + g′(x))

and that the indices of powerlikeness at t = 0 of any extension W of w are just μ_{t} and ν_{t}. Letting g(x) = λ + ε sin x, we get

μ_{t} = λ − |ε|\sqrt{2} and ν_{t} = λ + |ε|\sqrt{2}.

If g(x) = λ, then w is a power weight and μ_{t} = ν_{t} = λ.

The results of the next two paragraphs were established in [7] and [8] (also see [9]). If Γ is spiralic at t and W is an arbitrary weight in A_{p}(Γ), then the indicator functions

are α_{t}(x) = μ_{t} + δ_{t}x, β_{t}(x) = ν_{t} + δ_{t}x. (5.4)

Thus, Y (p; α_{t}, β_{t}) is the stripe

{z = x + iy ∈ C : 1/p + μ_{t} + δ_{t}x ≤ y ≤ 1/p + ν_{t} + δ_{t}x}

(which degenerates to a line in case μ_{t} = ν_{t}). The map z ↦ e^{2πz} maps the line {z = x + iy ∈ C : y = λ + δ_{t}x} to a logarithmic spiral, and the Möbius transform ζ ↦ ζ/(ζ − 1) maps this logarithmic spiral to a logarithmic double spiral wriggling out of 0 and scrolling up at 1. Thus, L^{0}(p; α_{t}, β_{t}) is the closed set between two such logarithmic double spirals. In [6], such sets were called spiralic horns.

Now suppose Γ is an arbitrary Carleson Jordan curve but W ∈ A_{p}(Γ) is powerlike at t. Then the indicator functions are given by

α_{t}(x) = λ_{t} + min {δ_{t}^{−}x, δ_{t}^{+}x}, β_{t}(x) = λ_{t} + max {δ_{t}^{−}x, δ_{t}^{+}x} (5.5)

where λ_{t} := μ_{t} = ν_{t}. The set Y (p; α_{t}, β_{t}) is now the double sector between two lines of slopes δ_{t}^{−} and δ_{t}^{+}. In [7], the corresponding leaves L^{0}(p; α_{t}, β_{t}) were called skew spiralic horns, in the terminology of [8], these leaves are logarithmic leaves with a separating point.

In summary, we know the leaves L_{t}(η) = L^{0}(p; α_{t}, β_{t}) in case Γ is spiralic at t or W is powerlike at t. In these two cases, L_{t}(η) is always bounded by pieces of at most two logarithmic double spirals. In the case where Γ is not spiralic at t and W is not powerlike at t, the situation is more delicate. Taking into account (5.4) and (5.5) one might guess that

α_{t}(x) = μ_{t} + min {δ_{t}^{−}x, δ_{t}^{+}x}, β_{t}(x) = ν_{t} + max {δ_{t}^{−}x, δ_{t}^{+}x},

but one of the big surprises of the story is that this guess is wrong ! It turns out that in the general case some kind of interference between the oscillation of the curve and the oscillation of the weight may result in a complete demolition of spirality: it may happen that no piece of the boundary of L_{t}(η) is a piece of some logarithmic double spiral. The leaves that may occur in the general case are completely described in [8]. This paper and [11] also contain pictures of several leaves.

5.2. Genesis of symbol matrices

Here now are a few remarks on the other aspect of Theorem 4.2, that is, on the structure of the symbol matrices Sym_{t,x}(A). It was once more Gohberg and Krupnik [30], [31], [32], [33] who first realized the possibility of a symbol calculus for algebras of singular integral operators and who also first understood the structure of the symbol matrices. They considered composed Lyapunov curves and power weights. In this case their theorem does the same as Theorem 4.2, but it is more complicated.

For the sake of simplicity, let us suppose that Γ is a Lyapunov Jordan curve. Gohberg and Krupnik define their symbol on the cylinder Γ × [0, 1] and put all information about the leaf L_{t}(η) into the function x ↦ Sym_{t,x}(A). This works well if the leaves themselves are curves (e.g. circular arcs), but it is clear that difficulties arise as soon as the leaves become heavy sets. Instead of this, we define the symbol on the leaf bundle M and hence, for each A, the map x ↦ Sym_{t,x}(A) is one and the same function. Redefining the Gohberg/Krupnik symbol to a symbol on M gives

(5.6)  Sym _{t,x}^{GK}(aI) = ( \begin{matrix}a(t + 0) & 0 \\ 0 & a(t − 0) \end{matrix} ) ,
(5.7)  Sym _{t,x}^{GK}(P_{Γ}) = ( \begin{matrix}x & \sqrt{x(1 − x)} \\ \sqrt{x(1 − x)} & 1 − x \end{matrix} ) ,

whereas Theorem 4.2 yields

Sym_{t,x}(aI) = ( \begin{matrix}a(t + 0) & 0 \\ 0 & a(t − 0) \end{matrix} ) , Sym_{t,x}(P_{Γ}) = ( \begin{matrix}x & x − 1 \\ −x & 1 − x \end{matrix} )

for x ∉ {0, 1}. The enigma’s resolution is in fact simple [50]: letting

D_{x} := diag ( ^{4} \sqrt{x/(1 − x)}, − ^{4} \sqrt{(1 − x)/x})

we have

Sym _{t,x}^{GK}(A) = D_{x}Sym_{t,x}(A)D_{x}^{−1}.

Thus, the differences between the Gohberg/Krupnik symbol and the symbol delivered by Theorem 4.2 are minor ones if Γ is a Jordan curve. However, the differences become serious when passing to composed curves.

In 1978, Douglas [22] studied the algebra of singular integral operators on L^{2} over the complex unit circle T. His approach was as follows. The algebra alg (ST, P C(T)) is generated by PT and {aI : a ∈ P C(T)}. Since PT is a projection, the element P := π_{t}(PT) is an idempotent element of the local algebra B_{t}(T), i.e. P^{2} = P. Every function a ∈ P C(T) may be written in the form

a(τ) = a(t + 0)χ_{t}(τ) + a(t − 0)(1 − χ_{t}(τ)) + φ(τ)

where χ_{t} is the characteristic function of some subarc of T starting at t and φ is continuous in a neighborhood of t and vanishes at t. Hence,

π_{t}(aI) = a(t + 0)π_{t}(χ_{t}I) + a(t − 0)(π_{t}(I) − π_{t}(χ_{t}I))
= a(t + 0)p + a(t − 0)(e − p),

where e := π_{t}(I) and p := π_{t}(χ_{t}I). The element p is also an idempotent. Thus, the smallest closed subalgebra A_{t}(T) of B_{t}(T) containing the set {π_{t}(A) : A ∈ alg (PT, P C(T))} is generated by the identity and two idempotents (“projections”). Douglas then employed a two projections theorem, which gave him a homomorphism

Sym _{t}^{D} : A_{t}(T) → C^{2×2}(L_{t})

such that e, p, P are sent to the matrix functions

( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}x & \sqrt{x(1 − x)} \\ \sqrt{x(1 − x)} & 1 − x \end{matrix} ) ,

respectively, and such that a ∈ A_{t}(T) is invertible in B_{t}(T) if and only if Sym _{t}^{D}(a) is invertible in C^{2×2}(L_{t}). The set L_{t} was the spectrum of pP p + (e − p)(e − P)(e − p) and thus, in principle, the spectrum of π_{t}(P_{η}^{+}), where η is the support of χ_{t}. In the case at hand, L_{t} = [0, 1]. Thus, in the end Douglas got exactly the matrices (5.6) and (5.7) for (Sym _{t}^{D}π_{t}(aI))(x) and (Sym _{t}^{D}π_{t}(PT))(x).

The local algebras A_{t}(T) introduced by Douglas were C^{∗}–algebras, and the two projections theorem one had at that time worked just for C^{∗}–algebras. Ten years later, in 1988, Roch and Silbermann [51] proved a two projections theorem for Banach algebras and derived the Gohberg/Krupnik symbol calculus on Lyapunov Jordan curves by Douglas’ elegant method.

If Γ is a composed Lyapunov curve, then the analogues A_{t}(Γ) of the local algebras A_{t}(T) are in general not generated by two idempotents. These algebras were studied by Duduchava, Krupnik, Nyaga, Costabel, Plamenevski, Senichkin, and others. Roch and Silbermann [52] (also see [36]) used techniques of locally straightening the curve Γ at t to a star comprised of N(t) rays and then employed Mellin convolutions techniques to obtain a theorem like Theorem 4.2. In [11] it was shown that the latter approach is also applicable to curves that may locally “straightened” to a star of N(t) logarithmic spirals. Notice in this connection that besides the unit circle T, the real half-line G_{0} := {r > 0} and the logarithmic spirals G_{δ} := {r^{1+iδ} : r > 0} (δ ≠ 0) are the only non-trivial Lie subgroups of the multiplication group C \ {0}. The methods of [52] and [11] nicely work for power weights and have only recently been extended by Rabinovich and two of the authors [12], [13] to large classes of general Muckenhoupt weights.

Immediately after the paper [58], which showed that circular arcs may metamorphose into horns, people started investigating algebras of singular integral operators over composed Lyapunov curves with general Muckenhoupt weights. However, this time one encountered a new obstacle. Suppose Γ is a Lyapunov Jordan curve and A ∈ alg (S_{Γ}, P C(Γ)). The two projections theorem tells us that the spectrum of π_{t}(A) in the local algebra A_{t}(Γ) is a horn. But to get sp_{ess} (A), we need the spectrum of π_{t}(A) in the local algebra B_{t}(Γ) ⊃ A_{t}(Γ). If B_{t}(Γ) and A_{t}(Γ) are C^{∗}–algebras (which was the case in Douglas’ situation) or if the horn degenerates to a circular arc, then the spectra of π_{t}(A) in A_{t}(Γ) and B_{t}(Γ) coincide. Since in general a horn is a heavy set, i.e. a set that may arise from another set by “filling in holes”, it is not clear whether the spectrum of π_{t}(A) in B_{t}(Γ) is a horn if this spectrum is a horn in A_{t}(Γ). These obstacles were overcome by Finck, Roch, Silbermann [26] and Gohberg, Krupnik [34], who elaborated the appropriate techniques for studying the inverse closedness of A_{t}(Γ) in B_{t}(Γ) and thus for constructing a symbol calculus for alg(S_{Γ}, P C(Γ)) on L^{p}(Γ, w) in case Γ is a Lyapunov Jordan curve and w is a weight in A_{p}(Γ). This symbol calculus was extended to Carleson Jordan curves in [8]. The case of a composed Lyapunov curve Γ and a weight w ∈ A_{p}(Γ) was finally disposed of in [35] by having recourse to the original approach of [30] and [32].

The difference between the methods of [26] and [35] motivated the search for an N projections theorem. Such a theorem was found in [50], and we will cite and make essential use of this theorem in Section 7. The paper [50] contains results close to Theorem 4.2. Nevertheless, Theorem 4.2 cannot be straightforwardly deduced from [50]; it still requires some results on the extension of Carleson curves and Muckenhoupt weights and on the behavior of leaves under such extensions. In a sense, almost all that will follow in the rest of this paper is devoted to proving and subsequently employing such results.

6. Extending Carleson stars and Muckenhoupt weights

In Section 4, we introduced the notion of a Carleson star γ = η_{1} ∪ … ∪ η_{N} (N ≥ 2) centered at a point t. We now also regard a simple Carleson arc γ = η as a Carleson star whose center is one of the endpoints of η.

A composed curve F is said to be a Carleson flower if it is a Carleson Jordan curve or if it is the union of a finite number N ≥ 2 of Carleson Jordan curves with exactly one common point. Thus, a Carleson flower F may be written as F = Γ_{1} ∪ … ∪ Γ_{N} where Γ_{1}, …, Γ_{N} are Carleson Jordan curves and Γ_{i} ∩ Γ_{j} = {t} for i ≠ j. If N ≥ 2, then the common point t is referred to as the center of the flower. Each point of a Jordan curve may be its center.

We always think of Jordan curves as being oriented counter-clockwise. Accordingly, a flower F consists of a finite number of counter-clockwise oriented Jordan curves. A star γ is comprised of a finite number of oriented simple arcs. When writing γ ⊂ F we always mean that the simple arcs of γ have the orientation inherited from F.

The behavior of the Cauchy singular integral operator on Carleson flowers is much better than on Carleson stars. The point is that if F is a Carleson flower, then S_{F}^{2} = I and hence P_{F} := (I + S_{F})/2 is a projection. This is not true for the Cauchy singular integral operator on a Carleson star. However, if γ is a Carleson star and F ⊃ γ is a Carleson flower, then many properties of S_{γ} are equivalent to the corresponding properties of the operator χ_{γ}S_{F} χ_{γ}I + χ_{F \γ}I, which belongs to alg (S_{F} , P C(F)). We are thus led to the problem of extending Carleson stars γ to Carleson flowers F and of extending weights w ∈ A_{p}(γ) to weights W ∈ A_{p}(F). The first of these two problems may be quickly disposed of by having recourse to the following result.

Theorem 6.1. Suppose Ω is a simply connected domain in C∪{∞} and its boundary ∂Ω is a Carleson curve. Let φ : D → Ω be a Riemann map of the unit disk D onto Ω. If σ is a Carleson curve in D, then φ(σ) is a Carleson curve in Ω.

This result is a special case of Proposition 4 in Zinsmeister’s paper [62]. The proof is based on the Hayman-Wu theorem (for which there are now a variety of proofs [37], [24], [27], [44]). Note that in [62] the term “Carleson domain” does not mean a domain bounded by a Carleson curve (although such a domain is an example of one), but something more general.

Theorem 6.2. If γ is a Carleson star, then there exists a Carleson flower F such that γ ⊂ F.

Proof. Suppose γ is made up of the simple arcs η_{1}, …, η_{N} with the common endpoint t and the other endpoint of η_{i} being y_{i} (i = 1, …, N). Put Ω := C \ γ and let φ : D → Ω be a Riemann map. Because ∂Ω = γ is rectifiable, it is certainly locally connected, and hence, φ extends continuously to D ∪ T. Each point y_{j} ∈ {y_{1}, …, y_{N}} has exactly one preimage z_{j} ∈ T, and the center t has exactly N preimages t_{1}, …, t_{N} on the circle T which separate the points z_{1}, …, z_{N}. We assume everything is labeled so that as we traverse T in the clockwise direction, we encounter the points in the order t_{1}, z_{1}, t_{2}, z_{2}, …, t_{N}, z_{N}. Put t_{N+1} := t_{1}.

For j ∈ {1, …, N}, let σ_{j} be the hyperbolic geodesic connecting z_{j} to t_{j+1} if η_{j} is directed away from t, and let σ_{j} stand for the hyperbolic geodesic connecting t_{j} to z_{j} in case η_{j} is an incoming arc. Then σ := σ_{1} ∪ … ∪ σ_{N} is clearly a Carleson curve and we may assume that no point of σ is mapped to ∞ (otherwise replace φ by φ ∘ ψ with an appropriate Möbius transform ψ : D → D). Thus, by Zinsmeister’s Theorem 6.1, φ(σ) ∪ γ is a Carleson flower containing γ. □

We should note that in the preceding proof, the use of hyperbolic geodesics was completely irrelevant and was only made for concreteness. Any choice of N simple Carleson arcs connecting the desired points without crossing each other would have worked just as well. Moreover, every possible completion of γ to a Carleson flower arises in this way. In fact, for any simply connected domain Ω ⊂ C ∪ {∞} whose boundary is a Carleson curve, any Riemann map φ of D onto Ω, and any Carleson curve Γ, the preimage φ^{−1}(Γ ∩ Ω) is always a Carleson curve in the disk. If Γ ⊂ Ω, this was established in [25], the general case is in [4].

We now turn to the problem of extending Muckenhoupt weights from Carleson stars to Muckenhoupt weights on Carleson flowers. The following construction also yields a self-contained proof of Theorem 6.2, that is, a proof which does not make use of Theorem 6.1.

The hyperbolic metric in the complex unit disk D and the upper complex half-plane C_{+} := {z ∈ C : Im z > 0} is given by

|dz|_{H} := \frac{2}{1 − |z|^{2}} |dz| and |dz|_{H} := \frac{1}{Im z} |dz|,

respectively. In what follows we denote by |Γ| and dist (z, Γ) the Euclidean length of Γ and the Euclidean distance between z and Γ, while |Γ|_{H} stands for the hyperbolic length of Γ.

Let Ω ⊂ C∪{∞} be a simply connected domain with boundary ∂Ω and let φ : D → Ω be a Riemann map. Koebe’s distortion theorem (see, e.g., [46, p. 22]) says that for all z ∈ D,

\frac{1}{4}(1 − |z|^{2})|φ′(z)| ≤ dist (φ(z), ∂Ω) ≤ (1 − |z|^{2})|φ′(z)|. (6.1)

Furthermore, if z, z_{0} ∈ D and the hyperbolic distance between z and z_{0} is h(z, z_{0}), then

\frac{e^{2h(z,z_{0})} − 1}{4e^{2h(z,z_{0})}} ≤ \frac{|φ(z) − φ(z_{0})|}{dist (φ(z_{0}), ∂Ω)} ≤ e^{2h(z,z_{0})} − 1 (6.2)

(see, e.g., [27, Lemma 3.1]). In the following we frequently write A_{τ} ≃ B_{τ} to indicate that the ratio A_{τ} /B_{τ} is bounded and bounded away from zero as τ ranges over some set. In particular, (6.1) implies that |φ′(z)|(1 − |z|^{2}) ≃ dist (φ(z), ∂Ω).

Suppose now that ∂Ω is rectifiable and equip ∂Ω with Lebesgue length measure. Given a measurable subset E ⊂ ∂Ω and a point z ∈ Ω, we denote by ω(z, E, Ω) the harmonic measure of E at z:

ω(z, E, Ω) := |φ_{z}^{−1}(E)|/(2π)

where φ_{z} : D → Ω is a Riemann map taking 0 to z. Equivalently, ω(z, E, Ω) may be defined as the value at z of the harmonic extension of the characteristic function of E into Ω. Finally, we say that {η_{n}}_{n∈Z} is a partition of the simple arc η if each η_{n} itself is a simple arc, if each pair η_{j}, η_{k} (j ≠ k) have at most an endpoint in common, and

if η = ⋃_{n∈Z} η_{n}.

Lemma 6.3. Let σ ⊂ D be a (maximal) hyperbolic geodesic and let τ ⊂ T be anyone of the two arcs of T between the endpoints of σ. There exist partitions {I_{n}}_{n∈Z} of σ and {J_{n}}_{n∈Z} of τ such that if we denote the endpoints of I_{n} by z_{n} and z_{n+1}, then

|I_{n}|_{H} = 1 for all n, (6.3) |J_{n}| ≃ dist (z_{n}, J_{n}) ≃ 1 − |z_{n}|, (6.4) ω(z_{n}, J_{n}, D) > 4/(10π). (6.5)

Proof. Without loss of generality assume that −1 is an endpoint of σ. Denote by C_{+} the upper half-plane and consider the Riemann maps

φ : D → C_{+}, z ↦ i(1 − z)/(1 + z),
ψ : C_{+} → D, w ↦ (i − w)/(i + w).

Clearly, φ(σ) is a half-line of the form {w ∈ C_{+} : Re w = a}. For n ∈ Z, put

w_{n} := a + ie^{n}, z_{n} := ψ(w_{n}),

and let I_{n} ⊂ σ be the arc with the endpoints z_{n} and z_{n+1}. Further, set

v_{n} := a + εe^{n}, t_{n} := ψ(v_{n}),

where ε = 1 if ψ maps {x ∈ R : x > a} to τ and ε = −1 if ψ maps {x ∈ R : x < a} to τ. Denote by J_{n} ⊂ T the arc whose endpoints are t_{n} and t_{n+1}.

It is easily verified that the line segment [w_{n}, w_{n+1}] has hyperbolic length 1. Since hyperbolic length is a conformal invariant, we so obtain (6.3). A straightforward computation gives

|J_{n}| ≃ |t_{n+1} − t_{n}| ≃ e^{−|n|}, 1 − |z_{n}| ≃ e^{−|n|}, dist (z_{n}, J_{n}) ≃ |z_{n} − t_{n}| ≃ e^{−|n|},

which proves (6.4). It is well known that πω(z_{n}, J_{n}, D) coincides with the angle at which the line segment [v_{n}, v_{n+1}] is seen at w_{n}. Thus,

ω(z_{n}, J_{n}, D) = \frac{1}{π} ( arctan e − \frac{π}{4} ) > \frac{4}{10π} .

□

Throughout what follows in this section, let γ be a Carleson star, let Ω := C \ γ, and let φ : D → Ω be a Riemann map. Choose the hyperbolic geodesics σ_{j} as in the proof of Theorem 6.2 and suppose that none of the arcs φ(σ_{j}) contains the point at infinity.

Lemma 6.4. Let {z_{n}} and {J_{n}} be as in Lemma 6.3. Then there exist a constant M < ∞ and measurable sets F_{n} ⊂ J_{n} such that

|F_{n}| ≥ (3/4)|J_{n}|, (6.6) ω(z_{n}, F_{n}, D) ≥ 3/(10π), (6.7) |φ(t) − φ(z_{n})| ≤ M|φ′(z_{n})|(1 − |z_{n}|^{2}) for all t ∈ F_{n}. (6.8)

Proof. Theorem 10.8 on p. 311 of [46] says that there exists a function K : (0, ∞) → (0, ∞) with the following property: if h : D → C is a univalent (analytic) function, h(0) = 0, h′(0) = 1, and if ε > 0 is any number, then there is a measurable subset E ⊂ T such that |E| < ε and

∫_{0}^{1} |h′(rτ)|dr ≤ K(ε)

for all τ ∈ T \ E.

Define ρ_{n} : D → D by ρ_{n}(z) := (z + z_{n})/(1 + \overline{z_{n}}z). Then ρ_{n} is a Möbius transform of D onto itself mapping 0 to z_{n}. Further, let σ_{n} : C → C be the linear map given by

σ_{n}(z) := (z − φ(z_{n}))/(φ′(z_{n})(1 − |z_{n}|^{2}).

The composition h_{n} := σ_{n} ∘ φ ∘ ρ_{n} is univalent on D, h_{n}(0) = 0, and h′_{n}(0) = 1. From what was said in the preceding paragraph we deduce that there are E_{n} ⊂ T such that |E_{n}| < ε and

|h_{n}(τ)| = |h_{n}(τ) − h_{n}(0)| ≤ ∫_{0}^{1} |h′_{n}(rτ)|dr ≤ K(ε)

for all τ ∈ T \ E_{n}. Consequently,

|φ(ρ_{n}(τ)) − φ(z_{n})| = |h_{n}(τ)||φ′(z_{n})|(1 − |z_{n}|^{2}) ≤ K(ε)|φ′(z_{n})|(1 − |z_{n}|^{2}) (6.9)

whenever τ ∉ E_{n}. Put F_{n} := J_{n} \ ρ_{n}(E_{n}). From (6.9) we infer that

|φ(t) − φ(z_{n})| ≤ K(ε)|φ′(z_{n})|(1 − |z_{n}|^{2}) for t ∈ F_{n}. (6.10)

We have

ω(z_{n}, ρ_{n}(E_{n}), D) = \frac{1}{2π} |ρ_{n}^{−1}(ρ_{n}(E_{n}))| = \frac{1}{2π} |E_{n}| < \frac{ε}{2π} ,

and since ω(z_{n}, J_{n}, D) > 4/(10π) by (6.5), it follows that ω(z_{n}, F_{n}, D) > 3/(10π) whenever ε ≤ 2/10. If t ∈ J_{n}, then |t − z_{n}| ≤ C(1 − |z_{n}|^{2}) with some C independent of n by virtue of (6.4). Thus, for t ∈ J_{n},

|(ρ_{n}^{−1})′(t)| = \frac{1 − |z_{n}|^{2}}{|1 − \overline{z_{n}}t|^{2}} = \frac{1 − |z_{n}|^{2}}{|t − z_{n}|^{2}} ≥ \frac{1}{C^{2}} \frac{1}{1 − |z_{n}|^{2}} ,

whence

ε > |E_{n}| = ∫_{ρ_{n}(E_{n})} |(ρ_{n}^{−1})′(t)||dt|

≥ ∫_{ρ_{n}(E_{n})∩J_{n}} |(ρ_{n}^{−1})′(t)||dt| ≥ \frac{1}{C^{2}} \frac{1}{1 − |z_{n}|^{2}} |ρ_{n}(E_{n}) ∩ J_{n}|.

Combining the latter inequality and (6.4), we get

|ρ_{n}(E_{n}) ∩ J_{n}| ≤ C^{2}ε(1 − |z_{n}|^{2}) ≤ Dε|J_{n}|

with some constant D independent of ε and n. Thus, |F_{n}| ≥ (1 − Dε)|J_{n}| ≥ (3/4)|J_{n}| if only ε ≤ 1/(4D). This completes the proof of (6.6) and (6.7). The estimate (6.10) gives (6.8) with M = K(ε_{0}) where ε_{0} ≤ min{2/10, 1/(4D)}. □

Given z_{n}, J_{n}, F_{n} (n ∈ Z), we put

A_{n} := φ(F_{n}), Γ_{n} := φ(I_{n}), x_{n} := φ(z_{n}), r_{n} := dist (x_{n}, γ).

We let D(x, r) stand for the disk {z ∈ C : |z − x| < r}. Our next objective is to prove Lemma 6.7. For this purpose, we need two more auxiliary results.

Lemma 6.5. There exists a function H : (1, ∞) → [0, 1) such that H(s) → 0 as s → ∞ and such that if Ω′ is any simply connected open subset of C with rectifiable boundary, x ∈ Ω′ and r = dist (x, ∂Ω′), then for all s > 1,

ω(x, ∂Ω′ \ D(x, sr), Ω′) ≤ H(s)

Proof. Let K : (0, ∞) → (0, ∞) be the function introduced in the beginning of the proof of Lemma 6.4. We may assume that K is monotonically decreasing and that K(ε) → +∞ as ε → 0 (otherwise replace K by K^{∗}(ε) := (1/ε) + sup{K(η) : η ≥ ε}). Denote by φ : D → Ω′ a Riemann map such that φ(0) = x and φ′(0) > 0. By Koebe’s distortion theorem, (1/4)φ′(0) ≤ r ≤ φ′(0). Put f(z) := (φ(z) − x)/φ′(0). Since f is univalent, f(0) = 0, f′(0) = 1, it follows from the definition of K that there is a measurable subset E ⊂ T such that |E| < ε and

|f(t) − f(0)| ≤ ∫_{0}^{1} |f′(rt)|dr ≤ K(ε)

whenever t ∈ T \ E. Thus,

|φ(t) − x| ≤ K(ε)φ′(0) ≤ 4K(ε)r (6.11)

for all t ∈ T \ E.

If s > 4K(ε) and |φ(t) − x| ≥ sr, then (6.11) implies that t ∈ E. Consequently,

ω(x, ∂Ω′ \ D(x, sr), Ω′) = \frac{1}{2π} |φ^{−1}(∂Ω′ \ D(x, sr))| ≤ \frac{1}{2π} |E| < \frac{ε}{2π} . (6.12)

For s > 1, define H(s) := inf {ε/(2π) : 4K(ε) < s}. With this definition, we may rewrite (6.12) in the form ω(x, ∂Ω′ \ D(x, sr), Ω′) ≤ H(s). It remains to prove that H(s) → 0 as s → ∞. The function H is monotonically decreasing because if s_{1} < s_{2} then

2πH(s_{1}) = inf{ε : 4K(ε) < s_{1}} ≥ inf{ε : 4K(ε) < s_{2}} = 2πH(s_{2}).

If H would not go to zero, there would exist a δ > 0 and an s_{0} ∈ (1, ∞) such that

δ ≤ inf{ε : 4K(ε) < s} for all s > s_{0}.

However, this is impossible since ⋃_{s>s_{0}}{ε : 4K(ε) < s} = (0, ∞). □

Lemma 6.6. Let Ω′ be a simply connected open subset of C with rectifiable boundary, let x ∈ Ω′, and let E be a measurable subset of ∂Ω′. If r = dist (x, ∂Ω′), then

(6.13) ω(x, E, Ω′) ≤ C \frac{log(|∂Ω′|/r)}{| log(|E|/r)| + 1}

with some universal constant C < ∞.

The estimate (6.13) is known as Lavrentiev’s inequality. A proof of Lemma 6.6 (which actually deserves the title “theorem”) is in [47, pp. 125-127] and also in [5].

Lemma 6.7. There are constants M_{1}, M_{2} < ∞ such that

A_{n} ⊂ γ ∩ D(x_{n}, M_{1}r_{n}) for all n ∈ Z, (6.14) (6.15) |Γ_{n}| ≤ M_{2}|A_{n}| for all n ∈ Z, |A_{n}| ≃ r_{n}. (6.16)

Proof. From (6.8) and (6.1) we infer that if t ∈ F_{n}, then

|φ(t) − φ(z_{n})| ≤ M|φ′(z_{n})|(1 − |z_{n}|^{2}) ≤ 4Mdist (x_{n}, γ) = 4Mr_{n},

whence A_{n} ⊂ D(x_{n}, 4Mr_{n}). Thus, we get (6.14) with M_{1} = 4M.

Since γ is a Carleson curve, we have |γ ∩ D(x, r)| ≤ C_{γ}r for all x ∈ γ. We claim that |γ ∩ D(x, r)| ≤ 2C_{γ}r for all x ∈ C. This is clear if γ ∩ D(x, r) = ∅. If there is a point t ∈ γ ∩ D(x, r), then |γ ∩ D(x, r)| ≤ |γ ∩ D(t, 2r)| ≤ 2C_{γ}r, which gives our claim. Using (6.14) we therefore get

|A_{n}| ≤ |γ ∩ D(x_{n}, M_{1}r_{n})| ≤ 2M_{1}C_{γ}r_{n}. (6.17)

We now show that |A_{n}| ≥ cr_{n} with some positive constant c independent of n. Let H be the function of Lemma 6.5 and choose N_{0} > 0 so that N_{0}M > 1 and H(N_{0}M_{1}) < 2/(10π).

Suppose first that r_{n} is smaller than (diam γ)/(2N_{0}M_{1}). Then the boundary of the disk D(x_{n}, N_{0}M_{1}r_{n}) necessarily hits γ. Hence, the connected component Ω′_{n} of Ω∩D(x_{n}, N_{0}M_{1}r_{n}) containing x_{n} is a (bounded) simply connected domain. We claim that ω(x_{n}, A_{n}, Ω′_{n}) ≥ ω(x_{n}, A_{n}, Ω) − ω(x_{n}, Σ_{n}, Ω′_{n}) (6.18)

where Σ_{n} := ∂Ω′_{n} \ γ. By the maximum principle for harmonic functions, (6.18) will follow as soon as we have shown that

ω^{∗}(ζ, A_{n}, Ω′_{n}) ≥ ω^{∗}(ζ, A_{n}, Ω) − ω^{∗}(ζ, Σ_{n}, Ω′_{n}) (6.19)

for almost all ζ ∈ ∂Ω′_{n}, where ω^{∗}(ζ, E, G) stands for the boundary value of ω(z, E, G) at ζ ∈ ∂G. If ζ ∈ A_{n}, then (6.19) reads 1 ≥ 1 − ω^{∗}(ζ, Σ_{n}, Ω′_{n}), which is certainly true. For ζ ∈ ∂Ω′_{n} ∩ (γ \ A_{n}), inequality (6.19) is 0 ≥ 0 − ω^{∗}(ζ, Σ_{n}, Ω′_{n}), which is also true. Finally, if ζ ∈ ∂Ω′_{n} \ γ, then (6.19) requires that 0 ≥ ω^{∗}(ζ, A_{n}, Ω) − 1, which is again satisfied. This proves our claim (6.18).

Since the points of Σ_{n} have distance at least N_{0}M_{1}r_{n} to x_{n}, we deduce from Lemma 6.5 that ω(x_{n}, Σ_{n}, Ω′_{n}) ≤ H(N_{0}M_{1}) < 2/(10π). (6.20)

From (6.7) and the conformal invariance of harmonic measure we see that ω(x_{n}, A_{n}, Ω) is at least 3/(10π). This together with (6.18) and (6.20) shows that

ω(x_{n}, A_{n}, Ω′_{n}) ≥ 1/(10π). (6.21)

Since γ is a Carleson curve, we have

|∂Ω′_{n}| ≤ |Σ_{n}| + |γ ∩ D(x_{n}, N_{0}M_{1}r_{n})| ≤ (2π + 2C_{γ})N_{0}M_{1}r_{n}. (6.22)

Combining (6.21) and (6.22) with Lemma 6.6 we obtain

\frac{1}{10π} ≤ C \frac{log ((2π + 2C_{γ})N_{0}M_{1})}{| log(|A_{n}|/r_{n})| + 1},

whence

\frac{|A_{n}|}{r_{n}} ≥ e((2π + 2C_{γ})N_{0}M_{1})^{−10πC} =: c_{1}. (6.23)

Clearly, there are only finitely many n for which r_{n} ≥ (diam γ)/(2N_{0}M_{1}). For these n we have |A_{n}|/r_{n} ≥ c_{2} > 0 with some constant c_{2} independent of n. This along with (6.23) gives the desired estimate |A_{n}| ≥ cr_{n} with c := min{c_{1}, c_{2}}. Taking into account (6.17), we arrive at (6.16).

We finally prove (6.15). If x ∈ Γ_{n}, then x = φ(z) with z ∈ I_{n}. By (6.3), the hyperbolic distance between z and z_{n} is at most 1. So (6.2) implies that

|φ(z) − φ(z_{n})| ≤ (e^{2} − 1)dist (φ(z_{n}), γ) = (e^{2} − 1)r_{n},

and because

dist (φ(z), γ) ≤ |φ(z) − φ(z_{n})| + dist (φ(z_{n}), γ) = |φ(z) − φ(z_{n})| + r_{n},

we arrive at the estimate

(6.24) dist (φ(z), γ) ≤ e^{2}r_{n} for all z ∈ I_{n}.

By virtue of (6.1),

|Γ_{n}| = ∫_{Γ_{n}} |dw| = ∫_{I_{n}} |φ′(z)||dz| ≃ ∫_{I_{n}} \frac{dist (φ(z), γ)}{1 − |z|^{2}}|dz|,

and hence, due to (6.24) and (6.3),

|Γ_{n}| ≤ c_{0}e^{2}r_{n} ∫_{I_{n}} \frac{2|dz|}{1 − |z|^{2}} = c_{0}e^{2}r_{n} (6.25)

with some c_{0} < ∞ independent of n. Combining (6.16) and (6.25) we get (6.15). □

Now extend γ to a flower as in the proof of Theorem 6.2. If γ is comprised of N simple Carleson arcs, γ = η_{1} ∪ … ∪ η_{N}, we have N subarcs τ_{1}, …, τ_{N} ⊂ T and N hyperbolic geodesics σ_{1}, …, σ_{N} ⊂ D. For each j ∈ {1, …, N}, we construct the sets I_{n}^{(j)}, J_{n}^{(j)}, Γ_{n}^{(j)}, A_{n}^{(j)} as above. Put

{Γ_{n}}_{n∈Z} := ⋃_{m∈Z} ⋃_{j=1}^{N}{Γ_{m}^{(j)}}, {A_{n}}_{n∈Z} := ⋃_{m∈Z} ⋃_{j=1}^{N}{A_{m}^{(j)}}

and suppose the enumeration is so that Γ_{n} is associated with A_{n}: if Γ_{n} = Γ_{m}^{(j)}, then A_{n} = φ(F_{m}^{(j)}). Further, for Γ_{n} = Γ_{m}^{(j)} let x_{n} := φ(z_{m}^{(j)}) and r_{n} := dist (x_{n}, γ). Thus,

γ ⊃ ⋃_{n∈Z} A_{n}, F \ γ = ⋃_{n∈Z} Γ_{n}.

For each n ∈ Z, set

w_{n} := (\frac{1}{|A_{n}|} ∫_{A_{n}} w(t)^{p}|dt|)^{1/p}

and then define a weight w̃ on F \ γ by w̃(x) := w_{n} for x ∈ Γ_{n}. Finally, let

W := wχ_{γ} + w̃χ_{F \γ}.

Obviously, W is an extension of w from γ to a weight on F. We are left with showing that W ∈ A_{p}(F).

Lemma 6.8. Let x ∈ F \ γ and let r ≤ (1/10) dist (x, γ). Then at most 2N of the arcs Γ_{n} hit D(x, r) and there are constants 0 < B_{1} ≤ B_{2} < ∞ independent of x and r such that

(6.26) B_{1} ≤ w_{n}/w_{k} ≤ B_{2} whenever Γ_{n} ∩ D(x, r) ≠ ∅ and Γ_{k} ∩ D(x, r) ≠ ∅.

Proof. We have x = φ(z_{0}) with some z_{0} in the hyperbolic geodesic σ ∈ {σ_{1}, …, σ_{N}}. If z is any point in D such that φ(z) ∈ D(x, r), then

|φ(z) − φ(z_{0})| < r ≤ (1/10) dist (φ(z_{0}), γ),

while (6.2) implies that

\frac{|φ(z) − φ(z_{0})|}{dist (φ(z_{0}), γ)} ≥ \frac{e^{2h(z,z_{0})} − 1}{4e^{2h(z,z_{0})}}.

Therefore,

\frac{e^{2h(z,z_{0})} − 1}{4e^{2h(z,z_{0})}} ≤ \frac{1}{10},

whence

h(z, z_{0}) ≤ \frac{1}{2} log \frac{10}{6} < \frac{1}{2} log e = \frac{1}{2}.

Since the arcs I_{n} ⊂ σ_{1} ∪ … ∪ σ_{N} all have hyperbolic length 1, it follows that at most 2N arcs Γ_{n} = φ(I_{n}) can hit D(x, r).

Now suppose Γ_{n} ∩ D(x, r) ≠ ∅. We claim that

r_{n} ≃ dist (x, γ). (6.27)

From (6.1) we know that

r_{n} ≃ |φ′(z_{n})|(1 − |z_{n}|), dist (x, γ) ≃ |φ′(z_{0})|(1 − |z_{0}|). (6.28)

Divide the (maximal) geodesics through the points z_{n} and z_{0} into arcs Ĩ_{m} as in the proof of Lemma 6.3. The arc I_{n} has the endpoints z_{n} and z_{n+1}, and at least one point of this arc lies in the hyperbolic disk {z : h(z, z_{0}) < 1/2}. As h(z_{n}, z_{n+1}) = 1, it follows that h(z_{0}, z_{n}) < 3/2. This shows that the points z_{n} and z_{0} belong to arcs Ĩ_{m} and Ĩ_{k} with |m − k| ≤ 2. Consequently,

1 − |z_{n}| ≃ e^{−|m|} ≃ e^{−|k|} ≃ 1 − |z_{0}|

(also see the proof of Lemma 6.3). Due to (6.28), the relation (6.27) will follow as soon as we have shown that |φ′(z_{n})| ≃ |φ′(z_{0})|.

Theorem 9.4 on page 269 of [46] with h(z) := (φ(z) − φ(0))/φ′(0) implies that log φ′(z) is a Bloch function, which means that

|\frac{d}{dz} log φ′(z)| ≤ \frac{2C}{1 − |z|^{2}} for all z ∈ D

with some constant C < ∞. It results that

| log |φ′(z_{n})| − log |φ′(z_{0})|| ≤ | log φ′(z_{n}) − log φ′(z_{0})|

= | ∫_{z_{0}}^{z_{n}} \frac{d}{dz} log φ′(z)dz| ≤ C ∫_{z_{0}}^{z_{n}} \frac{2|dz|}{1 − |z|^{2}}

= C ∫_{z_{0}}^{z_{n}} |dz|_{H} = Ch(z_{0}, z_{n}) < (3/2)C =: C_{0},

whence e^{−C_{0}} ≤ |φ′(z_{n})/φ′(z_{0})| ≤ e^{C_{0}}, which completes the proof of (6.27).

To prove (6.26), we will have recourse to the so-called A_{∞} condition. Two nonnegative Borel measures ν and μ on C are said to be comparable if there are numbers a, b ∈ (0, 1) so that if Δ is a disk and E ⊂ Δ is Borel, then

ν(E) < aν(Δ) ⟹ μ(E) < bμ(Δ).

Lemma 5 of [17] states that this is an equivalence relation on measures. Moreover, it states that if ν and μ are comparable, then there exist constants 0 < c < ∞ and δ > 0 independent of E and Δ such that

c(\frac{ν(E)}{ν(Δ)})^{δ} ≤ \frac{μ(E)}{μ(Δ)} ≤ 1 (6.29)

for all Borel subsets E ⊂ Δ. Define ν and μ by

ν(E) := |γ ∩ E|, μ(E) := ∫_{γ∩E} w(t)^{p}|dt|.

Since w ∈ A_{p}(γ), we may deduce from [59, Theorem 18 on p. 9] that ν and μ are comparable. Let A_{n} be the subset of γ associated with Γ_{n}. By (6.16) and (6.27), there exist constants 0 < M_{3} ≤ M_{4} < ∞ such that

M_{3}r_{x} ≤ |A_{n}| ≤ M_{4}r_{x}, M_{3}r_{x} ≤ r_{n} ≤ M_{4}r_{x} (6.30) where r_{x} := dist (x, γ). By (6.14),

A_{n} ⊂ γ ∩ D(x_{n}, M_{1}r_{n}) ⊂ γ ∩ D(x, |x_{n} − x| + M_{1}r_{n}),

and we have |x_{n} − x| < r + |Γ_{n}|, r ≤ r_{x}/10. From (6.15) and (6.30) we see that |Γ_{n}| ≤ M_{2}|A_{n}| ≤ M_{2}M_{4}r_{x}. Thus,

A_{n} ⊂ γ ∩ D(x, M_{5}r_{x})

with M_{5} := 1/10 + M_{2}M_{4} + M_{1}M_{4}. Since

ν(A_{n}) = |A_{n}| ≥ M_{3}r_{x} and ν(D(x, M_{5}r_{x})) = |γ ∩ D(x, M_{5}r_{x})| ≤ 2C_{γ}M_{5}r_{x},

we obtain from (6.29) that

c(\frac{M_{3}}{2C_{γ}M_{5}} )^{δ} ≤ c(\frac{ν(A_{n})}{ν(D(x, M_{5}r_{x}))})^{δ} ≤ \frac{μ(A_{n})}{μ(D(x, M_{5}r_{x}))} ≤ 1.

Hence, μ(A_{n}) ≃ μ(D(x, M_{5}r_{x})) for all n such that Γ_{n} ∩ D(x, r) ≠ ∅, while |A_{n}| ≃ r_{x} for these n by virtue of (6.30). This implies that

μ(A_{n}) ≃ μ(A_{k}), |A_{n}| ≃ |A_{k}| if Γ_{n} ∩ D(x, r) ≠ ∅ and Γ_{k} ∩ D(x, r) ≠ ∅,

whence

w_{n} := (\frac{μ(A_{n})}{|A_{n}|})^{1/p} ≃ (\frac{μ(A_{k})}{|A_{k}|})^{1/p} =: w_{k}.

□

Theorem 6.9. Let γ be a Carleson star, 1 < p < ∞, and w ∈ A_{p}(γ). Then there exist a Carleson flower F and a weight W ∈ A_{p}(F) such that γ ⊂ F and W |γ = w.

Proof. We have already constructed F and W. It remains to show that W ∈ A_{p}(F), which automatically implies that F is Carleson.

Suppose first that x ∈ F \ γ and that r ≤ (1/10) dist (x, γ). Then the point z_{0} := φ^{−1}(x) lies on some geodesic σ_{j} ∈ {σ_{1}, …, σ_{N}}. In the same way we proved (6.27), one can show that if φ(z) ∈ D(x, r), then

|φ′(z)|(1 − |z|^{2}) ≃ dist (x, γ) =: r_{x}. (6.31)

Moreover, for such z we have, by (6.2),

\frac{r}{r_{x}} > \frac{|φ(z) − φ(z_{0})|}{dist (φ(z_{0}), γ)} ≥ \frac{e^{2h(z,z_{0})} − 1}{4e^{2h(z,z_{0})}},

whence

h(z, z_{0}) < \frac{1}{2} log \frac{r_{x}}{r_{x}−4r} = \frac{1}{2} log (1 + \frac{4r}{r_{x}−4r} ) < \frac{1}{2} \frac{4r}{r_{x}−4r} < \frac{2r}{(1−4/10)r_{x}} = \frac{10}{3} \frac{r}{r_{x}} .

Thus, if φ(z) ∈ D(x, r), then z belongs to the hyperbolic disk D_{H} := {z : h(z, z_{0}) < 10r/(3r_{x})}. This observation together with (6.31) shows that

|φ(σ) ∩ D(x, r)| ≤ ∫_{σ∩D_{H}} |φ′(z)| |dz| = ∫_{σ∩D_{H}} |φ′(z)|(1 − |z|^{2})\frac{|dz|}{1 − |z|^{2}}

≤ M_{6}r_{x} ∫_{σ∩D_{H}} |dz|_{H} ≤ M_{6}r_{x} \frac{20}{3} \frac{r}{r_{x}} = \frac{20}{3}M_{6}r

with some M_{6} < ∞ independent of z and x. As we have N geodesics, we get

|F ∩ D(x, r)| ≤ (20/3)M_{6}Nr =: M_{7}r.

By Lemma 6.8, the weight W is piecewise constant on F ∩ D(x, r) and the ratio of the different values of the weight on F ∩ D(x, r) lies between B_{1} and B_{2}. Thus,

(\frac{1}{r} ∫_{F ∩D(x,r)} W (t)^{p}|dt|)^{1/p} = (\frac{1}{r} ∑_{n} ∫_{Γ_{n}∩D(x,r)≠∅} w_{n}^{p}|dt|)^{1/p} (6.32)

≤ (\frac{1}{r}B_{2}^{p}w_{n_{0}}^{p}|F ∩ D(x, r)|)^{1/p} ≤ M_{7}^{1/p}B_{2}w_{n_{0}}

and similarly,

(\frac{1}{r} ∫_{F ∩D(x,r)} W (t)^{−q}|dt|)^{1/q} ≤ M_{7}^{1/q}B_{1}^{−1}w_{n_{0}}^{−1}. (6.33)

Clearly, the product of (6.32) and (6.33) is uniformly bounded.

Now assume that either x ∈ γ or that x ∈ F \γ and r > (1/10) dist (x, γ). Obviously, it suffices to consider only the case where (F \ γ) ∩ D(x, r) ≠ ∅. Let {Γ_{n}} be the arcs which hit D(x, r) and let {A_{n}} be the corresponding subsets of γ. Pick anyone of the arcs Γ_{n}. There is a z ∈ I_{n} such that φ(z) ∈ D(x, r). If z ∈ I_{n} then, by (6.2) and by the definition of z_{n}, x_{n}, r_{n},

\frac{e^{2h(z,z_{n})} − 1}{4e^{2}} ≤ \frac{|φ(z) − x_{n}|}{r_{n}} ≤ e^{2h(z,z_{n})} − 1

and

\frac{e^{2h(z,z_{n})} − 1}{4e^{2}} ≤ \frac{|φ(z) − x_{n}|}{dist (φ(z), γ)} ≤ e^{2h(z,z_{n})} − 1,

whence

\frac{r_{n}}{4e^{2}} ≤ dist (φ(z), γ) ≤ 4e^{2}r_{n}. (6.34)

Since A_{n} ⊂ D(x_{n}, M_{1}r_{n}) due to (6.14) and |φ(z) − x_{n}| ≤ |Γ_{n}| ≤ M_{8}r_{n} by virtue of (6.15) and (6.16), it follows that

A_{n} ⊂ D(φ(z), (M_{1} + M_{8})r_{n}) =: D(φ(z), M_{9}r_{n}).

Because |φ(z) − x| < r, we so see that A_{n} ⊂ D(x, r + M_{9}r_{n}). From (6.34) and the assumption that r > (1/10) dist (x, γ) we infer that

r_{n} ≤ 4e^{2}dist (φ(z), γ) ≤ 4e^{2}(r + dist (x, γ)) ≤ 44e^{2}r.

Consequently, A_{n} ⊂ D(x, M_{10}r) with M_{10} := 1 + 44e^{2}M_{9}. Put

(F \ γ)(x, r) := (F \ γ) ∩ D(x, r), γ(x, r) := γ ∩ D(x, r).

We will prove that

∫_{(F \γ)(x,r)} w̃(t)^{p}|dt| ≤ M_{2} ∫_{γ(x,M_{10}r)} w(t)^{p}|dt|, (6.35)

∫_{(F \γ)(x,r)} w̃(t)^{−q}|dt| ≤ M_{2} ∫_{γ(x,M_{10}r)} w(t)^{−q}|dt|, (6.36)

where M_{2} is the constant from (6.15). These two estimates imply that

(\frac{1}{r} ∫_{F ∩D(x,r)} W (t)^{p}|dt|)^{1/p}(\frac{1}{r} ∫_{F ∩D(x,r)} W (t)^{−q}|dt|)^{1/q}

= (\frac{1}{r} ∫_{(F \γ)(x,r)}w̃(t)^{p}|dt| + \frac{1}{r} ∫_{γ(x,r)}w(t)^{p}|dt|)^{1/p}

×(\frac{1}{r} ∫_{(F \γ)(x,r)}w̃(t)^{−q}|dt| + \frac{1}{r} ∫_{γ(x,r)}w(t)^{−q}|dt|)^{1/q}

≤ (1 + M_{2})(\frac{1}{r} ∫_{γ(x,M_{10}r)} w(t)^{p}|dt|)^{1/p}(\frac{1}{r} ∫_{γ(x,M_{10}r)} w(t)^{−q}|dt|)^{1/q},

and this is uniformly bounded since w ∈ A_{p}(γ).

We have

∫_{(F \γ)(x,r)} w̃(t)^{p}|dt| ≤ ∑_{n} w_{n}^{p}|Γ_{n}| = ∑_{n} (\frac{1}{|A_{n}|} ∫_{A_{n}} w(t)^{p}|dt|)|Γ_{n}|

≤ M_{2} ∑_{n} ∫_{A_{n}} w(t)^{p}|dt| ≤ M_{2} ∫_{γ(x,M_{10}r)} w(t)^{p}|dt|,

which is (6.35). Further, Hölder’s inequality with the conjugate exponents (α + 1)/α and α + 1 shows that if m is a probability measure, then

1 = ( ∫ f^{\frac{α}{α+1}} f^{\frac{−α}{α+1}} dm)^{α+1} ≤ ( ∫ f dm)^{α}( ∫ f^{−α} dm),

whence

( ∫ f dm)^{−α} ≤ ∫ f^{−α} dm

for every α > 0. Taking into account the latter inequality we get

∫_{(F \γ)(x,r)} w̃(t)^{−q}|dt| ≤ ∑_{n} w_{n}^{−q}|Γ_{n}| = ∑_{n} (\frac{1}{|A_{n}|} ∫_{A_{n}} w(t)^{p}|dt|)^{−q/p}|Γ_{n}|

≤ ∑_{n} (\frac{1}{|A_{n}|} ∫_{A_{n}} w(t)^{−q}|dt|)|Γ_{n}| ≤ M_{2} ∑_{n} ∫_{A_{n}} w(t)^{−q}|dt|

≤ M_{2} ∫_{γ(x,M_{10}r)} w(t)^{−q}|dt|

and this is (6.36). □

7. Local algebras

Let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Fix t ∈ Γ and define the local algebra B_{t}(Γ) as in Section 3. Proposition 3.3 implies that B_{t}(Γ) contains π_{t}(S_{Γ}) and π_{t}(aI) for every a ∈ P C(Γ). Let A_{t}(Γ) be the smallest closed subalgebra of B_{t}(Γ) containing the set {π_{t}(S_{Γ})} ∪ {π_{t}(aI) : a ∈ P C(Γ)}. Clearly, if A ∈ alg (S_{Γ}, P C(Γ)) then π_{t}(A) ∈ A_{t}(Γ).

Suppose Γ_{0}(t, ε) = η_{1} ∪ … ∪ η_{N(t)} is as in (4.3). Put N := N(t) and abbreviate the limits (4.4) to a_{j} (j = 1, …, N). Let χ_{1}, …, χ_{N} denote the characteristic functions of the arcs η_{1}, …, η_{N}. It is easily seen that if φ is a function in P C(Γ), if φ is continuous at the point t and φ(t) = 0, then π(φI) ∈ J_{t}(Γ). We have

a = a_{1}χ_{1} + … + a_{N}χ_{N} + φ

where φ ∈ P C(Γ), φ is continuous at t and φ(t) = 0. It follows that

π(aI) − ∑_{j=1}^{N} a_{j}π(χ_{j}I) = π(φI) ∈ J_{t}(Γ)

and hence, π_{t}(aI) = a_{1}π_{t}(χ_{1}I) + … + a_{N}π_{t}(χ_{N}I). (7.1)

Because π_{t}(S_{Γ}) = 2π_{t}(P_{Γ}) − π_{t}(I), we see that A_{t}(Γ) is generated by P := π_{t}(P_{Γ}) and the N elements p_{1} := π_{t}(χ_{1}I), …, p_{N} := π_{t}(χ_{N}I). Since obviously χ_{j}^{2} = χ_{j}, we have p_{j}^{2} = p_{j} for all j. Thus, p_{1}, …, p_{N} are idempotents. In general, P is not an idempotent. However, it is well known that this happens if Γ = F is a Carleson flower. For the reader’s convenience, we will give a proof.

Theorem 7.1. If F is a Carleson flower then S_{F}^{2} = I and hence P_{F}^{2} = P_{F}.

Proof. Let F be comprised of the Carleson Jordan curves Γ_{1}, …, Γ_{N}. Denote by D_{j} the bounded connected component of C \ Γ_{j}, and let D_{∞} stand for the unbounded connected component of C\F. For j = 1, …, N, let R_{0}^{−}(D_{j}) be the set of all functions in R(F) which have no poles outside D_{j} and vanish at infinity. Further, let R^{−}(D_{∞}) stand for the functions in R(F) which have no poles outside D_{∞}.

Every function r ∈ R(F) is of the form r = r_{1} + … + r_{N} + r_{∞} with r_{j} ∈ R_{0}^{−}(D_{j}) for j = 1, …, N and r_{∞} ∈ R^{−}(D_{∞}). Denote by χ_{j}^{0} the characteristic function of Γ_{j}.

As in [33, Chapter 1, Theorem 1.1] one can show that if j = 1, …, N, then

(7.2) χ_{k}^{0}S_{F} χ_{i}^{0}r_{j} = \{ \begin{matrix}−χ_{k}^{0}r_{j} & if k = i = j, \\ χ_{k}^{0}r_{j} & if k = i ≠ j, \\ 0 & if k ≠ i ≠ j, \\ −2χ_{k}^{0}r_{j} & if k ≠ i = j, \end{matrix}

while

χ_{k}^{0}S_{F} χ_{i}^{0}r_{∞} = \{ \begin{matrix}χ_{k}^{0}r_{∞} & if k = i, \\ 0 & if k ≠ i; \end{matrix} (7.3)

we remark that in these equalities the notation χ_{k}^{0}f = χ_{k}^{0}g means that f(t) = g(t) for all t ∈ Γ_{k} at which Γ_{k} has a tangent – for rectifiable Γ_{k} this is the case almost everywhere. For j = 1, …, N, it follows from (7.2) that

S_{F} r_{j} = ∑_{k,i} χ_{k}^{0}S_{F} χ_{i}^{0}r_{j} = χ_{j}^{0}S_{F} χ_{j}^{0}r_{j} + ∑_{k=i≠j} χ_{k}^{0}S_{F} χ_{i}^{0}r_{j} + ∑_{k≠i=j} χ_{k}^{0}S_{F} χ_{i}^{0}r_{j}
(7.4)  = −χ_{j}^{0}r_{j} − 2 ∑_{k≠j} χ_{k}^{0}r_{j} + ∑_{k≠j} χ_{k}^{0}r_{j} = −r_{j}.

Thus, S_{F}^{2} r_{j} = r_{j} for j = 1, …, N. Similarly, using (7.3) one can show that S_{F} r_{∞} = r_{∞}, (7.5) whence S_{F}^{2} r_{∞} = r_{∞}. Consequently, S_{F} r = r for all r ∈ R(F). Theorems 2.1 and 2.2 finally give S_{F}^{2} f = f for all f ∈ L^{p}(F, w). □

The following theorem is our main tool for studying invertibility in the local algebras A_{t}(F) and B_{t}(F) at the center of a Carleson flower.

Theorem 7.2. Let B be a (complex) Banach algebra with identity e. Let p_{1}, p_{2}, …, p_{2N} and P be nonzero elements of B satisfying (7.6) p_{i}p_{j} = δ_{ij}p_{i} for all i, j and p_{1} + p_{2} + … + p_{2N} = e, where δ_{ij} is the Kronecker delta, and Q(p_{2j−1} + p_{2j})P = P (p_{2j} + p_{2j+1})Q = 0 for all j = 1, …, N, (7.7) where Q := e − P and p_{2N+1} := p_{1}. Denote by A the smallest closed subalgebra of B containing P and p_{1}, p_{2}, …, p_{2N}. Put

X := ∑_{j=1}^{N}(p_{2j−1}P p_{2j−1} + p_{2j}Qp_{2j}) (7.8)

and denote by sp_{B}(X) the spectrum of X in B. Suppose finally that (7.9) 0 and 1 are cluster points of sp_{B}(X).

Put E := diag (1, −1, 1, −1, …, 1, −1). For x ∈ C, define the map

σ_{x} : {P, p_{1}, p_{2}, …, p_{2N}} → C^{2N×2N} (7.10)

as follows. For x ∉ {0, 1}, let

σ_{x}(P) = E ( \begin{matrix}x & x − 1 & x − 1 & x − 1 & … & x − 1 & x − 1 \\ x & x − 1 & x − 1 & x − 1 & … & x − 1 & x − 1 \\ x & x & x & x − 1 & … & x − 1 & x − 1 \\ x & x & x & x − 1 & … & x − 1 & x − 1 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ x & x & x & x & … & x & x − 1 \\ x & x & x & x & … & x & x − 1 \end{matrix} ) , (7.11)

for x ∈ {0, 1}, let

σ_{x}(P) = diag (x, 1 − x, x, 1 − x, …, x, 1 − x), (7.12)

and for all x, let

(7.13) σ_{x}(p_{j}) = diag (0, …, 0, 1, 0, …, 0) with 1 at the jth position

(a) For each x ∈ sp_{B}(X), the map σ_{x} extends to a Banach algebra homomorphism σ_{x} of A into C^{2N×2N}.

(b) An element a ∈ A is invertible in B if and only if σ_{x}(a) is invertible in C^{2N×2N} for every x ∈ sp_{B}(X).

This theorem is a special case of the union of Theorem 9 and Proposition 9(d) of the paper [50]. □

Adding the second equalities in (7.7) and taking into account the second equality in (7.6) we get

0 = ∑_{j=1}^{N} P (p_{2j} + p_{2j+1})Q = P ( ∑_{i=1}^{2N} p_{i})Q = P Q,

whence 0 = P (e − P) = P − P^{2}. Thus, if the hypotheses of Theorem 7.2 are satisfied, then P must necessarily be an idempotent.

We will apply Theorem 7.2 to the case where B := B_{t}(F) and A := A_{t}(F) are the local algebras at the center t of a Carleson flower F. Let F = Γ_{1} ∪ … ∪ Γ_{N} with Carleson Jordan curves Γ_{1}, …, Γ_{N}. For sufficiently small ε > 0, the connected component F_{0}(t, ε) of F (t, ε) containing t may be written as

F_{0}(t, ε) = ⋃_{j=1}^{N}(η_{2j−1}^{∗} ∪ η_{2j}^{∗})

where η_{2j−1}^{∗} ⊂ Γ_{j} and η_{2j}^{∗} ⊂ Γ_{j} are outgoing and incoming arcs, respectively. Let χ_{i}^{∗} be the characteristic function of η_{i}^{∗} (i = 1, …, 2N) and put

p_{i} := π_{t}(χ_{i}^{∗}I), P := π_{t}(P_{F}), Q := π_{t}(I − P_{F}). (7.14)

Conditions (7.6) are obviously satisfied.

Lemma 7.3. If F is a Carleson flower and p_{i}, P, Q are defined by (7.14) then (7.7) holds.

We remark that for Lyapunov flowers (carrying a power weight) such a result is already in [30] and that just this result motivated the “axiom” (7.7) (see [50]).

Proof. We have χ_{2j−1}^{∗} + χ_{2j}^{∗} = χ_{j}^{0} + c_{j} where χ_{j}^{0} is the characteristic function of Γ_{j} and c_{j} ∈ P C(F) vanishes on F_{0}(t, ε). Hence, π(c_{j}I) ∈ J_{t}(F), π_{t}(c_{j}I) = 0, and

Q(p_{2j−1} + p_{2j})P = π_{t}((I − P_{F})(χ_{j}^{0} + c_{j})P_{F}) = π_{t}((I − P_{F})χ_{j}^{0}P_{F}). (7.15)

We claim that T := (I − P_{F})χ_{j}^{0}P_{F} = 0. (7.16)

Let r_{i} ∈ R_{0}^{−}(D_{i}) be as in the proof of Theorem 7.1. Then P_{F} r_{i} = (1/2)(r_{i} +S_{F} r_{i}) = 0 due to (7.4) and hence, T r_{i} = 0 for i = 1, 2, …, N. On the other hand, if r_{∞} ∈ R^{−}(D_{∞}), then (7.5) and (7.3) imply that

T r_{∞} = (I − P_{F})χ_{j}^{0}P_{F} r_{∞} = (I − P_{F})χ_{j}^{0}r_{∞} = χ_{j}^{0}r_{∞} − χ_{j}^{0}r_{∞} = 0.

This completes the proof of (7.16) and thus shows that (7.15) vanishes, as desired.

We now prove that P (p_{2j} + p_{2j+1})Q = 0 for every j = 1, …, N. Denote by η̃_{i}^{∗} (i = 1, …, 2N) the arc which results from η_{i}^{∗} by changing the orientation and consider the Carleson star

F̃_{0}(t, ε) := ⋃_{j=1}^{N}(η̃_{2j}^{∗} ∪ η̃_{2j+1}^{∗}).

By virtue of Theorem 5.2, we may extend F̃_{0}(t, ε) to a Carleson flower F̃ and w|F̃_{0}(t, ε) to a weight W ∈ A_{p}(F̃) so that F̃ = Γ̃_{1} ∪ … ∪ Γ̃_{N} with Carleson Jordan curves Γ̃_{1}, …, Γ̃_{N} and so that η̃_{2j}^{∗} and η̃_{2j+1}^{∗} are the outgoing and incoming arcs of Γ̃_{j}, respectively. From what was already proved we know that

π_{t}((I − PF̃))π_{t}(χ_{2j}^{∗}I + χ_{2j+1}^{∗}I)π_{t}(PF̃) = 0. (7.17)

Due to the change of orientation we have

π_{t}(PF̃) = π_{t}((I + SF̃)/2) = π_{t}((I − S_{F})/2) = π_{t}(I − P_{F}).

Consequently, (7.17) implies that

0 = π_{t}(P_{F})π_{t}(χ_{2j}^{∗}I + χ_{2j+1}^{∗}I)π_{t}(I − P_{F}) = P (p_{2j} + p_{2j+1})Q.
□

In summary, conditions (7.6) and (7.7) are satisfied if B = B_{t}(F) is the local algebra at the center of a flower F and p_{1}, p_{2}, …, p_{2N}, P are given by (7.14). Our next concern is to say something about the set

X_{t}(F) := sp_{B}(X) (7.18)

where X is the element (7.8).

8. Local spectra

In Section 4, we constructed the indicator leaf L^{0}(p; α_{t}, β_{t}) for each point t of a Carleson Jordan curve.

Theorem 8.1. If Γ is a Carleson Jordan curve, 1 < p < ∞, and w ∈ A_{p}(Γ), then for every t ∈ Γ,

X_{t}(Γ) = L^{0}(p; α_{t}, β_{t})

where α_{t}, β_{t} are the indicator functions of Γ, w at t.

Proof. If ε > 0 is sufficiently small, then Γ_{0}(t, ε) = η_{1}^{∗} ∪ η_{2}^{∗} with an outgoing arc η_{1}^{∗} and an incoming arc η_{2}^{∗}. The element (7.8) now is

X = p_{1}P p_{1} + p_{2}Qp_{2}

In [8] it is shown that for each x ∈ L^{0}(p; α_{t}, β_{t}) the map

Sym _{t,x}^{0} := {P, p_{1}, p_{2}} → C^{2×2}

sending P, p_{1}, p_{2} to

( \begin{matrix}x & \sqrt{x(1 − x)} \\ \sqrt{x(1 − x)} & 1 − x \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ,

respectively, extends to a Banach algebra homomorphism of A_{t}(Γ) to C^{2×2} and that an element a ∈ A_{t}(Γ) is invertible in B_{t}(Γ) if and only if Sym _{t,x}^{0}(a) is invertible for all x ∈ L^{0}(p; α_{t}, β_{t}). An easy computation yields

Sym _{t,x}^{0}(X − λe) = ( \begin{matrix}x − λ & 0 \\ 0 & x − λ \end{matrix} ) . (8.1)

Since X_{t}(Γ) is the set of all λ ∈ C for which the determinant (8.1) vanishes for some x ∈ L^{0}(p; α_{t}; β_{t}), it follows that X_{t}(Γ) = L^{0}(p; α_{t}, β_{t}). □

Lemma 8.2. Let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Let further t ∈ Γ and A ∈ Λ(Γ). Then π_{t}(A) = 0 if and only if there exist operators K_{n} ∈ K(Γ), E_{n} ∈ Λ(Γ), and functions c_{n} ∈ C(Γ) such that c_{n}(t) = 0 and

∥A − K_{n} − c_{n}E_{n}∥_{B(L^{p}(Γ,w))} → 0 as n → ∞.

Proof. The ideal J_{t}(Γ) is the closure in Λ(Γ)/K(Γ) of all finite sums

∑_{j=1}^{n} π(c_{j}E_{j}) with c_{j} ∈ C(Γ), c_{j}(t) = 0, E_{j} ∈ Λ(Γ). (8.2)

We claim that every sum (8.2) is actually of the form π(cE) with c ∈ C(Γ), c(t) = 0, E ∈ Λ(Γ). Obviously, it suffices to prove this for n = 2.

So let c_{1}, c_{2} ∈ C(Γ), c_{1}(t) = c_{2}(t) = 0. Put c := \sqrt{|c_{1}| + |c_{2}|}. Then c ∈ C(Γ) and c(t) = 0. Let N_{c} := {τ ∈ Γ : c(τ) = 0} and for j ∈ {1, 2}, define

g_{j}(τ) = \{ \begin{matrix}c_{j}(τ)/c(τ) & if τ ∉ N_{c}, \\ 0 & if τ ∈ N_{c}. \end{matrix}

If τ ∉ N_{c}, then

|g_{j}(τ)| = \frac{|c_{j}(τ)|}{c(τ)} = \frac{|c_{j}(τ)|}{|c_{1}(τ)| + |c_{2}(τ)|}c(τ) ≤ c(τ)

and hence, g_{j} ∈ C(Γ). Since c_{1} = cg_{1} and c_{2} = cg_{2}, it follows that c_{1}E_{1} + c_{2}E_{2} = c(g_{1}E_{1} + g_{2}E_{2}), which proves our claim.

We have π_{t}(A) = 0 if and only if π(A) ∈ J_{t}(Γ). From what was proved above, we infer that π(A) ∈ J_{t}(Γ) if and only if there are c_{n} ∈ C(Γ) and E_{n} ∈ Λ(Γ) such that c_{n}(t) = 0 and ∥π(A − c_{n}E_{n})∥ → 0 as n → 0. Because ∥π(B)∥ = inf{∥B + K∥ : K ∈ K(Γ)}, we arrive at the assertion. □

In what follows we have frequently to change curves and thus local algebras. This requires some caution, and we will therefore occasionally denote the homomorphism π_{t} : Λ(Γ) → B_{t}(Γ) by π_{t}^{Γ}.

Lemma 8.3. Let Γ_{1}, …, Γ_{N} be Carleson Jordan curves and let F = Γ_{1} ∪ … ∪ Γ_{N} be a Carleson flower. Denote the characteristic function of Γ_{j} by χ_{j}^{0} and let t be the center of F. Let further A_{j} ∈ B(Γ_{j}) := B(L^{p}(Γ_{j}, W |Γ_{j})) for j = 1, …, N and define A ∈ B(F) := B(L^{p}(F, W)) by

A = ∑_{j=1}^{n} χ_{j}^{0}A_{j}χ_{j}^{0}I.

Then the following hold.

(a) A ∈ K(F) ⟺ A_{j} ∈ K(Γ_{j}) for all j.

(b) A ∈ Λ(F) ⟺ A_{j} ∈ Λ(Γ_{j}) for all j.

(c) sp π_{t}^{F} (A) = ⋃_{j=1}^{N} sp π_{t}^{Γ_{j}}(A_{j}).

Remark. In this lemma and in several similar situations that will follow, we identify the operator A_{j} ∈ B(Γ_{j}) with A_{j} ⊕ 0 ∈ B(F), i.e. with its extension by zero to all of the flower F.

Proof. (a) Obvious.

(b) Let A ∈ Λ(F). If ψ ∈ C(Γ_{j}), then ψ = χ_{j}^{0}φ with some φ ∈ C(F) and hence

A_{j}ψI − ψA_{j} = χ_{j}^{0}(Aχ_{j}^{0}φI − χ_{j}^{0}φA)χ_{j}^{0}I
= χ_{j}^{0}(φAχ_{j}^{0}I + K − χ_{j}^{0}φA)χ_{j}^{0}I = χ_{j}^{0}Kχ_{j}^{0}

with some K ∈ K(F). So A_{j} ∈ Λ(Γ_{j}) by virtue of (a). Conversely, suppose A_{j} ∈ Λ(Γ_{j}) for all j. If φ ∈ C(F), then φ|Γ_{j} ∈ C(Γ_{j}) and thus

AφI − φA = ∑_{j=1}^{N} χ_{j}^{0}(A_{j}φI − φA_{j})χ_{j}^{0}I ∈ Λ(F)

again due to (a).

(c) Suppose π_{t}^{F} (A−λI) is invertible. Then there is a B ∈ Λ(F) such that π_{t}^{F} (B(A− λI) − I) = 0. From Lemma 8.2 we obtain K_{n} ∈ K(F), E_{n} ∈ Λ(F), c_{n} ∈ C(F) such that c_{n}(t) = 0 and

∥B(A − λI) − I − K_{n} − c_{n}E_{n}∥_{B(F)} → 0.

It follows that

∥χ_{j}^{0}B(A − λI)χ_{j}^{0}I − χ_{j}^{0}I − χ_{j}^{0}K_{n}χ_{j}^{0}I − c_{n}χ_{j}^{0}E_{n}χ_{j}^{0}I∥_{B(Γ_{j})} → 0.

We have χ_{j}^{0}K_{n}χ_{j}^{0}I ∈ K(Γ_{j}) and χ_{j}^{0}E_{n}χ_{j}^{0}I ∈ Λ(Γ_{j}) by (a) and (b). Further,

χ_{j}^{0}B(A − λI)χ_{j}^{0}I = χ_{j}^{0}B ∑_{k=1}^{N} χ_{k}^{0}(A_{k} − λI)χ_{k}^{0}χ_{j}^{0}I = χ_{j}^{0}Bχ_{j}^{0}(A_{j} − λI)χ_{j}^{0}I.

Thus, we deduce from Lemma 8.2 that

π_{t}^{Γ_{j}}(χ_{j}^{0}Bχ_{j}^{0}I)π_{t}^{Γ_{j}}(A_{j} − λI) = π_{t}^{Γ_{j}}(I).

Analogously one can show that π_{t}^{Γ_{j}}(A_{j} − λI) is invertible from the right.

Conversely, assume π_{t}^{Γ_{j}}(A_{j} − λI) is invertible for all j. Then there are B_{j} ∈ Λ(Γ_{j}) such that π_{t}^{Γ_{j}}(B_{j}(A_{j} − λI) − I) = 0 and from Lemma 8.2 we get K_{n}^{(j)} ∈ K(Γ_{j}), E_{n}^{(j)} ∈ Λ(Γ_{j}), c_{n}^{(j)} ∈ C(Γ_{j}) such that c_{n}^{(j)}(t) = 0 and

∥B_{j}(A_{j} − λI) − I − K_{n}^{(j)} − c_{n}^{(j)}E_{n}^{(j)}∥_{B(Γ_{j})} → 0.

Put c_{n} := ∑_{j=1}^{N}c_{n}^{(j)}χ_{j}^{0} and

B := ∑_{j=1}^{N}χ_{j}^{0}B_{j}χ_{j}^{0}I, K_{n} := ∑_{j=1}^{N}χ_{j}^{0}K_{n}^{(j)}χ_{j}^{0}I, E_{n} := ∑_{j=1}^{N}χ_{j}^{0}E_{n}^{(j)}χ_{j}^{0}I.

Clearly, c_{n} ∈ C(F) and c_{n}(t) = 0. From (a) and (b) we deduce that K_{n} ∈ K(F) and E_{n} ∈ Λ(F). Since

∥∑_{j=1}^{N}χ_{j}^{0}B_{j}χ_{j}^{0}(A_{j} − λI)χ_{j}^{0} − I − K_{n} − c_{n}E_{n}∥_{B(F)} → 0

and, as above,

∑_{j=1}^{N}χ_{j}^{0}B_{j}χ_{j}^{0}(A_{j} − λI)χ_{j}^{0}I = B(A − λI),

Lemma 8.2 implies that π_{t}^{F} (B)π_{t}^{F} (A − λI) = π_{t}^{F} (I). □

Theorem 8.4. Let Γ_{1}, …, Γ_{N} be Carleson Jordan curves and let F = Γ_{1} ∪ … ∪ Γ_{N} be a Carleson flower with the center t. Suppose 1 < p < ∞ and W ∈ A_{p}(F). Then

X_{t}(F) = L^{0}(p; α_{t}^{1}, β_{t}^{1}) ∪ … ∪ L^{0}(p; α_{t}^{N}, β_{t}^{N}) (8.3)

where α_{t}^{j}, β_{t}^{j} are the indicator functions of Γ_{j}, W |Γ_{j} at t.

Proof. By (7.18), (7.8), (7.14),

X_{t}(F) = sp π_{t}^{F} (∑_{j=1}^{N}(χ_{2j−1}^{∗}P_{F} χ_{2j−1}^{∗}I + χ_{2j}^{∗}Q_{F} χ_{2j}^{∗}I))

= sp π_{t}^{F} (∑_{j=1}^{N}χ_{j}^{0}(χ_{2j−1}^{∗}P_{Γ_{j}}χ_{2j−1}^{∗}I + χ_{2j}^{∗}Q_{Γ_{j}}χ_{2j}^{∗}I)χ_{j}^{0}I).

Lemma 8.3(c) therefore gives

X_{t}(F) = ⋃_{j=1}^{N} sp π_{t}^{Γ_{j}}(χ_{2j−1}^{∗}P_{Γ_{j}}χ_{2j−1}^{∗}I + χ_{2j}^{∗}Q_{Γ_{j}}χ_{2j}^{∗}I) = ⋃_{j=1}^{N} X_{t}(Γ_{j}),

and Theorem 8.1 then implies (8.3). □

We will show that actually L^{0}(p; α_{t}^{1}, β_{t}^{1}) = … = L^{0}(p; α_{t}^{N}, β_{t}^{N}). At the present moment we only remark that, by (4.13) and (4.15), each leaf L^{0}(p; α_{t}^{j}, β_{t}^{j}) is a connected set containing the points 0 and 1, so that (7.9) is satisfied in view of (8.3).

9. Symbol calculus

Let Γ be a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). We fix a point t ∈ Γ and denote by γ := Γ_{0}(t, ε) the connected component of the portion Γ(t, ε) containing t. If ε > 0 is small enough, then γ is a Carleson star comprised of N(t) simple Carleson arcs η_{1}, …, η_{N(t)}. By Theorem 6.9, we may find a Carleson flower F consisting of N(t) Carleson Jordan curves Γ_{1}, …, Γ_{N(t)} and a weight W ∈ A_{p}(F) such that γ ⊂ F and W |γ = w|γ. We may assume that η_{j} ⊂ Γ_{j}. Write (Γ_{j})_{0}(t, ε) = η_{2j−1}^{∗} ∪ η_{2j}^{∗} where η_{2j−1}^{∗} is the outgoing arc and η_{2j}^{∗} is the incoming arc. Thus, if η_{j} is outgoing (⟺ ε_{j}(t) = 1) then η_{j} = η_{2j−1}^{∗} and η_{2j}^{∗} is not contained in γ, while if η_{j} is incoming (⟺ ε_{j}(t) = −1) then η_{j} = η_{2j}^{∗} and η_{2j−1}^{∗} is not a subset of γ. Clearly,

γ = Γ_{0}(t, ε) = η_{1} ∪ … ∪ η_{N(t)},

F_{0}(t, ε) = ⋃_{j=1}^{N(t)}(Γ_{j})_{0}(t, ε) = ⋃_{j=1}^{N(t)}(η_{2j−1}^{∗} ∪ η_{2j}^{∗}).

We have the three local algebras

B_{t}(Γ) := B_{t}(Γ, p, w), B_{t}(γ) := B_{t}(γ, p, w|γ), B_{t}(F) := B_{t}(F, p, W)

and their subalgebras A_{t}(Γ), A_{t}(γ), A_{t}(F). The results of Sections 7 and 8 tell us a lot about A_{t}(F). We now connect A_{t}(Γ) with A_{t}(F) via A_{t}(γ).

Lemma 9.1. The map

Φ_{t}^{π} : B_{t}(Γ) → B_{t}(γ), π_{t}^{Γ}(A) ↦ π_{t}^{γ}(χ_{γ}Aχ_{γ}I)

is a well-defined Banach algebra isomorphism.

Proof. As in the proof of Lemma 8.3(b) one can verify that χ_{γ}Aχ_{γ}I ∈ Λ(γ) whenever A ∈ Λ(Γ). Clearly, the map

Φ : Λ(Γ) → B_{t}(γ), A ↦ π_{t}^{γ}(χ_{γ}Aχ_{γ}I)

is a bounded linear operator. We claim that Φ(AB) = Φ(A)Φ(B), i.e. that

π_{t}^{γ}(χ_{γ}ABχ_{γ}I) = π_{t}^{γ}(χ_{γ}Aχ_{γ}Bχ_{γ}I)

for all A, B ∈ Λ(Γ). Let φ ∈ C(Γ) be any function such that φ(t) = 1 and φ|Γ \ γ = 0. Then π_{t}^{γ}(φI) = π_{t}^{γ}(I), and since χ_{γ}φ = φ ∈ C(Γ), we get

π_{t}^{γ}(χ_{γ}Aχ_{γ}Bχ_{γ}I) = π_{t}^{γ}(χ_{γ}Aχ_{γ}Bχ_{γ}φI) = π_{t}^{γ}(χ_{γ}Aχ_{γ}φBχ_{γ}I)
= π_{t}^{γ}(χ_{γ}ABχ_{γ}^{2}φI) = π_{t}^{γ}(χ_{γ}ABχ_{γ}I),

which proves our claim and shows that Φ is a Banach algebra homomorphism. Obviously, K(Γ) ⊂ Ker Φ. Therefore

Φ^{π} : Λ(Γ)/K(Γ) → B_{t}(γ), π^{Γ}(A) ↦ π_{t}^{γ}(χ_{γ}Aχ_{γ}I)

is a well-defined Banach algebra homomorphism. It is readily seen that J_{t}(Γ) ⊂ Ker Φ^{π}, which implies that Φ_{t}^{π} is a well-defined Banach algebra homomorphism.

Every element in B_{t}(γ) is of the form π_{t}^{γ}(B) with B ∈ Λ(γ). Define A ∈ B(Γ) by A := φBφI where φ is as above. If ψ ∈ C(Γ), then

AψI − ψA = (φB)(ψφI) − (ψφ)(BφI) ∈ K(Γ)

and hence, A ∈ Λ(Γ). Because π_{t}^{γ}(χ_{γ}Aχ_{γ}I) = π_{t}^{γ}(φBφI) = π_{t}^{γ}(B), we conclude that Φ_{t}^{π} is surjective.

Now assume π_{t}^{γ}(χ_{γ}Aχ_{γ}I) = 0 for some A ∈ Λ(Γ). Lemma 8.2 gives K_{n} ∈ K(γ), E_{n} ∈ Λ(γ), c_{n} ∈ C(γ) such that c_{n}(t) = 0 and

∥χ_{γ}Aχ_{γ}I − K_{n} − c_{n}E_{n}∥_{B(γ)} → 0.

Again let φ ∈ C(Γ) be as above. Then

∥φ^{2}χ_{γ}Aχ_{γ}φI − φ^{2}K_{n}φI − φc_{n}φE_{n}φ∥_{B(Γ)} → 0,

and because φ^{2}K_{n}φI ∈ K(Γ), φE_{n}φI ∈ Λ(Γ), φc_{n} ∈ C(Γ), (φc_{n})(t) = 0, we deduce from Lemma 8.2 that

0 = π_{t}^{Γ}(φ^{2}χ_{γ}Aχ_{γ}φI) = π_{t}^{Γ}(A).

This proves that Φ_{t}^{π} is injective. □

Lemma 9.2. The map

Ψ_{t}^{π} : B_{t}(γ) → B_{t}(F), π_{t}^{γ}(A) ↦ π_{t}^{F} (χ_{γ}Aχ_{γ}I)

is a well-defined injective Banach algebra homomorphism enjoying the property that π_{t}^{γ}(A) is invertible in B_{t}(γ) if and only if π_{t}^{F} (χ_{γ}Aχ_{γ}I) + π_{t}^{F} (χ_{F \γ}I) is invertible in B_{t}(F).

Proof. The map

Ψ : Λ(γ) → B_{t}(F), A ↦ π_{t}^{F} (χ_{γ}Aχ_{γ}I)

is a bounded linear operator. If A, B ∈ Λ(γ), then χ_{γ}ABχ_{γ}I = χ_{γ}Aχ_{γ}Bχ_{γ}I and thus, Ψ is a Banach algebra homomorphism. Taking into account that K(γ) ⊂ Ker Ψ, we see that

Ψ^{π} : Λ(γ)/K(γ) → B_{t}(F), π^{γ}(A) ↦ π_{t}^{F} (χ_{γ}Aχ_{γ}I)

is a well-defined Banach algebra homomorphism. Since every function c ∈ C(γ) vanishing at t may be extended to a function c ∈ C(F) taking on the value zero at t, it follows that J_{t}(γ) ⊂ Ker Ψ^{π} and hence that Ψ_{t}^{π} is a well-defined Banach algebra homomorphism. Notice, however, that Ψ_{t}^{π} maps the identity of B_{t}(γ) to π_{t}^{F} (χ_{γ}I), which is not the identity of B(F).

To show that Ψ_{t}^{π} is injective, assume A ∈ Λ(γ) and π_{t}^{F} (χ_{γ}Aχ_{γ}I) = 0. By Lemma 8.2., there are K_{n} ∈ K(F), E_{n} ∈ Λ(F), c_{n} ∈ C(F) such that c_{n}(t) = 0 and

∥χ_{γ}Aχ_{γ}I − K_{n} − c_{n}E_{n}∥_{B(F)} → 0.

It follows that

∥χ_{γ}Aχ_{γ}I − χ_{γ}K_{n}χ_{γ}I − c_{n}χ_{γ}Eχ_{γ}I∥_{B(γ)} → 0,

and because χ_{γ}K_{n}χ_{γ}I ∈ K(γ) and χ_{γ}Eχ_{γ}I ∈ Λ(γ), we infer from Lemma 8.2 that

0 = π_{t}^{γ}(χ_{γ}Aχ_{γ}I) = π_{t}^{γ}(A).

This proves the injectivity of Ψ_{t}^{π}.

Now suppose A ∈ Λ(γ) and π_{t}^{γ}(A) is invertible. Then there is an operator B ∈ Λ(γ) such that π_{t}^{γ}(BA) = π_{t}^{γ}(I). We claim that

π_{t}^{F} (χ_{γ}Bχ_{γ}I + χ_{F \γ}I)π_{t}^{F} (χ_{γ}Aχ_{γ}I + χ_{F \γ}I) = π_{t}^{F} (I).

Indeed,

π_{t}^{F} (χ_{γ}Bχ_{γ}I + χ_{F \γ}I)π_{t}^{F} (χ_{γ}Aχ_{γ}I + χ_{F \γ}I) − π_{t}^{F} (I)
= π_{t}^{F} (χ_{γ}Bχ_{γ}Aχ_{γ}I + χ_{F \γ}I − χ_{F} I)
= π_{t}^{F} (χ_{γ}Bχ_{γ}Aχ_{γ}I − χ_{γ}I)
= π_{t}^{F} (χ_{γ}Bχ_{γ}I)π_{t}^{F} (χ_{γ}Aχ_{γ}I) − π_{t}^{F} (χ_{γ}I),

and this equals zero because the map π_{t}^{γ}(C) ↦ π_{t}^{F} (χ_{γ}Cχ_{γ}I) was shown to be a homomorphism and π_{t}^{γ}(B)π_{t}^{γ}(A) − π_{t}^{γ}(I) = 0 by assumption. Right invertibility can be disposed of analogously.

Conversely, suppose now that A ∈ Λ(γ) and that π_{t}^{F} (χ_{γ}Aχ_{γ}I + χ_{F \γ}I) is invertible. Then there exists a B ∈ Λ(F) such that

π_{t}^{F} (B)π_{t}^{F} (χ_{γ}Aχ_{γ}I + χ_{F \γ}I) = π_{t}^{F} (I).

It results that

0 = π_{t}^{F} (χ_{γ}I)π_{t}^{F} (B(χ_{γ}Aχ_{γ}I + χ_{F \γ}I) − χ_{F} I)
= π_{t}^{F} (χ_{γ}Bχ_{γ}Aχ_{γ}I − χ_{γ}I).

Since the map π_{t}^{γ}(C) ↦ π_{t}^{F} (χ_{γ}Cχ_{γ}I) is injective, we obtain that

0 = π_{t}^{γ}(χ_{γ}Bχ_{γ}Aχ_{γ}I − χ_{γ}I) = π_{t}^{γ}(χ_{γ}Bχ_{γ}I)π_{t}^{γ}(A) − π_{t}^{γ}(I),

which shows that π_{t}^{γ}(χ_{γ}Bχ_{γ}I) is a left inverse of π_{t}^{γ}(A). In the same way one can show that π_{t}^{γ}(A) is invertible from the right. □

Put D_{t} := diag (δ_{1}(t), δ_{2}(t), …, δ_{2N(t)}(t)) (9.1)

where δ_{i}(t) = 1 if η_{i}^{∗} ⊂ γ and δ_{i}(t) = 0 if η_{i}^{∗} is not contained in γ. In other words, if η_{j} is outgoing (⟺ ε_{j}(t) = 1) then δ_{2j−1}(t) = 1 and δ_{2j}(t) = 0, while if η_{j} is incoming (⟺ ε_{j}(t) = −1) then δ_{2j−1}(t) = 0 and δ_{2j}(t) = 1. Recall that X_{t}(F) is given by formula (8.3).

Lemma 9.3. (a) For each x ∈ X_{t}(F), the map

H_{t,x} : {P_{Γ}} ∪ {aI : a ∈ P C(Γ)} → C^{2N(t)×2N(t)}

given by

(9.2)  H_{t,x}(aI) = D_{t} diag (a_{1}(t), a_{1}(t), …, a_{2N(t)}(t), a_{2N(t)}(t))D_{t},
(9.3)  H_{t,x}(P_{Γ}) = D_{t}M_{x}D_{t},

M_{x} being the matrix (7.11), (7.12), extends to a Banach algebra homomorphism of the algebra alg (S_{Γ}, P C(Γ)) to C^{2N(t)×2N(t)}.

(b) For A ∈ alg (S_{Γ}, P C(Γ)) and x ∈ X_{t}(F), put

Sym _{t,x}^{0}(A) = H_{t,x}(A) + I − D_{t}. (9.4)

Then π_{t}^{Γ}(A) is invertible in B_{t}(Γ) if and only if Sym _{t,x}^{0}(A) is invertible in C^{2N(t)×2N(t)} for all x ∈ X_{t}(F).

Proof. Combining Lemmas 9.1 and 9.2 we see that the map

H_{t} := Ψ_{t}^{π} ∘ Φ_{t}^{π} ∘ π_{t}^{Γ} : Λ(Γ) → B_{t}(F), A ↦ π_{t}^{F} (χ_{γ}Aχ_{γ}I)

is a Banach algebra homomorphism with the property that π_{t}^{Γ}(A) is invertible in B_{t}(Γ) if and only if H_{t}(A) + π_{t}^{F} (χ_{F \γ}I) is invertible in B_{t}(F). Clearly, H_{t} maps alg (S_{Γ}, P C(Γ)) into A_{t}(F).

The algebra A_{t}(F) is generated by P := π_{t}^{F} (P_{F}) and p_{i} := π_{t}^{F} (χ_{i}^{∗}I) where χ_{i}^{∗} is the characteristic function of η_{i}^{∗} (i = 1, 2, …, 2N(t)). Lemma 7.3 and the remark at the end of Section 8 show that the hypotheses of Theorem 7.2 are satisfied for B := B_{t}(F). We have sp _{B}(X) = X_{t}(F) by the definition (7.18) and we have some preliminary information about X_{t}(F) from Theorem 8.4. Thus, Theorem 7.2 implies that for each x ∈ X_{t}(F) the map

σ_{t,x} : {P, p_{1}, p_{2}, …, p_{2N(t)}} → C^{2N(t)×2N(t)}

which sends p_{i} and P to the matrices (7.13) and (7.11), respectively, extends to a Banach algebra homomorphism

σ_{t,x} : A_{t}(F) → C^{2N(t)×2N(t)}

and that π_{t}^{F} (A) is invertible in B_{t}(F) if and only if σ_{t,x}(π_{t}^{F} (A)) is invertible in C^{2N(t)×2N(t)} for all x ∈ X_{t}(F).

Define

(9.5)  H_{t,x} := σ_{t,x} ∘ H_{t} : alg (S_{Γ}, P C(Γ)) → C^{2N(t)×2N(t)},
A ↦ σ_{t,x}(π_{t}^{F} (χ_{γ}Aχ_{γ}I))

and

(9.6)  Sym _{t,x}^{0} : alg (S_{Γ}, P C(Γ)) → C^{2N(t)×2N(t)},
A ↦ H_{t,x}(A) + σ_{t,x}(π_{t}^{F} (χ_{F \γ}I)).

We know that H_{t,x} is a Banach algebra homomorphism and we know that π_{t}^{Γ}(A) is invertible in B_{t}(Γ) if and only if Sym _{t,x}^{0}(A) is invertible in C^{2N(t)×2N(t)} for all x ∈ X_{t}(F). The lemma will follow once we have shown that the map H_{t,x} defined by (9.5) sends aI and P_{Γ} to the matrices (9.2) and (9.3), respectively, and that (9.6) is nothing but the map (9.4).

Obviously,

σ_{t,x}(π_{t}^{F} (χ_{γ}I)) = D_{t}, σ_{t,x}(π_{t}^{F} (χ_{F \γ}I)) = I − D_{t}

where D_{t} is the matrix (9.1). Moreover, σ_{t,x}(π_{t}^{F} (χ_{γ}aχ_{γ}I)) equals

∑_{j=1}^{N(t)} σ_{t,x}(π_{t}^{F} (δ_{2j−1}(t)a_{j}(t)χ_{2j−1}^{∗}I + δ_{2j}(t)a_{j}(t)χ_{2j}^{∗}I))

= D_{t} diag (a_{1}(t), a_{1}(t), a_{2}(t), a_{2}(t), …, a_{N(t)}(t), a_{N(t)}(t)),

which is (9.2), and σ_{t,x}(π_{t}^{F} (χ_{γ}P_{Γ}χ_{γ}I)) is equal to

σ_{t,x}(π_{t}^{F} (χ_{γ}P_{F} χ_{γ}I)) = σ_{t,x}(π_{t}^{F} (χ_{γ}I))σ_{t,x}(π_{t}^{F} (P_{F}))σ_{t,x}(π_{t}^{F} (χ_{γ}I)),

which coincides with (9.3). □

The previous lemma is already almost the thing we are looking for. Its flaws are the following: although the maps H_{t,x} are homomorphisms, they do not answer for invertibility; the maps Sym _{t,x}^{0} answer for invertibility but are no homomorphisms; the “symbol matrices” are of order 2N(t) and not of order N(t); the set X_{t}(F) is not yet identified with the leaf L_{t}(Γ). The first three of these spots will be remedied by the following corollary. The equality X_{t}(F) = L_{t}(Γ) will be shown in Section 10.

Corollary 9.4. (a) For each x ∈ X_{t}(F), the map

Sym_{t,x} : {S_{Γ}} ∪ {aI : a ∈ P C(Γ)} → C^{N(t)×N(t)}

sending S_{Γ} and aI to the matrices (4.5), (4.6) and (4.7), respectively, extends to a Banach algebra homomorphism of alg (S_{Γ}, P C(Γ)) to C^{N(t)×N(t)}.

(b) If A ∈ alg (S_{Γ}, P C(Γ)) then

sp π_{t}^{Γ}(A) = ⋃_{x∈X_{t}(F)} sp Sym_{t,x}(A).

In short, Theorem 4.2 holds with L_{t}(Γ) replaced by X_{t}(F).

Proof. From (9.2) and (9.3) we infer that

H_{t,x}(A) = D_{t}H_{t,x}(A)D_{t} (9.7)

for every A ∈ alg (S_{Γ}, P C(Γ)). Appropriately changing rows and columns of H_{t,x}(A),

one can produce a matrix of the form ( \begin{matrix}∗ & 0 \\ 0 & 0 \end{matrix} ) where ∗ is an N(t) × N(t) matrix.

To be more precise, let Π_{t} be the permutation matrix for which

Π_{t}D_{t}Π_{t}^{−1} = ( \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} ) .

From (9.7) we then get

Π_{t}H_{t,x}(A)Π_{t}^{−1} = ( \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} ) Π_{t}H_{t,x}(A)Π_{t}^{−1} ( \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} ) .

Denote the nonzero N(t) × N(t) block of the latter matrix by Sym_{t,x}(A). Thus,

Π_{t}H_{t,x}(A)Π_{t}^{−1} = ( \begin{matrix}Sym_{t,x}(A) & 0 \\ 0 & 0 \end{matrix} ) .

Since H_{t,x} is a Banach algebra homomorphism, the map

Sym_{t,x} : alg (S_{Γ}, P C(Γ)) → C^{N(t)×N(t)}

must also be a Banach algebra homomorphism. Because

Π_{t}Sym _{t,x}^{0}(A)Π_{t}^{−1} = Π_{t}(H_{t,x}(A) + I − D_{t})Π_{t}^{−1}

= ( \begin{matrix}Sym_{t,x}(A) & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}0 & 0 \\ 0 & I \end{matrix} ) = ( \begin{matrix}Sym_{t,x}(A) & 0 \\ 0 & I \end{matrix} ) ,

it follows from Lemma 9.3(b) that π_{t}^{Γ}(A − λI) is invertible in B_{t}(Γ) if and only if the matrix Sym_{t,x}(A − λI) is invertible in C^{N(t)×N(t)} for all x ∈ X_{t}(F).

What we are left with is to show that if x ∉ {0, 1}, then Sym_{t,x}(S_{Γ}) and Sym_{t,x}(aI) are just the matrices (4.5) and (4.7) respectively. This is easy for aI:

Π_{t}H_{t,x}(aI)Π_{t}^{−1} = diag (a_{1}(t), …, a_{N(t)}(t), 0, …, 0)

and thus, Sym_{t,x}(aI) = diag (a_{1}(t), …, a_{N(t)}(t)). A little computation shows that

Π_{t}H_{t,x}(P_{Γ})Π_{t}^{−1} = Π_{t}D_{t}Π_{t}^{−1}Π_{t}M_{x}Π_{t}^{−1}Π_{t}D_{t}Π_{t}^{−1}

has the north-west N(t) × N(t) block

(9.8) Sym_{t,x}(P_{Γ}) = diag (ε_{1}(t), ε_{2}(t), …, ε_{N(t)}(t)) ×

× ( \begin{matrix}x + \frac{ε_{1}(t)−1}{2} & x − 1 & … & x − 1 \\ x & x + \frac{ε_{2}(t)−1}{2} & … & x − 1 \\ ⋮ & ⋮ & ... & ⋮ \\ x & x & … & x + \frac{ε_{N(t)}(t)−1}{2} \end{matrix} ) .

This implies that Sym_{t,x}(S_{Γ}) = 2Sym_{t,x}(P_{Γ}) − I is the matrix (4.5). □

10. Identification of leaves

We are now in a position to complete the proofs of the four theorems stated in Section 4.

Proof of Theorem 4.3. Part (a) is a special case of Theorem 6.9. So let the situation be as in part (b). By definition,

L_{t}(η) = sp π_{t}^{η}(P_{η}^{+}) = sp π_{t}^{η}((I + S_{η}^{+})/2).

Lemma 9.2 (with γ := η and F := Γ) tells us that

\{1\} ∪ sp π_{t}^{η}(\frac{I + S_{η}^{+}}{2} ) = sp π_{t}^{Γ}(χ_{η} \frac{I + S_{η}^{+}}{2} χ_{η}I + χ_{Γ\η}I) =: sp π_{t}^{Γ}(A_{η}).

The (counter-clockwise) orientation of Γ induces an orientation on η. If t is the starting point of η with respect to this orientation, then

A_{η} = χ_{η} \frac{I + S_{Γ}}{2} χ_{η}I + χ_{Γ\η}I, (10.1)

while if t is the terminating point of η in this orientation, we have

A_{η} = χ_{η} \frac{I − S_{Γ}}{2} χ_{η}I + χ_{Γ\η}I. (10.2)

For x ∈ X_{t}(Γ), we construct Sym_{t,x}(A_{η}) as in Corollary 9.4. Let x ∉ {0, 1}. In the case (10.1), Sym_{t,x}(A_{η}) equals

\frac{1}{2} ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) [ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ( \begin{matrix}2x − 1 & 2x − 2 \\ 2x & 2x − 1 \end{matrix} ) ] ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) + ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ,

and in the case (10.2), Sym_{t,x}(A_{η}) is equal to

\frac{1}{2} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) [ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) − ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ( \begin{matrix}2x − 1 & 2x − 2 \\ 2x & 2x − 1 \end{matrix} ) ] ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) .

The latter two matrices are

( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 \\ 0 & x \end{matrix} ) ,

respectively. In either case, the eigenvalues of the resulting matrices are x and 1. The same is also true if x ∈ {0, 1}. Thus, by Corollary 9.4(b),

sp π_{t}^{Γ}(A_{η}) = ⋃_{x∈X_{t}(F)}{x, 1} = X_{t}(Γ) ∪ {1}.

Because X_{t}(Γ) = L^{0}(p; α_{t}, β_{t}) due to Theorem 8.1, we see that

L_{t}(η) ∪ {1} = L^{0}(p; α_{t}, β_{t}) ∪ {1} = L^{0}(p; α_{t}, β_{t}).

The connectedness of L^{0}(p, α_{t}, β_{t}) and the closedness of L_{t}(η) finally show that L_{t}(η) = L^{0}(p, α_{t}, β_{t}). □

Proof of Theorem 4.1. By virtue of Theorem 6.9, there exists a Carleson Jordan curve Γ and a weight W ∈ A_{p}(Γ) such that η_{1} ∪ η_{2} ⊂ Γ and W |η_{1} ∪ η_{2} = w. Theorem 4.3(b) implies that L_{t}(η_{1}) = L^{0}(p; α_{t}, β_{t}) and L_{t}(η_{2}) = L^{0}(p; α_{t}, β_{t}). □

Proof of Theorem 4.2. As already said, part (c) is a consequence of part (b) and Propositions 3.2 and 3.3. Let the notation be as in the first paragraph of Section 9. Corollary 9.4 tells us that parts (a) and (b) of Theorem 4.2 are true with L_{t}(η) replaced by X_{t}(F). From Theorem 8.4 we deduce that

X_{t}(F) = L^{0}(p; α_{t}^{1}, β_{t}^{1}) ∪ … ∪ L^{0}(p; α_{t}^{N(t)}, β_{t}^{N(t)}).

Theorem 4.3(b) then implies that X_{t}(F) = L_{t}(η_{1}) ∪ … ∪ L_{t}(η_{N(t)}), and since w ∈ A_{p}(η_{i} ∪ η_{j}) for each pair (i, j) by the boundedness of S_{η_{i}∪η_{j}}, we infer from Theorem 4.1 that L_{t}(η_{1}) = … = L_{t}(η_{N(t)}) =: L_{t}(Γ). Thus, X (F) = L_{t}(Γ). □.

Proof of Theorem 4.4. Put ε_{j} := ε_{j}(t) and denote the matrix (4.5) by A_{x}. From Theorem 4.3 we deduce that

sp π_{t}(S_{Γ}) = ⋃_{x∈L^{0}(p;α_{t},β_{t})} sp A_{x}. (10.3)

Let Δ_{N}(ε_{1}, …, ε_{N}; λ) := det(A_{x} − λI). If ε_{j} + ε_{j+1} = 0 for some j, we first add the jth row of A_{x} − λI to the j + 1st row and then subtract the j + 1st column from the jth column to obtain

Δ_{N}(ε_{1}, …, ε_{N}, λ) = (λ^{2} − 1)Δ_{N−2}(ε_{1}, …, ε_{j−1}, ε_{j+2}, …, ε_{N}; λ).

Repeating this procedure we get the formula

Δ_{N}(ε_{1}, …, ε_{N}, λ) = \{ \begin{matrix}(λ^{2} − 1)^{(N−ε)/2}Δ_{ε}(1, …, 1; λ) if ε > 0, \\ (λ^{2} − 1)^{(N−|ε|)/2}Δ_{|ε|}(−1, …, −1; λ) if ε < 0. \end{matrix} (10.4)

If ε = 0, then Δ_{N}(ε_{1}, …, ε_{N}; λ) = (λ^{2} − 1)^{N/2}, which gives (4.19). Assume ε > 0. Using the identity

| \begin{matrix}c & a & a & … & a \\ b & c & a & … & a \\ … & … & … & … & … \\ b & b & b & … & c \end{matrix} |_{(ε×ε)} = \frac{a}{a − b}(c − b)^{ε} + \frac{b}{b − a}(c − a)^{ε}

we arrive at the equality

Δ_{ε}(1, …, 1; λ) = x(1 − λ)^{ε} − (x − 1)(−1 − λ)^{ε}. (10.5)

If x = 0 then (10.5) vanishes for λ = −1, and if x = 1 then (10.5) is zero for λ = 1. Hence, sp A_{0} ∪ sp A_{1} = {−1, 1}. (10.6)

So let x ∉ {0, 1}. Then (10.4) and (10.5) give

\{−1, 1\} ∪ sp A_{x} = \{−1, 1\} ∪ \{λ ∈ C : (\frac{λ + 1}{λ − 1})^{ε} = \frac{x}{x − 1}\}. (10.7)

We have

\{\frac{x}{x − 1} : x ∈ L^{0}(p; α_{t}, β_{t}) \ \{0, 1\}\} = \{e^{2πz} : z ∈ Y (p; α_{t}, β_{t})\}

and therefore, by (10.3), (10.6), (10.7),

sp π_{t}(S_{Γ}) = {−1, 1} ∪ {sp A_{x} : x ∈ L^{0}(p; α_{t}, β_{t}) \ {0, 1}}

= \{−1, 1\} ∪ ⋃_{k=0}^{ε−1} \{λ ∈ C : \frac{λ + 1}{λ − 1} = e^{2π(z+ik)/ε} for some z ∈ Y (p; α_{t}, β_{t})\}.

Since 1/p + α_{t}(x) ≤ y ≤ 1/p + β_{t}(x) if and only if

\frac{1}{p} + (\frac{1}{pε} − \frac{1}{p} + \frac{k}{ε} + \frac{α_{t}(εx/ε)}{ε}) ≤ \frac{y + k}{ε} ≤ \frac{1}{p} + (\frac{1}{pε} − \frac{1}{p} + \frac{k}{ε} + \frac{β_{t}(εx/ε)}{ε}),

we see that z ∈ Y (p; α_{t}, β_{t}) if and only if

\frac{z + ik}{ε} ∈ Y (p; \frac{1}{ε}(\frac{1}{p} − \frac{ε}{p} + k + T_{ε}α_{t}), \frac{1}{ε}(\frac{1}{p} − \frac{ε}{p} + k + T_{ε}β_{t})) =: Y_{k}.

Because

\{λ ∈ C : \frac{λ + 1}{λ − 1} = e^{2πζ} for some ζ ∈ Y_{k}\}

= \{λ ∈ C : λ = 2\frac{e^{2πζ}}{e^{2πζ} − 1} − 1 for some ζ ∈ Y_{k}\},

we finally get (4.20). The case ε < 0 may be tackled analogously or may be reduced to the case ε > 0 by changing the orientation of Γ and taking into account that S_{−Γ} = −S_{Γ}. □

11. Matrix case and semi-Fredholmness

Let L^{p,n}(Γ, w) be the direct sum of n copies of L^{p}(Γ, w) and provide L^{p,n}(Γ, w) with the norm

∥f∥ = ∥(f_{1}, …, f_{n})∥ := (∥f_{1}∥^{p} + … + ∥f_{n}∥^{p})^{1/p}

or any norm equivalent to this one. The operator S_{Γ} is defined on L^{p,n}(Γ, w) elementwise, that is,

S_{Γ}(f_{1}, …, f_{n}) := (S_{Γ}f_{1}, …, S_{Γ}f_{n}).

Denote by P C^{n}(Γ) the algebra of all n×n matrix functions with entries in P C(Γ). We think of L^{p,n}(Γ, w) as a space of column vectors and may therefore identify elements a ∈ P C^{n}(Γ) with the multiplication operators they induce on L^{p,n}(Γ, w):

(af)(t) = a(t)f(t) a.e. on Γ.

Let alg (S_{Γ}, P C^{n}(Γ)) stand for the closed algebra generated by S_{Γ} and P C^{n}(Γ). Let also B_{t}^{n}(X), A_{t}^{n}(X) be similar modifications of B_{t}(X), A_{t}(X) (with X ∈ {Γ, F, γ}).

Theorem 7.2 can be extended to a theorem applicable to the case B = B_{t}^{n}(F), A = A_{t}^{n}(F). Namely, Theorem 13 of [50] implies the following.

Theorem 11.1. Let B be a (complex) Banach algebra with identity e containing a subalgebra G isomorphic to C^{n×n} and nonzero elements p_{1}, …, p_{2N}, P such that

p_{i}g = gp_{i} and gP = P g for all i = 1, …, 2N and g ∈ G,
p_{i}p_{j} = δ_{ij}p_{i} for all i, j and p_{1} + p_{2} + … + p_{2N} = e,

where δ_{ij} is the Kronecker delta, and

Q(p_{2j−1} + p_{2j})P = P (p_{2j} + p_{2j+1})Q = 0 for all j = 1, …, N,

where Q := e − P and p_{2N+1} := p_{1}. Denote by A the smallest closed subalgebra of B containing G ∪ {P, p_{1}, p_{2}, …, p_{2N}}. Put

X := ∑_{j=1}^{N}(p_{2j−1}P p_{2j−1} + p_{2j}Qp_{2j})

and let sp_{B}(X) denote the spectrum of X in B. Also suppose that

0 and 1 are cluster points of sp_{B}(X).

Let φ : G → C^{n×n} be any isomorphism and let 0 and I stand for the n × n zero and identity matrices, respectively. Put E := diag (I, −I, I, −I, …, I, −I), and given x ∈ C, define the map

σ_{x} : G ∪ {P, p_{1}, p_{2}, …, p_{2N}} → C^{2Nn×2Nn}

by

σ_{x}(g) = diag (φ(g), …, φ(g)) (g ∈ G),
σ_{x}(p_{j}) = diag (0, …, 0, I, 0, …, 0), the I located at the jth position,

for all x and by

σ_{x}(P) = E ( \begin{matrix}xI & (x−1)I & (x−1)I & (x−1)I & … & (x−1)I & (x−1)I \\ xI & (x−1)I & (x−1)I & (x−1)I & … & (x−1)I & (x−1)I \\ xI & xI & xI & (x−1)I & … & (x−1)I & (x−1)I \\ xI & xI & xI & (x−1)I & … & (x−1)I & (x−1)I \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ xI & xI & xI & xI & … & xI & (x−1)I \\ xI & xI & xI & xI & … & xI & (x−1)I \end{matrix} )

if x ∉ {0, 1} and

σ_{x}(P) = diag (xI, (1 − x)I, xI, (1 − x)I, …, xI, (1 − x)I)

if x ∈ {0, 1}.

(a) For each x ∈ sp_{B}(X), the map σ_{x} extends to a Banach algebra homomorphism σ_{x} of A into C^{2Nn×2Nn}.

(b) An element a ∈ A is invertible in B if and only if σ_{x}(a) is invertible in C^{2Nn×2Nn} for every x ∈ sp_{B}(X).

Lemmas 9.1 - 9.3 also extend to the matrix case with obvious adjustments (for example, the Banach algebra homomorphism in Lemma 9.3 acts now from alg(S_{Γ}, P C^{n}(Γ)) to C^{2N(t)n×2N(t)n}). Hence, our main result (Theorem 4.2) carries over to the matrix setting.

Theorem 11.2. Suppose Γ is a composed Carleson curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Given t ∈ Γ, let E(t) := diag (ε_{1}(t)I, …, ε_{N(t)}(t)I). For x ∈ C, define the map

Sym_{t,x} : {S_{Γ}} ∪ {aI : a ∈ P C^{n}(Γ)} → C^{N(t)n×N(t)n}

by

Sym_{t,x}(aI) := diag (a_{1}(t), …, a_{N(t)}(t))

for all x and by

Sym_{t,x}(S_{Γ}) := E(t) ( \begin{matrix}(2x − 1)I & (2x − 2)I & (2x − 2)I & … & (2x − 2)I \\ 2xI & (2x − 1)I & (2x − 2)I & … & (2x − 2)I \\ 2xI & 2xI & (2x − 1)I & … & (2x − 2)I \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 2xI & 2xI & 2xI & … & (2x − 1)I \end{matrix} )

if x ∉ {0, 1} and

Sym_{t,x}(S_{Γ}) := (2x − 1) diag (ε_{1}(t)I, …, ε_{N(t)}(t)I)

if x ∈ {0, 1}.

(a) For each x ∈ L_{t}(Γ), the map Sym_{t,x} extends to a Banach algebra homomorphism

Sym_{t,x} : alg (S_{Γ}, P C^{n}(Γ)) → C^{N(t)n×N(t)n}.

(b) If A ∈ alg (S_{Γ}, P C^{n}(Γ)), then

sp π_{t}(A) = ⋃_{x∈L_{t}(Γ)} sp Sym_{t,x}(A).

(c) An operator A ∈ alg (S_{Γ}, P C^{n}(Γ)) is Fredholm on L^{p,n}(Γ, w) if and only if Sym_{t,x}(A) is an invertible matrix for all (t, x) in the leaf bundle

M := M(Γ, p, w) := ⋃_{t∈Γ} ({t} × L_{t}(Γ)).

Equivalently,

sp_{ess} A = ⋃_{(t,x)∈M} sp Sym_{t,x}(A) = ⋃_{t∈Γ} ⋃_{x∈L_{t}(Γ)} sp Sym_{t,x}(A).

Theorems 4.2 and 11.2 give necessary and sufficient conditions for operators from the algebras alg (S_{Γ}, P C(Γ)) and alg (S_{Γ}, P C^{n}(Γ)) to be Fredholm. The next natural question to ask is when an operator A from these algebras is semi-Fredholm, which means that Im A := A(X) is closed and at least one of the dimensions dim Ker A := dim{x ∈ X : Ax = 0}, dim CokerA := dim X/Im A is finite. Of course, any Fredholm operator is always semi-Fredholm, and the converse is obviously not true. Moreover, there exist singular integral operators on L^{p}(T) which are semi-Fredholm but not Fredholm: for example, if a is an infinite Blaschke product and

P_{Γ} := (I + S_{Γ})/2, Q_{Γ} := (I − S_{Γ})/2,

then P_{Γ} + aQ_{Γ} is surjective with an infinite dimensional kernel while aP_{Γ} + Q_{Γ} is injective and has a closed image of infinite codimension. However, as the following theorem shows, this phenomenon does not occur for operators with piecewise continuous symbols.

Theorem 11.3. All semi-Fredholm operators in alg (S_{Γ}, P C^{n}(Γ)) are Fredholm.

For piecewise Lyapunov curves, this result was established in [35]. There the general case was first reduced to the case of operators of the form aP_{Γ}+Q_{Γ} on Jordan curves Γ, and in the latter case it had already been shown in [58] that semi-Fredholmness implies Fredholmness. Due to the extension results for Carleson curves and Muckenhoupt weights established in Section 6, the first part of the proof remains literally the same. The second part, however, was based on the fact that for each a invertible in P C(Γ) the set of all p ∈ (1, ∞) for which aP_{Γ} +Q_{Γ} is Fredholm on L^{p}(Γ, w) has cluster points at 1 and ∞. Interestingly enough, this property fails for general Carleson Jordan curves.

Theorem 11.4. Let Γ be a Carleson Jordan curve, let t ∈ Γ, and let a be an invertible function in P C(Γ) continuous on Γ \ {t}. Then

(a) aP_{Γ} + Q_{Γ} is Fredholm on L^{p}(Γ) for all but maybe one value of p ∈ (1, ∞) if δ_{t}^{−} = δ_{t}^{+};

(b) aP_{Γ} + Q_{Γ} is not Fredholm on any L^{p}(Γ, w) (1 < p < ∞, w ∈ A_{p}(Γ)) if δ_{t}^{+} ≠ δ_{t}^{−} and |a(t − 0)/a(t + 0)| ≥ e^{4π/(δ_{t}^{+}−δ_{t}^{−})}.

Proof. Taking into account (4.8) and (4.10), we see that for every (τ, ξ) ∈ Γ×L_{τ} (Γ) such that ξ ∉ {0, 1} we have

Sym_{τ,ξ}(aP_{Γ} + Q_{Γ}) = Sym_{τ,ξ}(\frac{a + 1}{2}I + \frac{a − 1}{2}S_{Γ})

= ( \begin{matrix}(a(τ + 0) − 1)ξ + 1 & (a(τ + 0) − 1)(ξ − 1) \\ (1 − a(τ − 0))ξ & (1 − a(τ − 0))ξ + a(τ − 0) \end{matrix} ) .

The determinant of this matrix is a(τ + 0)ξ + a(τ − 0)(1 − ξ). For ξ ∈ {0, 1} we get the same determinant. If τ ∈ Γ \ {t}, then

a(τ + 0)ξ + a(τ − 0)(1 − ξ) = a(τ)ξ + a(τ)(1 − ξ) ≠ 0

because a is invertible. Thus, Theorem 4.2(c) tells us that aP_{Γ} + Q_{Γ} is Fredholm on L^{p}(Γ, w) if and only if

a(t + 0)ξ + a(t − 0)(1 − ξ) ≠ 0 for all ξ ∈ L_{t}(Γ). (11.1)

Combining (4.18), (4.17), and (4.16), we arrive at the conclusion that (11.1) holds if and only if

\frac{a(t + 0)e^{2πz} − a(t − 0)}{e^{2πz} − 1} ≠ 0 for all z ∈ Y (p; α_{t}, β_{t}),

which is equivalent to saying that

(11.2) \frac{1}{2π} log \frac{a(t − 0)}{a(t + 0)} ∉ Y (p; α_{t}, β_{t}) for all values of the log.

We now consider the two cases separately.

(a) Suppose δ_{t}^{−} = δ_{t}^{+} =: δ_{t}. Because L^{p}(Γ) involves no weight, we have μ_{t} = ν_{t} = 0. Therefore α_{t}(x) = β_{t}(x) = δ_{t}x by virtue of (5.4), and consequently, (11.2) assumes the form

\frac{1}{p} ≠ F (\frac{1}{2π} arg \frac{a(t − 0)}{a(t + 0)} − δ_{t} \frac{1}{2π} log |\frac{a(t − 0)}{a(t + 0)}| )

where F (x) stands for the fractional part of x. This gives the assertion.

(b) Now let δ_{t}^{−} ≠ δ_{t}^{+}. Consider the sector

Δ := {z = x + iy ∈ C : 1 + δ_{t}^{−}x ≤ y ≤ δ_{t}^{+}x}.

Theorem 3.2 of [8], which contains more precise information about the indicator functions than (4.13) to (4.15), says that there exist numbers μ_{t}^{−} and ν_{t}^{+} in (−1/p, 1/q) such that if x + iy ∈ Δ, then

\frac{1}{p} + α_{t}(x) ≤ \frac{1}{p} + μ_{t}^{−} + δ_{t}^{−}x < 1 + δ_{t}^{−}x ≤ y ≤ δ_{t}^{+}x < \frac{1}{p} + ν_{t}^{+} + δ_{t}^{+}x ≤ \frac{1}{p} + β_{t}(x).

This shows that Δ ⊂ Y. Put

Δ_{0} := {z = x + iy ∈ Δ : x ≥ 2/(δ_{t}^{+} − δ_{t}^{−})}.

Since Δ_{0} ⊂ Δ, it follows from (11.2) that the operator aP_{Γ} + Q_{Γ} cannot be Fredholm for any p ∈ (1, ∞) and w ∈ A_{p}(Γ) provided that

\frac{1}{2π} log \frac{a(t − 0)}{a(t + 0)} ∈ Δ_{0}.

Separating the real and imaginary part in the latter condition, we may rewrite it as

\frac{1}{2π} log |\frac{a(t − 0)}{a(t + 0)}| = x ≥ 2/(δ_{t}^{+} − δ_{t}^{−}), (11.3)

\frac{1}{2π} arg \frac{a(t − 0)}{a(t + 0)} = y ∈ [1 + δ_{t}^{−}x, δ_{t}^{+}x] + Z. (11.4)

Condition (11.3) is satisfied by assumption. Since the length of the interval [1 + δ_{t}^{−}x, δ_{t}^{+}x] is at least 1 if (11.3) holds, condition (11.4) is also satisfied in the case at hand. □

Because of Theorem 11.4, the proof of Theorem 11.3 proposed in [35] goes through if and only if δ_{t}^{+} = δ_{t}^{−} for all t ∈ Γ, that is, for so-called spiralic curves. The result itself, however, remains valid in the most general setting. As already observed, we need prove it only for operators aP_{Γ} + Q_{Γ} acting on Jordan curves. This is done by the following lemma.

Lemma 11.5. Let Γ be a Carleson Jordan curve, 1 < p < ∞, and w ∈ A_{p}(Γ). Let also a be an invertible function in P C(Γ) such that the operator aP_{Γ} + Q_{Γ} is semi-Fredholm on L^{p}(Γ, w). Then aP_{Γ} + Q_{Γ} is Fredholm on L^{p}(Γ, w).

Proof. Suppose that aP_{Γ} + Q_{Γ} is semi-Fredholm but not Fredholm. Then one of the numbers dim Ker (aP_{Γ} + Q_{Γ}), dim Coker(aP_{Γ} + Q_{Γ}) is infinite. Without loss of generality we may consider the case when dim Ker (aP_{Γ} + Q_{Γ}) = ∞; otherwise, we may pass to the adjoint operator

(aP_{Γ} + Q_{Γ})^{∗} = Q_{Γ}\overline{a} + P_{Γ} = (Q_{Γ} + P_{Γ}\overline{a}^{−1})\overline{a}

acting on L^{q}(Γ, w^{−1}) and may have recourse to the identity

bP_{Γ} + Q_{Γ} = (I − P_{Γ}bQ_{Γ})(Q_{Γ} + P_{Γ}b)(I + Q_{Γ}bP_{Γ}),

in which (I − P_{Γ}bQ_{Γ})^{−1} = I + P_{Γ}bQ_{Γ} and (I + Q_{Γ}bP_{Γ})^{−1} = I − Q_{Γ}bP_{Γ}. Thus, assume aP_{Γ} + Q_{Γ} ∈ Φ^{−}, where Φ^{−} is the set of all semi-Fredholm operators with infinite dimensional kernel. The set Φ^{−} is open, and hence we may suppose that a has only finitely many jumps, say at t_{1}, …, t_{M}. Write a = a_{1} ⋯ a_{M} where a_{j} is continuous on Γ \ {t_{j}}. Since aP_{Γ} + Q_{Γ} = (P_{Γ}aP_{Γ} + Q_{Γ})(I + Q_{Γ}aP_{Γ}) (11.5) and I + Q_{Γ}aP_{Γ} has the inverse I − Q_{Γ}aP_{Γ}, the operator aP_{Γ} + Q_{Γ} is Φ^{−} if and only if so is P_{Γ}aP_{Γ} + Q_{Γ}. With an appropriate continuous partition of the unity, we deduce from Proposition 3.3 that the difference

P_{Γ}aP_{Γ} + Q_{Γ} − (P_{Γ}a_{1}P_{Γ} + Q_{Γ}) ⋯ (P_{Γ}a_{M}P_{Γ} + Q_{Γ})

is compact and that the operators on the right commute modulo compact operators. This implies that all operators P_{Γ}a_{j}P_{Γ} + Q_{Γ} are semi-Fredholm and that at least one of them belongs to Φ^{−}. Again using (11.5), we may assume that at least one of the operators a_{j}P_{Γ} + Q_{Γ} is in Φ^{−}. Considering the latter operator instead of the original aP_{Γ} + Q_{Γ}, we may suppose that a itself has exactly one discontinuity, say at t_{0}. Since a(t_{0} ± 0) ≠ 0, we may find a power function t^{α} continuous on Γ \ {t_{0}} such that f(t) := a(t)/t^{α} is continuous and nonzero on Γ (we here suppose for simplicity that 0 lies in the interior D_{+} of Γ; otherwise we would have to choose a point z_{0} ∈ D_{+} and consider (t − z_{0})^{α} instead of t^{α}). The difference

aP_{Γ} + Q_{Γ} − (t^{α}P_{Γ} + Q_{Γ})(fP_{Γ} + Q_{Γ}) = (t^{α} − 1)Q_{Γ}fP_{Γ}

is compact due to Proposition 3.3, and since fP_{Γ} + Q_{Γ} is Fredholm, it follows that the operator t^{α}P_{Γ} + Q_{Γ} belongs to Φ^{−} simultaneously with aP_{Γ} + Q_{Γ}. In particular, dim Ker (t^{α}P_{Γ} + Q_{Γ}) = ∞, that is, there exist infinitely many linearly independent functions ϕ^{+} ∈ P L^{p}(Γ, w) such that t^{α}ϕ^{+} = ϕ^{−} ∈ QL^{p}(Γ, w). To show that this is not possible, we use the factorization

t^{α} = (t − t_{0})^{α}(1 − t_{0}/t)^{−α}

in which (t − t_{0})^{α} is any branch analytic in D_{+}, and (1 − t_{0}/t)^{−α} is the corresponding branch analytic in the exterior D_{−} of Γ. The equality ϕ^{−} = t^{α}ϕ^{+} can then be rewritten as

(t − t_{0})^{α}ϕ^{+}(t) = (1 − t_{0}/t)^{α}ϕ^{−}(t), t ∈ Γ,

or (t − t_{0})^{α+m}ϕ^{+}(t) = (1 − t_{0}/t)^{α}(t − t_{0})^{m}ϕ^{−}(t), t ∈ Γ (11.6)

for any m ∈ Z. Since Γ is a Carleson curve, we have

arg(t − t_{0})/(− log |t − t_{0}|) = O(1) as t → t_{0}, t ∈ Γ

(see [53]). As in [6, Proof of Lemma 6.1] we conclude from here that

arg(z − t_{0})/(− log |z − t_{0}|) = O(1) as z → t_{0}, z ∈ D_{+},

so that | arg(z − t_{0})| ≤ −M log |z − t_{0}| with a certain constant M < ∞ whenever z ∈ D^{+} is sufficiently close to t_{0}. Thus, for these z we have

|(z − t_{0})^{α+m}| = |z − t_{0}|^{m+Re α}e^{−Im α arg(z−t_{0})} ≤ |z − t_{0}|^{m+Re α−M|Im α|}.

It follows that for every m > M|Im α| − Re α the function (z − t_{0})^{α+m} is analytic and bounded in D_{+}. Similarly, the function (1 − t_{0}/z)^{α}(z − t_{0})^{m} is analytic and bounded on bounded subsets of D_{−} if only m is large enough. At infinity the latter function has a pole of order m. What results is that the function on the left of (11.6) belongs to P_{Γ}L^{p}(Γ, w) together with ϕ^{+} and that, up to a polynomial summand q of degree at most m − 1, the function on the right of (11.6) lies in Q_{Γ}L^{p}(Γ, w) together with ϕ^{−}. Consequently, after subtracting q(t) from both sides of (11.6) we get a function in P_{Γ}L^{p}(Γ, w) ∩ Q_{Γ}L^{p}(Γ, w) = {0}. Therefore both sides of (11.6) are a polynomial of degree at most m − 1, and since

ϕ^{+}(t) = (t − t_{0})^{−α−m}q(t),

we obtain that dim Ker (t^{α}P_{Γ} + Q_{Γ}) ≤ m < ∞. This contradiction completes the proof. □

Theorem 11.3 in its full generality follows from Lemma 11.5 and the reasoning of the paper [35].

12. Index formulas

Once Theorems 4.2 and 11.2 are available, one can establish index formulas for Fredholm operators in alg (S_{Γ}, P C(Γ)) and alg (S_{Γ}, P C^{n}(Γ)) using the method developed by Gohberg and Krupnik for operators on composed Lyapunov curves. This method is now standard, and therefore we restrict ourselves to the main steps and omit the details. For composed Lyapunov curves, index formulas are already in [30, 32, 33] (power weights) and in [35] (general weights). These formulas differ from ours, since they are based on representations like (5.6), (5.7) and Theorems 7.2 and 11.1 were not available at that time.

Let X be a Banach space. An operator A ∈ B(X) is Fredholm if and only if its image Im A := A(X) is closed and both its kernel Ker A := {x ∈ X : Ax = 0} and its cokernel Coker A := X/Im A are finite-dimensional. In that case the index of A is defined as

Ind A := dim Ker A − dim Coker A.

Several properties of the index can be found in [33]. We only remark that the index is stable under small or compact perturbations and that Ind (AB) = Ind A + Ind B whenever A and B are Fredholm.

12.1. Structure of the index formula

Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). We may write Γ = η_{1} ∪ … ∪ η_{m} where η_{1}, …, η_{m} are simple oriented arcs each pair of which have at most endpoints in common. A point t ∈ Γ is called a node if t is endpoint of at least one of the arcs η_{1}, …, η_{m}. The set of all nodes is denoted by T = T (Γ). Note that the partition of Γ into simple oriented arcs is not unique. Hence, neither is the set T. For example, if Γ is a Jordan curve, we may represent it as the union of m simple oriented arcs for every number m ≥ 2. In what follows we always assume that we are given a fixed representation Γ = η_{1} ∪ … ∪ η_{m}. In that case T is uniquely determined. Of course, in practice one strives to keep m minimal.

Denote by Φ(S_{Γ}, P C^{n}(Γ)) the set of all operators in alg (S_{Γ}, P C^{n}(Γ)) which are Fredholm on L^{p,n}(Γ, w). We will construct functions φ_{η_{j}} (j = 1, …, m) and φ_{t} (t ∈ T) of Φ(S_{Γ}, P C^{n}(Γ)) into R such that

Ind A = ∑_{j=1}^{m} φ_{η_{j}}(A) + ∑_{t∈T} φ_{t}(A) (12.1)

for every A ∈ Φ(S_{Γ}, P C^{n}(Γ)). It should be emphasized that in fact φ_{η_{j}} and φ_{t} (and thus also Ind A) depend on Γ, p, w:

φ_{η_{j}} = φ_{η_{j},Γ,p,w}, φ_{t} = φ_{t,Γ,p,w}.

We also remark that φ_{η_{j}}(A) and φ_{t}(A) need not be integers. The sum on the right of (12.1) will howewer always be an integer.

Let first η ∈ {η_{1}, …, η_{m}}, denote the endpoints of η by t_{1} and t_{2}, and put η^{0} := η \ {t_{1}, t_{2}}. For A ∈ Φ(S_{Γ}, P C^{n}(Γ)), define Sym_{t,x}(A) as in Theorem 11.2. If

(t, x) ∈ M(η^{0}, p, w) := ⋃_{t∈η^{0}} ({t} × L_{t}(Γ)),

then Sym_{t,x}(A) is a 2n × 2n matrix and hence

Sym_{t,x}(A) = ( \begin{matrix}A_{11}(t, x) & A_{12}(t, x) \\ A_{21}(t, x) & A_{22}(t, x) \end{matrix} )

with n × n matrices A_{ij}(t, x). The matrices Sym_{t,0}(A) and Sym_{t,1}(A) are blockdiagonal and hence A_{11}(t, 0) and A_{22}(t, 1) are invertible whenever A ∈ Φ(S_{Γ}, P C^{n}(Γ)). For (t, x) ∈ M(η^{0}, p, w), put

(12.2) A(t, x) := \frac{det Sym_{t,x}(A)}{det (A_{11}(t, 0)A_{22}(t, 1))}

and A_{0}(t) := A(t, 0). The function A_{0} is an invertible function in P C(η). Since Sym_{t,0}(A) and Sym_{t,1}(A) are always block-diagonal, we conclude that

(12.3) A(t, 0) = \frac{det A_{22}(t, 0)}{det A_{22}(t, 1)}, A(t, 1) = \frac{det A_{11}(t, 1)}{det A_{11}(t, 0)}

and that the maps

\begin{matrix}A ↦ A_{22}(t, 0), & A ↦ A_{11}(t, 1), \\ A ↦ A_{22}(t, 1), & A ↦ A_{11}(t, 0) \end{matrix}

are Banach algebra homomorphisms. If A ∈ {S_{Γ}} ∪ {aI : a ∈ P C^{n}(Γ)}, it can be straightforwardly verified that

\begin{matrix}A_{22}(t − 0, 0) = A_{22}(t, 0), & A_{11}(t + 0, 1) = A_{11}(t, 1), \\ A_{22}(t − 0, 1) = A_{22}(t, 1), & A_{11}(t + 0, 0) = A_{11}(t, 0). \end{matrix}

It follows that the latter equalities hold for all A ∈ alg (S_{Γ}, P C^{n}(Γ)) and therefore (12.3) gives

A_{0}(t − 0) = A(t − 0, 0) = A(t, 0), A_{0}(t + 0) = A(t + 0, 0) = A(t, 1) (12.4)

for all t ∈ η^{0} and all A ∈ Φ(S_{Γ}, P C^{n}(Γ)). Denote by Λ[A_{0}] the points of η^{0} at which A_{0} has a jump. Theorem 11.2 implies that if t ∈ Λ[A_{0}], then

A(t, x) ≠ 0 for all x ∈ L_{t}(Γ) = L^{0}(p; α_{t}, β_{t}). (12.5)

Let L_{t}^{#} be any continuous curve joining 0 to 1 and staying within L^{0}(p; α_{t}, β_{t}). For example, we can take L_{t}^{#} = L^{0}(p; ψ_{t}, ψ_{t}) where ψ_{t} : R → R is any continuous function such that α_{t}(x) ≤ ψ_{t}(x) ≤ β_{t}(x) for all x ∈ R. Note that ψ_{t} = (α_{t} + β_{t})/2 is such a function. For each t ∈ Λ[A_{0}] join the endpoints (12.3) of the jump of A_{0} at t by the curve {A(t, x) : x ∈ L_{t}^{#}} and orient this curve from A(t, 0) to A(t, 1). This construction gives us a continuous and oriented curve

A_{η}^{#} := A_{0}(η \ Λ[A_{0}]) ∪ ⋃_{t∈Λ[A_{0}]}{A(t, x) : x ∈ L_{t}^{#}}

joining

A_{0}(t_{1} + 0) := lim_{t→t_{1}} A_{0}(t) and A_{0}(t_{2} − 0) := lim_{t→t_{2}} A_{0}(t)

(see [39] for the continuity of the curve). The origin does not belong to A_{η}^{#}. Let {arg z}_{z∈A_{η}^{#}} denote the increment of any continuous branch of the argument as A_{η}^{#} is traced out from A_{0}(t_{1} + 0) to A_{0}(t_{2} − 0). We finally put

φ_{η}(A) := −\frac{1}{2π} \{arg z\}_{z∈A_{η}^{#}} (12.6)

and have thus defined the functions φ_{η_{j}} in (12.1).

Now let t ∈ T. If A ∈ Φ(S_{Γ}, P C^{n}(Γ)), then the N(t)n × N(t)n matrix Sym_{t,x}(A) is invertible for all x ∈ L_{t}(Γ) = L^{0}(p; α_{t}, β_{t}) by virtue of Theorem 11.2. Define a curve L_{t}^{#} ⊂ L^{0}(p; α_{t}, β_{t}) joining 0 to 1 as in the preceding paragraph. Then

A_{t}^{#} := {det Sym_{t,x}(A) : x ∈ L_{t}^{#}}

is a continuous curve joining det Sym_{t,0}(A) to det Sym_{t,1}(A) which does not pass through the origin (for the continuity of the curve we refer again to [39]). Denote by {arg z}_{z∈A_{t}^{#}} the increment of any continuous argument as A_{t}^{#} is traced out from det Sym_{t,0}(A) to det Sym_{t,1}(A) and define

φ_{t}(A) := −\frac{1}{2π} \{arg z\}_{z∈A_{t}^{#}}. (12.7)

We so also know the functions φ_{t} in (12.1).

Theorem 12.1. Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). If A ∈ alg (S_{Γ}, P C^{n}(Γ)) is Fredholm on L^{p,n}(Γ, w), then the index of A is given by (12.1), (12.6), (12.7).

The (outline of the) proof will be divided into several steps: we first show Theorem 12.1 for Jordan curves, then for flowers and stars, and finally for general composed curves.

12.2. Jordan curves

In the case where Γ is a Jordan curve, the right-hand side of (12.1) may be written in another form. Given A ∈ Φ(S_{Γ}, P C^{n}(Γ)), define A(t, x) for (t, x) ∈ M(Γ, p, w) by (12.2) and put A_{0}(t) := A(t, 0) for t ∈ Γ. Denote by A_{Γ}^{#} the closed, continuous, and naturally oriented curve resulting from the essential range of A_{0} by filling in the curve

{A(t, x) : x ∈ L_{t}^{#}}

between A(t, 0) = A_{0}(t − 0) and A(t, 1) = A_{0}(t + 0) for each t ∈ Λ[A_{0}]. The curve A_{Γ}^{#} does not pass through the origin, and hence its winding number about the origin, wind (A_{Γ}^{#}, 0), is well-defined. Note that both A_{Γ}^{#} and wind (A_{Γ}^{#}, 0) actually depend also on p and w.

Proposition 12.2. Let Γ be a Carleson Jordan curve, p ∈ (1, ∞), w ∈ A_{p}(Γ), and A ∈ Φ(S_{Γ}, P C^{n}(Γ)). If Γ = η_{1} ∪ … ∪ η_{m} is any partition of Γ into oriented simple arcs, then the right-hand side of (12.1) equals −wind (A_{Γ}^{#}, 0).

Proof. For simplicity, choose two different points t_{1} and t_{2} on Γ, let η_{1} be the arc from t_{1} to t_{2}, and denote by η_{2} the arc from t_{2} to t_{1}. Then T = {t_{1}, t_{2}}. We have to show that φ_{η_{1}}(A) + φ_{η_{2}}(A) + φ_{t_{1}}(A) + φ_{t_{2}}(A) = −wind (A_{Γ}^{#}, 0). (12.8)

For t ∈ Γ, put c_{t} := det (A_{11}(t, 0)A_{22}(t, 1)). The curve A_{η_{1}}^{#} joins A_{0}(t_{1} + 0) = A(t_{1}, 1) to A_{0}(t_{2} − 0) = A(t_{2}, 0) and coincides with A_{Γ}^{#} between these points. The curve A_{t_{2}}^{#} joins det Sym_{t_{2},0}(A) = c_{t_{2}}A(t_{2}, 0) to det Sym_{t_{2},1}(A) = c_{t_{2}}A(t_{2}, 1) along

{det Sym_{t_{2},x}(A) : x ∈ L_{t_{2}}^{#}} = {c_{t_{2}}A(t_{2}, x) : x ∈ L_{t_{2}}^{#}},

while A_{Γ}^{#} joins A_{0}(t_{2} −0) = A(t_{2}, 0) to A_{0}(t_{2} +0) = A(t_{2}, 1) along {A(t_{2}, x) : x ∈ L_{t_{2}}^{#}}. Clearly, the increment of arg z as z traces out {c_{t_{2}}A(t_{2}, x) : x ∈ L_{t_{2}}^{#}} is equal to the increment of arg z as moves on {A(t_{2}, x) : x ∈ L_{t_{2}}^{#}}. The arc η_{2} and the point t_{1} may be considered analogously. Consequently, the left-hand side of (12.8) is

−\frac{1}{2π} \{arg z\}_{z∈A_{Γ}^{#}} = −wind (A_{Γ}^{#}, 0).

□

We now consider a special case. Let A = aP_{Γ} + Q_{Γ} with a (scalar-valued) function a ∈ P C(Γ). For x ∉ {0, 1} we have

Sym_{t,x}(A) = ( \begin{matrix}a(t + 0)x + (1 − x) & (1 − a(t + 0))(1 − x) \\ (1 − a(t − 0))x & x + a(t − 0)(1 − x) \end{matrix} ) ,

hence A(t, x) = a(t + 0)x + a(t − 0)(1 − x), and thus A_{Γ}^{#} is the closed, continuous, and naturally oriented curve resulting from the essential range of a by filling in the curve

{a(t + 0)x + a(t − 0)(1 − x) : x ∈ L_{t}^{#}}

between a(t − 0) and a(t + 0) for each t at which a has a jump.

Proposition 12.3. Let Γ be a Carleson Jordan curve, p ∈ (1, ∞), w ∈ A_{p}(Γ), and a ∈ P C(Γ). The operator A := aP_{Γ} + Q_{Γ} is Fredholm on L^{p}(Γ, w) if and only if 0 ∉ A_{Γ}^{#}, in which case (12.9) Ind A = −wind (A_{Γ}^{#}, 0).

Proof. This is Theorem 3.1 of [8] (note that the Fredholm criterion is also an immediate consequence of Theorem 4.2). □

Now let A := aP_{Γ} + Q_{Γ} with a ∈ P C^{n}(Γ). Then

Sym_{t,x}(A) = ( \begin{matrix}a(t + 0)x + (1 − x)I & (I − a(t + 0))(1 − x) \\ (I − a(t − 0))x & Ix + a(t − 0)(1 − x) \end{matrix} ) ,

for x ∉ {0, 1}, and it is easily seen that

A(t, x) = det Sym_{t,x}(A) = det(a(t + 0)x + a(t − 0)(1 − x)).

Consequently, A_{Γ}^{#} is now obtained from the essential range of det a by filling in the curve

{det(a(t + 0)x + a(t − 0)(1 − x)) : x ∈ L_{t}^{#}}

whenever a(t − 0) ≠ a(t + 0).

Proposition 12.4. Let Γ be a Carleson Jordan curve, p ∈ (1, ∞), w ∈ A_{p}(Γ), a ∈ P C^{n}(Γ). The operator A := aP_{Γ}+Q_{Γ} is Fredholm on L^{p,n}(Γ, w) if and only if 0 ∉ A_{Γ}^{#}. In that case (12.9) holds.

Proof. Assume first that the set Λ(a) of the points t ∈ Γ at which a has a jump is finite. Let A be Fredholm. Then 0 ∉ A_{Γ}^{#} by Theorem 11.2. In particular, a ∈ P C^{n}(Γ) is invertible. By a result of [29] (also see [15, p. 171] or [42, p. 199]), we may represented a in the form a = fbg (12.10)

where f and g are invertible matrix functions in C^{n}(Γ) and b is an invertible uppertriangular matrix function in P C^{n}(Γ) whose diagonal entries b_{1}, …, b_{n} are continuous on Γ \ Λ(a). As in the proof of Lemma 11.5 we get

(12.11) P_{Γ}aP_{Γ} + Q_{Γ} = (P_{Γ}fP_{Γ} + Q_{Γ})(P_{Γ}bP_{Γ} + Q_{Γ})(P_{Γ}gP_{Γ} + Q_{Γ}) + K

with K ∈ K(Γ). An invertible matrix function h ∈ C^{n}(Γ) is homotopic to the diagonal matrix diag (det h, 1, …, 1) within the invertible matrix functions of C^{n}(Γ) (see, e.g., [43, Appendix 6 by B. Bojarski]). This implies that the operators P_{Γ}fP_{Γ} + Q_{Γ} and P_{Γ}gP_{Γ} + Q_{Γ} are Fredholm of index −wind (det f, 0) and −wind (det g, 0), respectively. Since P_{Γ}aP_{Γ} + Q_{Γ} is Fredholm together with aP_{Γ} + Q_{Γ} and both operators have the same index (again recall the proof of Lemma 11.5), we see from (12.11) that B := bP_{Γ} + Q_{Γ} is Fredholm and that

(12.12) Ind A = −wind (det f, 0) + Ind B − wind (det g, 0).

Put B_{j} := b_{j}P_{Γ} + Q_{Γ} (j = 1, …, n). Because

det Sym_{t,x}(B) ≠ 0 for all (t, x) ∈ M,

it follows that

det Sym_{t,x}(B_{j}) ≠ 0 for all (t, x) ∈ M and all j.

Thus, the operators B_{j} are Fredholm and

Ind B = ∑_{j=1}^{n} Ind B_{j}.

By Proposition 12.3, Ind B_{j} = −wind ((B_{j})_{Γ}^{#}, 0). Consequently,

(12.13) Ind B = − ∑_{j=1}^{n} wind ((B_{j})_{Γ}^{#}, 0) = −wind (B_{Γ}^{#}, 0).

(the last equality of (12.13) needs a justification we renounce to give here). Combining (12.12) and (12.13) we arrive at (12.9).

For general Λ[a] the assertion follows from the stability of Ind A and −wind (A_{Γ}^{#}, 0) under perturbations by operators with small norm. □

By virtue of Proposition 12.2, the following theorem is Theorem 12.1 for Jordan curves.

Theorem 12.5. Let Γ be a Carleson Jordan curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). If A ∈ Φ(S_{Γ}, P C^{n}(Γ)) then 0 ∉ A_{Γ}^{#} and the index of A on L^{p,n}(Γ, w) is given by (12.9).

Proof. Assume first that A = aP_{Γ} + bQ_{Γ} with a, b ∈ P C^{n}(Γ). Since then

det (A_{11}(t, 0)A_{22}(t, 1)) = det (b(t + 0)b(t − 0)),

we get A(t, x) = C(t, x) where C := cP_{Γ} + Q_{Γ} with c := b^{−1}a. Hence, by Proposition 12.4,

Ind A = Ind C = −wind (C_{Γ}^{#}, 0) = −wind (A_{Γ}^{#}, 0).

Let now

A = ∑_{i=1}^{k} A_{i1} … A_{ir} (12.14)

where A_{ij} = a_{ij}P_{Γ} + b_{ij}Q_{Γ} and a_{ij}, b_{ij} ∈ P C^{n}(Γ). Following [31], we associate an operator Ã = ãP_{Γ} + b̃Q_{Γ} with A such that ã, b̃ ∈ P C^{n(k(r+1)+1)}(Γ) and

(12.15) Ind A = Ind Ã, det Sym_{t,x}(A) = det Sym_{t,x}(Ã),

(12.16) det A_{11}(t, 0) = det Ã_{11}(t, 0), det A_{22}(t, 1) = det Ã_{22}(t, 1).

The operator Ã is called the linear dilation of A, and it will explicitly be given below. The equalities (12.15), (12.16) imply that A(t, x) = Ã(t, x) and A_{Γ}^{#} = Ã_{Γ}^{#}. As Proposition 12.4 proves the index formula for Ã, we therefore get it for A.

Here is the linear dilation of A. Denote by O_{m} and I_{m} the m × m zero and identity matrix, respecively. For l = 1, …, r, let B_{l} be the kn × kn matrix

B_{l} = diag (A_{1l}, A_{2l}, …, A_{kl}),

then define the kn(r + 1) × kn(r + 1) matrix Z by

Z = ( \begin{matrix}I_{kn} & B_{1} & O_{kn} & … & O_{kn} \\ O_{kn} & I_{kn} & B_{2} & … & O_{kn} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ O_{kn} & O_{kn} & O_{kn} & … & B_{r} \\ O_{kn} & O_{kn} & O_{kn} & … & I_{kn} \end{matrix} ) ,

put

X = column(\underbrace{O_{n}, …, O_{n}}_{kr}, \underbrace{−I_{n}, …, −I_{n}}_{k}),
Y = (\underbrace{I_{n}, …, I_{n}}_{k}, \underbrace{O_{n}, …, O_{n}}_{kr}),

and finally, let

Ã = ( \begin{matrix}Z & X \\ Y & O_{n} \end{matrix} ) .

To establish (12.15) and (12.16), put

M_{0} = (\underbrace{I_{n}, …, I_{n}}_{k}),

for l ∈ {1, …, r} let

M_{l} = (A_{11}A_{12} … A_{1l}, A_{21}A_{22} … A_{2l}, …, A_{k1}A_{k2} … A_{kl}),

and define W = (M_{0}, M_{1}, …, M_{r}). It can be verified straightforwardly that

( \begin{matrix}I_{nk(r+1)} & O \\ W & I_{n} \end{matrix} ) ( \begin{matrix}I_{nk(r+1)} & O \\ O & A \end{matrix} ) ( \begin{matrix}Z & X \\ O & I_{n} \end{matrix} ) = ( \begin{matrix}Z & X \\ Y & O_{n} \end{matrix} ) =: Ã. (12.17)

The outer matrices on the left of (12.17) are obviously invertible. This gives (12.15) and (12.16).

To complete the proof, we remark that an operator A ∈ alg (S_{Γ}, P C^{n}(Γ)) can be approximated by operators of the form (12.14) and that Ind A as well as wind (A_{Γ}^{#}, 0) are stable in the operator norm; the stability of wind (A_{Γ}^{#}, 0) is shown in [39]. □

12.3. Flowers

Let F be a Carleson flower comprised of N ≥ 2 counter-clockwise oriented Jordan curves Γ_{1}, …, Γ_{N} and let t be the center of F. On each Γ_{j} choose a point t_{j} ∈ Γ_{j} \{t}, let η_{2j−1}^{∗} ⊂ Γ_{j} be the positively oriented arc from t to t_{j}, and denote by η_{2j}^{∗} ⊂ Γ_{j} the positively oriented arc from t_{j} to t. Let further p ∈ (1, ∞) and w ∈ A_{p}(Γ).

Let Π^{n}(F) stand for the set of all matrix functions a on F which are continuous on F \ {t} and have finite limits

a_{i} := lim_{τ→t,τ∈η_{i}^{∗}} a(τ) (i = 1, …, 2N).

Lemma 12.6. If a, b ∈ Π^{n}(F) and A := aP_{F} + bQ_{F} is Fredholm on L^{p,n}(F, w) then

(12.18) det Sym_{t,x}(A) = det (b_{1}a_{2}b_{3}a_{4} … b_{2N−1}a_{2N}) ×

× det (xb_{1}^{−1}a_{1}a_{2}^{−1}b_{2} … b_{2N−1}^{−1}a_{2N−1}a_{2N}^{−1}b_{2N} + (1 − x)I)

for all x ∈ L_{t}(F).

Proof. Since

Sym_{t,0}(A) = diag (b_{1}, a_{2}, b_{3}, …, a_{2N}), Sym_{t,1}(A) = diag (a_{1}, b_{2}, a_{3}, …, b_{2N}),

and these matrices are invertible in view of Theorem 11.2, the equality (12.18) is trivial for x ∈ {0, 1}. For x ∈ L_{t}(F) \ {0, 1}, the matrix Sym_{t,x}(A) is

( \begin{matrix}a_{1}x + b_{1}(1 − x) & (a_{1} − b_{1})(x − 1) & (a_{1} − b_{1})x & … & (a_{1} − b_{1})(x − 1) \\ (b_{2} − a_{2})x & a_{2}(1 − x) + b_{2}x & (b_{2} − a_{2})x & … & (b_{2} − a_{2})(x − 1) \\ (a_{3} − b_{3})x & (a_{3} − b_{3})x & a_{3}x + b_{3}(1 − x) & … & (a_{3} − b_{3})(x − 1) \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ (b_{2N} − a_{2N})x & (b_{2N} − a_{2N})x & (b_{2N} − a_{2N})x & … & a_{2N}(1 − x) + b_{2N}x \end{matrix} ) .

Put

N_{x} = ( \begin{matrix}I & O & O & … & O & (1 − x)I \\ −I & I & O & … & O & O \\ O & −I & I & … & O & O \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ O & O & O & … & I & O \\ O & O & O & … & −I & xI \end{matrix} ) .

Obviously, det N_{x} = 1. A straightforward calculation yields

Sym_{t,x}(A) · N_{x} = ( \begin{matrix}a_{1} & O & O & … & O & b_{1}(1 − x) \\ −a_{2} & b_{2} & O & … & O & O \\ O & −b_{3} & a_{3} & … & O & O \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ O & O & O & … & a_{2N−1} & O \\ O & O & O & … & −a_{2N} & b_{2N}x \end{matrix} ) , (12.19)

and the determinant of the matrix on the right of (12.19) coincides with the right-hand side of (12.18). □

Proposition 12.7. Let a, b ∈ Π^{n}(F) and suppose A := aP_{F} + bQ_{F} is Fredholm on L^{p,n}(F, w). If F = η_{1} ∪ … ∪ η_{m} is any partition of F into oriented simple arcs, then the right-hand side of (12.1) equals the sum of

− \frac{1}{2π} ∑_{j=1}^{N}\{arg det(b^{−1}a)\}_{Γ_{j}\\{t\}} (12.20)

and

− \frac{1}{2π} \{arg det(xb_{1}^{−1}a_{1}a_{2}^{−1}b_{2} … b_{2N−1}^{−1}a_{2N−1}a_{2N}^{−1}b_{2N} + (1 − x)I)\}_{L_{t}^{#}}. (12.21)

Proof. Since A is Fredholm, det Sym_{τ,x}(A) ≠ 0 for all (τ, x) ∈ M(F, p, w). If τ ∈ Γ \ {t}, then

A(τ, 0) = \frac{det Sym_{τ,0}(A)}{det (A_{11}(τ, 0)A_{22}(τ, 1))} = \frac{det (b(τ)a(τ))}{det (b(τ)b(τ))} = det (b^{−1}(τ)a(τ)).

This shows that the first sum in (12.1) is independent of the partition F = η_{1} ∪…∪η_{m} and always equals (12.20). Taking F = η_{1}^{∗} ∪ … ∪ η_{2N}^{∗} as above, we see that the second sum in (12.1) reduces to

−\frac{1}{2π} \{arg det Sym_{t,x}(A)\}_{L_{t}^{#}},

which, by Lemma 12.6, is just (12.21). □

Put

Π_{+}^{n}(F) := {a ∈ Π^{n}(F) : a_{1} = a_{2}, a_{3} = a_{4}, …, a_{2N−1} = a_{2N}},
Π_{−}^{n}(F) := {a ∈ Π^{n}(F) : a_{2} = a_{3}, a_{4} = a_{5}, …, a_{2N} = a_{1}}.

Of course, Π_{+}^{n}(F) ∪ Π_{−}^{n}(F) = C(F). The following result was established in [35] for nice curves. Note that this result also proves Lemma 7.3.

Lemma 12.8. If a ∈ Π_{+}^{n}(F) then Q_{F} aP_{F} is compact, and if a ∈ Π_{−}^{n}(F) then P_{F} aQ_{F} is compact.

Proof. Let a ∈ Π_{+}^{n}(F). Then a = ∑_{j=1}^{N} a_{2j}χ_{j}^{0} + c where χ_{j}^{0} is the characteristic function of Γ_{j}, c ∈ C(F), and c(t) = 0. Since Q_{F} χ_{j}^{0}P_{F} = 0 due to (7.16), we obtain that Q_{Γ}aP_{F} = Q_{Γ}cP_{F}. As cS_{F} −S_{F} cI is compact for every c ∈ C^{n}(F) and Q_{F} P_{F} = 0 (Theorem 7.1), it follows that Q_{F} cP_{F} = (cS_{F} − S_{F} cI)P_{F} /2 is compact.

In the case where a ∈ Π_{−}^{n}(F) we can proceed as in the proof of Proposition 1.2 of [33, Chapter 10]; notice that in this proof the equality g = g_{1} … g_{N} must be replaced by g = g_{1} … g_{N−1}. □

Lemma 12.9. Formula (12.1) holds if A := aP_{F} + bQ_{Γ} is Fredholm on L^{p,n}(F, w) and a, b ∈ Π_{+}^{n}(F) or a, b ∈ Π_{−}^{n}(F).

Proof. If a, b ∈ Π_{+}^{n}(F) or a, b ∈ Π_{−}^{n}(F), then the determinant in (12.21) is identically 1 and hence, by Proposition 12.7, we are left with verifying that

(12.22) Ind A = −\frac{1}{2π} ∑_{j=1}^{N}\{arg det(b^{−1}a)\}_{Γ_{j}\\{t\}}.

This can be done as in the proof of Theorem 1.1 of [33, Chapter 10] (also see [32]). □

Proposition 12.10. If a, b ∈ Π^{n}(F) and A := aP_{F} + bQ_{F} is Fredholm then the index formula (12.1) is true.

Proof. Due to Proposition 12.7, we have to show that Ind A is given by the sum of (12.20) and (12.21).

Suppose A is Fredholm. Then a and b are invertible matrix functions. Put c := b^{−1}a. In [32, Lemma 4.1] and [35, Lemma 2.4] it is shown that if y ∈ Π^{n}(F) is any invertible matrix function such that y_{1}y_{2}^{−1} … y_{2N−1}y_{2N}^{−1} is similar to c_{1}c_{2}^{−1} … c_{2N−1}c_{2N}^{−1}, then there is a representation c = fyg with invertible matrix functions f ∈ Π_{−}^{n}(F) and g ∈ Π_{+}^{n}(F). Thus, if y ∈ Π^{n}(F) is any invertible matrix function such that

y|Γ \ Γ_{1} = I, y_{2} = I, y_{1} = c_{1}c_{2}^{−1} … c_{2N−1}c_{2N}^{−1}, (12.23)

then c = fyg with invertible f ∈ Π_{−}^{n}(F) and g ∈ Π_{+}^{n}(F). We have

A = b(cP_{F} + Q_{F}) = b(fygP_{F} + Q_{F})
= b[f(yP_{F} + Q_{F})(gP_{F} + f^{−1}Q_{F}) + (fy − f)(Q_{F} gP_{F} − P_{F} f^{−1}Q_{F})].

The operators Q_{F} gP_{F} and P_{F} f^{−1}Q_{F} are compact by Lemma 12.9. Thus, (yP_{F} + Q_{F})(gP_{F} + f^{−1}Q_{F}) is Fredholm together with A and has the same index as A. The operator gP_{F} + f^{−1}Q_{F} is Fredholm, a regularizer being g^{−1}P_{F} + fQ_{F}. It follows that yP_{F} + Q_{F} is Fredholm and that

(12.24) Ind A = Ind (yP_{F} + Q_{F}) + Ind (gP_{F} + f^{−1}Q_{F}).

Since y = I on F \ Γ_{1} and the operator yP_{F} + Q_{F} is Fredholm on L^{p,n}(Γ, w), its restriction yP_{Γ_{1}} + Q_{Γ_{1}} to L^{p,n}(Γ_{1}, w|Γ_{1}) is Fredholm and

Ind (yP_{F} + Q_{F}) = Ind (yP_{Γ_{1}} + Q_{Γ_{1}}).

Proposition 12.4 in conjunction with (12.23) shows that

Ind (yP_{Γ_{1}} + Q_{Γ_{1}}) = −\frac{1}{2π} \{arg det y\}_{Γ_{1}\\{t\}} − \frac{1}{2π} \{arg det(xy_{1} + (1 − x)y_{2})\}_{L_{t}^{#}}

= −\frac{1}{2π} ∑_{j=1}^{N}\{arg det y\}_{Γ_{j}\\{t\}} − \frac{1}{2π} \{arg det(xc_{1}c_{2}^{−1} … c_{2N−1}c_{2N}^{−1} + (1 − x)I)\}_{L_{t}^{#}}.

We now consider the second term on the right of (12.24). By Lemma 12.8,

gP_{F} + f^{−1}Q_{F} = (gP_{F} + Q_{F})(P_{F} + f^{−1}Q_{F}) + K_{1}
= (P_{F} + f^{−1}Q_{F})(gP_{F} + Q_{F}) + K_{2}

with compact operators K_{1}, K_{2}. Thus, since gP_{F} + f^{−1}Q_{F} is Fredholm, so also are gP_{F} + Q_{F} and P_{F} + f^{−1}Q_{F}. Because g ∈ Π_{+}^{n}(F) and f ∈ Π_{−}^{n}(F), we can use Lemma 12.9 and (12.22) to get

Ind (gP_{F} + f^{−1}Q_{F}) = Ind (gP_{F} + Q_{F}) + Ind (P_{F} + f^{−1}Q_{F})
= −\frac{1}{2π} ∑_{j=1}^{N} (\{arg det g\}_{Γ_{j}\\{t\}} + \{arg det f\}_{Γ_{j}\\{t\}}) .

Inserting the expressions for Ind (yP_{Γ_{1}} + Q_{Γ_{1}}) and Ind (gP_{F} + f^{−1}Q_{F}) we have obtained in (12.24), we obtain that Ind A equals

−\frac{1}{2π} ∑_{j=1}^{N}\{arg det(fyg)\}_{Γ_{j}\\{t\}} − \frac{1}{2π} ∑_{j=1}^{N}\{arg det(xc_{1}c_{2}^{−1} … c_{2N−1}c_{2N}^{−1} + (1 − x)I)\}_{L_{t}^{#}}.

Since fyg = c = b^{−1}a, this is the sum of (12.20) and (12.21). □

12.4. General composed curves

Lemma 12.11. Let γ = ⋃_{j=1}^{N} η_{j} (N ≥ 1) be a Carleson star centered at t, let p ∈ (1, ∞) and w ∈ A_{p}(γ). Suppose a, b ∈ P C^{n}(γ) are continuous on γ \ {t} and a(t_{j}) = b(t_{j}) = I at the endpoints t_{j} of η_{j} different from t. If the operator A := aP_{γ} + bQ_{γ} is Fredholm on L^{p,n}(γ, w) then

Ind A = −\frac{1}{2π} ∑_{j=1}^{N}\{arg det(b^{−1}a)\}_{η_{j}} − \frac{1}{2π} \{arg det Sym_{t,x}(A)\}_{L_{t}^{#}}.

P r o o f. By Theorem 6.9, we can extend the Carleson star γ and the weight w ∈ A_{p}(γ) to a Carleson flower F = ⋃_{j=1}^{N} Γ_{j} and a weight W ∈ A_{p}(F). Assume η_{j} ⊂ Γ_{j}. Let

ã(τ) := \{ \begin{matrix}a(τ) & if τ ∈ γ \\ I & if τ ∈ F \ γ \end{matrix} , b̃(τ) := \{ \begin{matrix}b(τ) & if τ ∈ γ \\ I & if τ ∈ F \ γ \end{matrix} .

Then ã and b̃ belong to Π^{n}(F). It is clear that A = aP_{γ} + bQ_{γ} is Fredholm on L^{p,n}(γ, w) if and only if Ã := ãP_{F} + b̃Q_{F} is Fredholm on L^{p,n}(F, W). In that case Ind A = Ind Ã. One can easily verify that

det Sym_{t,x}(Ã) = \{ \begin{matrix}det Sym_{t,x}(A) & if (t, x) ∈ M(γ, p, w) \\ 1 & if (t, x) ∈ M(F, p, W) \ M(γ, p, w) \end{matrix} .

Thus, taking into account that a(t_{j}) = b(t_{j}) = I, we obtain from Proposition 12.10 that

Ind A = Ind Ã = −\frac{1}{2π} ∑_{j=1}^{N}\{arg det(b̃^{−1}ã)\}_{Γ_{j}\\{t\}} − \frac{1}{2π} \{arg det Sym_{t,x}(Ã)\}_{L_{t}^{#}}

= −\frac{1}{2π} ∑_{j=1}^{N}\{arg det(b^{−1}a)\}_{η_{j}} − \frac{1}{2π} \{arg det Sym_{t,x}(A)\}_{L_{t}^{#}}. □

Proof of Theorem 12.1. We follow the proof of Theorem 4.2 of [35]. Suppose first that A := cP_{Γ} + Q_{Γ} with an invertible matrix function c ∈ P C^{n}(Γ).

Let T_{0} = {z_{k} : k = 1, …, r} be a subset of Γ which contains the set T of all nodes and exactly one point t_{j} ∈ η_{j} \ T for each j = 1, …, m. For z_{k} ∈ T_{0}, choose open neighborhoods γ_{k} and γ′_{k} on Γ such that \overline{γ′_{k}} ⊂ γ_{k}, γ_{k} is a star with the center z_{k}, γ_{k} ∩ (T_{0} \ {z_{k}}) = ∅, and Γ ⊂ γ′_{1} ∪ … ∪ γ′_{r}.

Assume c is continuous on Γ \ T. Let c^{(1)} be any invertible matrix function in P C^{n}(Γ) which equals c on γ′_{1}, is I on Γ\γ_{1}, and is continuous on the closure of γ_{1} \γ′_{1}. Of course, c^{(1)} is continuous on Γ \ {z_{1}}, its limits c_{i}^{(1)}(z_{1}) at z_{1} coincide with the limits c_{i}(z_{1}), and the limits of the matrix function (c^{(1)})^{−1} c at the points z_{2}, …, z_{r} are equal to the corresponding limits of c. For points in γ′_{1}, the latter limits are I. Repeating the construction with the point z_{2} and the matrix (c^{(1)})^{−1} c in place of c and then with the points z_{3}, …, z_{r}, we arrive at a representation c = c^{(1)}c^{(2)} … c^{(r)} in which c^{(k)} is continuous and invertible on Γ \ {z_{k}},

c^{(k)}|Γ \ γ_{k} = I, (12.25)

and c^{(k)}|γ′_{k} = c|γ′_{k}.

Put A_{k} := c^{(k)}P_{Γ} + Q_{Γ}. Since the matrices c^{(1)}, c^{(2)}, …, c^{(r)} do not have common points of discontinuity, the Fredholmness of A implies that the operators A_{1}, A_{2}, …, A_{r} are Fredholm and that

Ind A = Ind A_{1} + … + Ind A_{r}.

From (12.25) we deduce that A′_{k} := A_{k}|L^{p,n}(γ_{k}, w|γ_{k}) is Fredholm and that Ind A_{k} = Ind A′_{k} for each k. Lemma 12.11 is applicable to the operators A′_{k} and gives

(12.26) Ind A′_{k} = −\frac{1}{2π} ∑_{j=1}^{N(z_{k})}\{arg det c^{(k)}\}_{η_{j}} − \frac{1}{2π} \{arg det Sym_{z_{k},x}(A′_{k})\}_{L_{z_{k}}^{#}} ,

where N(z_{k}) is the number of simple arcs composing the star γ_{k}. In (12.26) we may delete the primes and replace N(z_{k}) by m. Adding the resulting equalities we get

Ind A = −\frac{1}{2π} ∑_{j=1}^{m}\{arg det c\}_{η_{j}} − \frac{1}{2π} ∑_{k=1}^{r}\{arg det Sym_{z_{k},x}(A_{k})\}_{L_{z_{k}}^{#}} .

Since Sym_{z_{k},x}(A_{k}) = Sym_{z_{k},x}(A) for all x ∈ L_{z_{k}}(Γ) and c is continuous at the points of T_{0} \ T, we therefore have

Ind A = −\frac{1}{2π} ∑_{j=1}^{m}\{arg det c\}_{η_{j}} − \frac{1}{2π} ∑_{t∈T}\{arg det Sym_{t,x}(A)\}_{L_{t}^{#}}.

Equality (12.1) now follows from the observation that

{arg det c}_{η_{j}} = {arg z}_{z∈A_{η_{j}}^{#}} .

This completes the proof for operators A = cP_{Γ} + Q_{Γ} whose coefficient is continuous on Γ \ T.

If c has a finite number Λ′_{c} of jumps at points of Γ \ T, we include Λ′_{c} into the set of nodes and may proceed as above to get the index formula. The case of infinitely many jumps can be disposed of by the usual approximation and stability argument.

The case where A = aP_{Γ} + bQ_{Γ} with invertible matrix function a, b ∈ P C^{n}(Γ) may be reduced to the above case by writing A = b(b^{−1}aP_{Γ} + Q_{Γ}). If A is of the form (12.14), one can construct a linear dilation Ã = ãP_{Γ} + b̃Q_{Γ} which leaves both sides of (12.1) invariant (recall the proof of Theorem 12.5). Finally, taking into account that (12.1) is stable under small perturbations (see [39] for the case of Jordan curves), one gets the assertion for general A ∈ alg (S_{Γ}, P C^{n}(Γ)). □

13. Coburn’s lemma

The following result is known as Coburn’s lemma. It is in Coburn’s paper [16] for L^{p}(Γ, w) = L^{2}(T). For piecewise Lyapunov Jordan curves Γ, it was established by Simonenko [55, 57] (who did not know of the Coburn paper), and for composed Lyapunov curves Γ one can find the theorem in Gohberg and Krupnik’s book [33, Chapter 7.5]. Gohberg and Krupnik also conjectured that the result is true in the general case.

Theorem 13.1. Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). Suppose a and b are functions in L^{∞}(Γ). Then the operator aP_{Γ} + bQ_{Γ} is invertible on L^{p}(Γ, w) if and only if it is Fredholm of index zero.

This theorem divides the problem of deciding whether an operator of the form aP_{Γ} + bQ_{Γ} is invertible into two ”halves”: into finding out whether it is Fredholm and into computing its index. Unfortunately, a theorem like Theorem 13.1 is not true for operators with matrix coefficients nor for generic operators in the algebra alg (S_{Γ}, P C(Γ)).

Theorem 13.1 can be proved by the arguments of [33, Chapter 7.5] provided Theorem 13.3 below is at our disposal. To prove this theorem, we need one more result on the extension of Muckenhoupt weights.

Theorem 13.2. Let γ = η_{1} ∪ … ∪ η_{N} be a Carleson star and let F = Γ_{1} ∪ … ∪ Γ_{N} be any Carleson flower such that the arc η_{k} is contained in the Jordan curve Γ_{k} (k = 1, …, N). If w ∈ A_{p}(γ), then there exists a weight W ∈ A_{p}(F) such that W |γ = w.

Thus, while in Theorem 6.9 we extend γ and w to F and W, we now are given F and have to extend w from γ to F. Obviously, Theorem 6.2 and Theorem 13.2 imply Theorem 6.9. Since Theorem 6.9 is just the result we needed to prove the theorems of Section 4, we accompanied it with a detailed proof. As for the proof of Theorem 13.2, we confine ourselves to sketching the basic idea.

Proof. (outline) Using the arguments of Section 6, all we need to do is to show that F \ γ can be divided into pieces Γ_{k} and that to each Γ_{k} we can associate a set A_{k} ⊂ γ so that the sets A_{k} have bounded overlaps, diam (Γ_{k}) ≃ dist (Γ_{k}, γ), |A_{k}| ≥ C_{1}diam (Γ_{k}), diam (A_{k}) ≤ C_{2}diam (Γ_{k}), dist (A_{k}, F \ γ) ≤ C_{3}diam (Γ_{k}),

where C_{1}, C_{2}, C_{3} are finite positive constants independent of k. Given this, the definition of w̃ and Hölder’s inequality argument go through exactly as in Section 6.

Put Ω := C \ γ and let {Q_{j}} be a Whitney decomposition of Ω, i.e. suppose each Q_{j} is a square, Q_{j} and Q_{k} have at most pieces of edges in common if j ≠ k, Ω = ⋃_{j} Q_{j}, and diam (Q_{j}) ≃ dist (Q_{j}, ∂Ω). Let λQ_{j} denote the concentric square with diam (λQ_{j}) = λ diam (Q_{j}).

Now consider just the Whitney squares Q_{j} which hit F \γ. Let Γ_{k} be any measurable pairwise disjoint subsets of F \ γ such that

F \ γ = ⋃_{k} Γ_{k} and Γ_{k} ⊂ (F \ γ) ∩ \frac{4}{3}Q_{k}.

Given Γ_{k}, denote by C_{k} the Whitney squares Q_{j} ≠ Q_{k} such that diam (Q_{j}) ≤ diam (Q_{k}) and 10Q_{j} ∩ 10Q_{k} ≠ ∅. Define

n(x) := ∑_{Q_{j}∈C_{k}} χ_{γ∩10Q_{j}}(x) and A_{k} := {x ∈ γ ∩ 10Q_{j} : n(x) < λ}.

One can show that if λ > 0 is large enough, then the sets Γ_{k} and A_{k} we have constructed possess all the desired properties. □

Given a Jordan curve Γ, denote by D(Γ) the bounded connected component of C\Γ and let \overline{D(Γ)} := D(Γ) ∪ Γ. Recall that we always assume that a Jordan curve is oriented counter-clockwise. We call a composed curve G a loop curve if it may be represented in the form G = Γ_{1} ∪ … ∪ Γ_{m} where Γ_{1}, …, Γ_{m} are Jordan curves such that \overline{D(Γ_{j})} and \overline{D(Γ_{k})} have at most finitely many points in common whenever j ≠ k. Obviously, flowers are examples of loop curves. Here is a generalizations of Theorem 6.9.

Theorem 13.3. Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). There exists a Carleson loop curve G and a weight W ∈ A_{p}(G) such that Γ ⊂ G and W |Γ = w.

Proof. Write Γ = η_{1} ∪ … ∪ η_{m} with simple oriented arcs having at most endpoints in common, put E := {η_{1}, …, η_{m}}, and denote by T the set of nodes of Γ (recall the first paragraph of Section 12.1). Let Ω be an arbitrary connected component of C \ Γ. We say that an arc η_{j} ∈ E as of type Ω_{+} if η_{j} ⊂ ∂Ω and there exists an arc η′_{j} ⊂ Ω such that η_{j} ∪ η′_{j} is a Jordan curve. Otherwise we call η_{j} an arc of type Ω_{−}. Note that clearly every η_{j} may be extended to a (counter-clockwise oriented!) Jordan curve η_{j} ∩ η′_{j}, but that it is precisely the arcs of type Ω_{+} for which this is possible with η′_{j} ⊂ Ω. Since Ω is a simply connected domain in C ∪ {∞}, we can use Theorem 6.1 and proceed as in the proof of Theorem 6.2 to extend each arc of type Ω_{+} to Carleson Jordan curve η_{j} ∪ η′_{j} such that η′_{j} ⊂ Ω and the closures of D(η_{i} ∪ η′_{i}) and D(η_{l} ∪ η′_{l}) have at most finitely many points in common if i ≠ l.

Each arc η_{j} ∈ E is of type Ω_{+} for exactly one connected component of C \ Γ. Thus, doing the above construction for each connected component of C \ Γ we arrive at a Carleson loop curve G = (η_{1} ∪ η′_{1}) ∪ … ∪ (η_{m} ∪ η′_{m}) containing Γ.

Let η_{j} ∈ E and denote by t_{1} and t_{2} the endpoints of η_{j}. We divide the associated arc η′_{j} into three disjoint subarcs

η′_{j} = η′_{j}(t_{1}) ∪ μ′_{j} ∪ η′_{j}(t_{2})

so that t_{1} ∈ η′_{j}(t_{1}) and t_{2} ∈ η′_{j}(t_{2}).

We now extend w ∈ A_{p}(Γ) to a weight W ∈ A_{p}(G). Let t ∈ T. Denote by η_{i_{1}}, …, η_{i_{n}} the arcs in E having the endpoint t. Then γ_{t} := η_{i_{1}} ∪ … ∪ η_{i_{n}} is a Carleson star and

F_{t} := (η_{i_{1}} ∪ η′_{i_{1}}) ∪ … ∪ (η_{i_{n}} ∪ η′_{i_{n}})

is a Carleson flower containing γ_{t}. By Theorem 13.2, we can extend w|γ_{t} to a weight W_{t} ∈ A_{p}(F_{t}) such that W_{t}|γ_{t} = w|γ_{t}. Denote by W′_{t} the restriction of W_{t} to

γ_{t} ∪ η′_{i_{1}}(t) ∪ … ∪ η′_{i_{n}}(t).

Repeating this construction for each node t ∈ T, we get a weight W′ on

G′ := Γ ∪ ⋃_{j=1}^{m}(η′_{j} \ μ′_{j})

such that W′|Γ = w and W′ ∈ A_{p}(G′).

It remains to define W on the middle pieces μ′_{j} of the arcs η′_{j} we have added. If η_{j} has the endpoints t_{1} and t_{2}, then γ_{j} := η′_{j}(t_{1}) ∪ η_{j} ∪ η′_{j}(t_{2}) is a simple arc (and thus a star) and η_{j} ∪ η′_{j} = γ_{j} ∪ μ′_{j} is a Jordan curve (and thus a flower) extending γ_{j}. By virtue of Theorem 13.2, the weight W′|γ_{j} can be extended to a weight W_{j} on γ_{j} ∪ μ′_{j} such that W_{j} ∈ A_{p}(γ_{j} ∪ μ′_{j}). Clearly, the weight W on G defined by W |η_{j} ∪ η′_{j} = W_{j} extends w and belongs to A_{p}(G). □

As already said, once Theorem 13.3 is available, Theorem 13.1 can be proved by the methods developed in [33] for the case of composed Lyapunov curve. Given A := aP_{Γ} + bQ_{Γ}, one first extends Γ and w to a Carleson loop curve G and a weight W ∈ A_{p}(G) and then one considers Ã := ãP_{G} + b̃Q_{G} ∈ B(L^{p}(G, W)) with

ã := aχ_{Γ} + χ_{G\Γ}, b̃:= bχ_{Γ} + χ_{G\Γ}.

One can easily see that

dim Ker A = dim Ker Ã and dim Ker A^{∗} = dim Ker Ã^{∗},

where the adjoint operators act on L^{q}(Γ, w^{−1}) and L^{q}(G, W^{−1}) (1/p + 1/q = 1). This reduces the proof of Theorem 13.1 to the proof of this theorem for the case where Γ = G is a loop curve. Proceeding as in the proof of Theorem 7.1 one obtains that if G is a Carleson loop curve, then S_{G}^{2} = I and thus P_{G}^{2} = P_{G}. Taking into account that

Ã^{∗} = H_{G}(Q_{G}ã + P_{G}b̃)H_{G} on L^{q}(G, W^{−1}),

where (H_{G}f)(t) := e^{−iΘ_{G}(t)}\overline{f(t)} and Θ_{G}(t) is the angle between the (almost everywhere existing) tangent to G at t and the positively oriented real axis, and using the Lusin-Privalov theorem, one finally gets the following theorem.

Theorem 13.4. Let Γ be a composed Carleson curve, p ∈ (1, ∞), w ∈ A_{p}(Γ). Let further a and b be functions in L^{∞}(Γ), consider the operator A := aP_{Γ} + bQ_{Γ} on L^{p}(Γ, w) and its adjoint operator A^{∗} = H_{Γ}(Q_{Γ}a + P_{Γ}b)H_{Γ} on L^{q}(Γ, w^{−1}). If neither a nor b vanishes identically and

|{t ∈ Γ : a(t) = b(t) = 0}| = 0,

then dim Ker A = 0 or dim Ker A^{∗} = 0.

If a, b ∈ L^{∞}(Γ) and aP_{Γ} + bQ_{Γ} is Fredholm, then a and b can be shown to be invertible (this is the Hartman-Winter-Simonenko theorem). Thus, Theorem 13.4 implies Theorem 13.1.

14. Spectrum of the Cauchy singular integral operator

Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). The essential spectrum of S_{Γ} on L^{p}(Γ, w) is given by Theorem 4.4. Since S_{Γ} = P_{Γ} − Q_{Γ}, we may invoke Theorems 12.1 and 13.1 to determine the spectrum of S_{Γ} on L^{p}(Γ, w):

sp S_{Γ} = sp_{ess} S_{Γ} ∪ {λ ∈ C \ sp_{ess} S_{Γ} : Ind (S_{Γ} − λI) ≠ 0}.

So let us compute the index of S_{Γ} − λI.

An indicator leaf L^{0}(p; α, β) is always a connected compact set containing the points 0 and 1. Given points z, w ∈ C, we put

L^{0}(z, w; p; α, β) := {(1 − x)z + xw : x ∈ L^{0}(p; α, β)}. (14.1)

For t ∈ Γ, let L_{t}^{#} ⊂ L_{t}(Γ) = L^{0}(p; α_{t}, β_{t}) be a curve as in Section 12.1. For z and w in C, we define

L_{t}^{#}(z, w) := {(1 − x)z + xw : x ∈ L_{t}^{#}},

and we think of L_{t}^{#}(z, w) as being oriented from z to w.

Write Γ = η_{1} ∪ … ∪ η_{m} as in Section 12.1 and denote by T = {t_{1}, …, t_{n}} the set of nodes. Let ε_{j} be the valency of t_{j}, put σ_{0} = 0 and σ_{j} = ε_{1} +…+ε_{j} for j ∈ {1, …, n}. The points −1 and 1 always belong to sp_{ess} S_{Γ}. So let λ ∈ C \ {−1, 1} and put μ := (λ − 1)/(λ + 1). Consider the set

Φ_{Γ}^{#}(λ) := Φ_{Γ,w,λ}^{#}(λ) := ⋃_{j=1}^{n} L^{0}(μ^{σ_{j−1}}, μ^{σ_{j}}; p; α_{t_{j}}, β_{t_{j}})

and the curve

C_{Γ}^{#}(λ) := C_{Γ,p,w}^{#}(λ) := ⋃_{j=1}^{n} L_{t_{j}}^{#}(μ^{σ_{j−1}}, μ^{σ_{j}}).

Obviously, the curve C_{Γ}^{#}(λ) is entirely contained in the “festoon” Φ_{Γ}^{#}(λ). The curve C_{Γ}^{#}(λ) is a continuous oriented curve joining 1 = μ^{0} via μ^{σ_{1}}, μ^{σ_{2}}, …, μ^{σ_{n−1}} to μ^{σ_{n}}. Since σ_{n} = ε_{1} + … + ε_{n} = 0, it follows that C_{Γ}^{#}(λ) is a closed curve starting and terminating at the point 1.

Theorem 14.1. Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). A point λ ∈ C \ {−1, 1} lies in the complement of the essential spectrum of S_{Γ} on L^{p}(Γ, w) if and only if Φ_{Γ}^{#}(λ) does not contain the origin. In that case

(14.2) Ind (S_{Γ} − λI) = −wind (C_{Γ}^{#}(λ), 0).

Proof. Let first t ∈ Γ \ T. Then, by Theorem 4.2,

Sym_{t,x}(S_{Γ} − λI) = ( \begin{matrix}2x − 1 − λ & 2x − 2 \\ −2x & 1 − 2x − λ \end{matrix} )

and the determinant of this matrix is λ^{2} − 1. Thus, Sym_{t,x}(S_{Γ} − λI) is invertible for all x ∈ L_{t}(Γ) and from (12.2) we get

(S_{Γ} − λI)(t, x) = \frac{λ^{2} − 1}{(−1 − λ)(−1 − λ)} = \frac{λ − 1}{λ + 1} =: μ,

whence (S_{Γ} − λI)_{0}(t) = μ for all t ∈ Γ \ T. This shows that the first sum in (12.1) is zero.

Now pick t_{j} ∈ T. From the proof of Theorem 4.4 we know that det Sym_{t_{j},x}(S_{Γ} −λI) equals

\begin{matrix}(λ^{2} − 1)^{(N_{j}−ε_{j})/2}(x(1 − λ)^{ε_{j}} + (1 − x)(−1 − λ)^{ε_{j}}) & if ε_{j} > 0, \\ (λ^{2} − 1)^{(N_{j}+ε_{j})/2}((1 − x)(1 − λ)^{−ε_{j}} + x(−1 − λ)^{−ε_{j}}) & if ε_{j} < 0, \\ (λ^{2} − 1)^{N_{j}/2} & if ε_{j} = 0. \end{matrix}

Thus, if ε_{j} > 0, letting Δ_{j}(λ) := (λ^{2} − 1)^{(N_{j}−ε_{j})/2} we see that (S_{Γ} − λI)_{t_{j}}^{#} is equal to

{det Sym_{t_{j},x}(S_{Γ} − λI) : x ∈ L_{t_{j}}^{#}} (14.3)

= Δ_{j}(λ) {x(1 − λ)^{ε_{j}} + (1 − x)(−1 − λ)^{ε_{j}} : x ∈ L_{t_{j}}^{#}}
= Δ_{j}(λ)(−1 − λ)^{ε_{j}} {xμ^{ε_{j}} + (1 − x) · 1 : x ∈ L_{t_{j}}^{#}}
= Δ_{j}(λ)(−1 − λ)^{ε_{j}}μ^{−σ_{j−1}} {xμ^{σ_{j}} + (1 − x)μ^{σ_{j−1}} : x ∈ L_{t_{j}}^{#}}
=: Δ_{j}(λ)(−1 − λ)^{ε_{j}}μ^{−σ_{j−1}}L_{t_{j}}^{#}(μ^{σ_{j−1}}, μ^{σ_{j}}).

Analogously, for ε_{j} > 0 we deduce from Theorem 4.2(b) that

λ ∉ sp π_{t_{j}}(S_{Γ}) ⟺ 0 ∉ Δ_{j}(λ)(−1 − λ)^{ε_{j}}μ^{−σ_{j−1}}L^{0}(μ^{σ_{j−1}}, μ^{σ_{j}}; p; α_{t_{j}}, β_{t_{j}}). (14.4)

Hence, if λ ∉ sp_{ess} S_{Γ} then (14.3) gives

− \frac{1}{2π} \{arg z\}_{z∈(S_{Γ}−λI)_{t_{j}}^{#}} = −\frac{1}{2π} \{arg z\}_{z∈L_{t_{j}}^{#} (μ^{σ_{j−1}} ,μ^{σ_{j}})}. (14.5)

Both (14.4) and (14.5) also hold for ε_{j} ≤ 0. From (14.4) we deduce that S_{Γ} − λI is Fredholm if and only if 0 ∉ Φ_{Γ}^{#}(λ), and from (14.5) and (12.1), (12.7) we obtain

Ind (S_{Γ} − λI) = −\frac{1}{2π} ∑_{j=1}^{n}\{arg z\}_{z∈L_{t_{j}}^{#} (μ^{σ_{j−1}} ,μ^{σ_{j}})} = −wind (C_{Γ}^{#}(λ), 0).

□

The following theorem provides another index formula. For a node t ∈ T, let L_{t}(Γ) = L^{0}(p; α_{t}, β_{t}) be as above and for k = 0, 1, …, n − 1, define

L_{t,k,n}^{0} := L_{k,n}^{0} := L^{0} (p; \frac{1}{n}(\frac{1}{p} − \frac{n}{p} + k + T_{n}α_{t}), \frac{1}{n}(\frac{1}{p} − \frac{n}{p} + k + T_{n}β_{t})) (14.6)

as in Theorem 4.4. Given z, w ∈ C, we put

L_{t,k,n}^{#}(z, w) := {(1 − x)z + xw : x ∈ L_{t,k,n}^{#}} ,

where L_{t,k,n}^{#} ⊂ L_{t,k,n}^{0} is a curve as in Section 12.1, and we think of L_{t,k,n}^{#} as being oriented from z to w.

Theorem 14.2. Let Γ be a composed Carleson curve, p ∈ (1, ∞), and w ∈ A_{p}(Γ). Denote by T_{∗} the set of all nodes t ∈ T with valency ε(t) ≠ 0. Then

C_{Γ}^{#} := C_{Γ,p,w}^{#} := ⋃_{t∈T_{∗}} ⋃_{k=0}^{|ε(t)|−1} L_{t,k,|ε(t)|}^{#}(−sign ε(t), sign ε(t))

is a closed, continuous, and oriented curve which is contained in sp_{ess} S_{Γ}. If λ ∈ C \ sp_{ess} S_{Γ}, then (14.7) Ind (S_{Γ} − λI) = −wind (C_{Γ}^{#}, λ).

Remark. Since ∑_{t∈T_{∗}} ε(t) = 0, the number of arcs of C_{Γ}^{#} joining −1 to 1 is equal to the number of arcs of C_{Γ}^{#} going from 1 to −1. Thus, the arcs of C_{Γ}^{#} may be traced out in a succession so that C_{Γ}^{#} becomes a closed, continuous, and oriented curve.

Proof. It is immediate from Theorem 4.4 that C_{Γ}^{#} ⊂ sp_{ess} S_{Γ}. Let λ ∈ C \ sp_{ess} S_{Γ}. Then the equality (14.2) of Theorem 14.1 holds. Fix a node t_{j} ∈ T. We have σ_{j} = σ_{j−1} + ε_{j}, where ε_{j} is the valency of t_{j}. If ε_{j} = 0, then L_{t_{j}}^{#}(μ^{σ_{j−1}}, μ^{σ_{j}}) is a point and does not contribute to the winding number. We therefore need only consider nodes t_{j} ∈ T_{∗}.

Suppose ε_{j} > 0. We have

L_{t_{j}}^{#}(μ^{σ_{j−1}}, μ^{σ_{j}}) = μ^{σ_{j}}L_{t_{j}}^{#}(μ^{−ε_{j}}, 1) = μ^{σ_{j}}L_{t_{j}}^{#} ((\frac{λ + 1}{λ − 1})^{ε_{j}}, 1) ,

and hence, the increment of arg z along L_{t_{j}}^{#}(μ^{σ_{j−1}}, μ^{σ_{j}}) is equal to the increment of arg z along

L_{t_{j}}^{#} ((\frac{λ + 1}{λ − 1})^{ε_{j}}, 1) := \{(\frac{λ + 1}{λ − 1})^{ε_{j}}(1 − x) + x : x ∈ L_{t_{j}}^{#}\} , (14.8)

where

L_{t_{j}}^{#} := L^{0}(p; ψ_{t_{j}}, ψ_{t_{j}}) := \{\frac{e^{2πγ}}{e^{2πγ} − 1} : γ ∈ Y (p; ψ_{t_{j}}, ψ_{t_{j}})\} ∪ \{0, 1\}.

If ε is a positive integer, γ ∈ C, and x/(x − 1) = e^{2πγ}, then

(\frac{λ + 1}{λ − 1})^{ε} (1 − x) + x = (1 − x) ∏_{k=0}^{ε−1} (\frac{λ + 1}{λ − 1} − e^{2πki/ε} (\frac{x}{x − 1})^{1/ε}) (14.9)

= \frac{1}{1 − e^{2πγ}} · ∏_{k=0}^{ε−1} (\frac{λ + 1}{λ − 1} − e^{2π(γ+ki)/ε})

= (∏_{k=0}^{ε−1} \frac{1}{1 − e^{2π(γ+ki)/ε}} ) · ∏_{k=0}^{ε−1} (\frac{λ + 1}{λ − 1} − e^{2π(γ+ki)/ε})

= ∏_{k=0}^{ε−1} (\frac{λ + 1}{λ − 1} · \frac{1}{1 − e^{2π(γ+ki)/ε}} + \frac{e^{2π(γ+ki)/ε}}{e^{2π(γ+ki)/ε} − 1}) .

If γ traces out Y (p; ψ_{t}, ψ_{t}), then (γ + ki)/ε traverses

Y (p; \frac{1}{ε}(\frac{1}{p} − \frac{ε}{p} + k + T_{ε}ψ_{t}), \frac{1}{ε}(\frac{1}{p} − \frac{ε}{p} + k + T_{ε}ψ_{t}))

(recall the proof of Theorem 4.4), and therefore

w := \frac{e^{2π(γ+ki)/ε}}{e^{2π(γ+ki)/ε} − 1}

describes the curve L_{t,k,n}^{#}. Since the kth factor of the product (14.9) is \frac{λ+1}{λ−1}(1−w)+w, we obtain from (14.8) and (14.9) that the increment of arg z along L_{t_{j}}^{#}(μ^{σ_{j−1}}, μ^{σ_{j}}) is equal to the sum of the increments of arg z along the arcs

L_{t_{j},k,ε_{j}}^{#} (\frac{λ + 1}{λ − 1}, 1) := \{\frac{λ + 1}{λ − 1}(1 − w) + w : w ∈ L_{t_{j},k,ε_{j}}^{#}\} , (14.10)

k ranging over {0, 1, …, ε_{j} − 1}. Because (14.10) is

\frac{1}{1 − λ} \{(−1 − λ)(1 − w) + (1 − λ)w : w ∈ L_{t_{j},k,ε_{j}}^{#}\} ,

the increment of arg z along the arc (14.10) coincides with the increment of arg z along L_{t_{j},k,ε_{j}}^{#}(−1 − λ, 1 − λ) and therefore with the increment of arg(z − λ) as z traces out

L_{t_{j},k,ε_{j}}^{#}(−1, 1) = L_{t_{j},k,ε_{j}}^{#}(−sign ε_{j}, sign ε_{j}).

The case ε_{j} < 0 may be disposed of analogously. □

Thus, in order to decide whether a given connected component Ω of C \ sp_{ess} S_{Γ} is contained in sp S_{Γ}, we need only pick a particular λ_{0} ∈ Ω and apply Theorems 14.1 or 14.2 to compute Ind (S_{Γ} − λ_{0}I). The advantage of formula (14.2) is that it is in terms of the original leaves L_{t}, while formula (14.7) involves the ”root leaves” L_{t,k,|ε(t)|}^{0} (k = 0, 1, …, |ε(t)| − 1). On the other hand, formula (14.2) is very inconvenient in order to find the spectrum of S_{Γ}. In contrast to this, formula (14.7) enables us to see the spectrum of S_{Γ} at a glance once sp_{ess} S_{Γ} has been plotted.

To illustrate Theorems 14.1 and 14.2, suppose Γ = η_{1}∪η_{2} where η_{1} and η_{2} are simple arcs with exactly one common endpoint and with no other common points. Denote the common endpoint by t_{0}, and let t_{1} and t_{2} be the other endpoints of η_{1} and η_{2}. Suppose η_{1} and η_{2} are oriented from t_{0} to t_{1} and t_{2}, respectively. We have ε(t_{0}) = 2 and ε(t_{1}) = ε(t_{2}) = −1. Let α_{t} and β_{t} be any functions such that L_{t}(Γ) = L^{0}(p; α_{t}, β_{t}) and choose continuous functions ψ_{t} such that α_{t} ≤ ψ_{t} ≤ β_{t}. By Theorem 4.4,

sp_{ess} S_{Γ} = L_{t_{1}}^{0}(1, −1) ∪ L_{t_{2}}^{0}(1, −1) ∪ L_{t_{0},0,2}^{0}(−1, 1) ∪ L_{t_{0},1,2}^{0}(−1, 1)

where

L_{t}^{0}(1, −1) := 1 − 2L^{0}(p; α_{t}, β_{t}) = L^{0}(1, −1; p; α_{t}, β_{t}),
L_{t,k,2}^{0}(−1, 1) := 2L_{k,2}^{0} − 1 = L^{0}(−1, 1; p; α_{t}^{(k)}, β_{t}^{(k)}),

and α_{t}^{(k)}, β_{t}^{(k)} are given by

α_{t}^{(k)} := \frac{1}{n} (\frac{1}{p} − \frac{n}{p} + k + T_{n}α_{t}) , β_{t}^{(k)} := \frac{1}{n} (\frac{1}{p} − \frac{n}{p} + k + T_{n}β_{t})

with n := |ε(t)|. We have

L_{t}^{#} := L^{0}(p; ψ_{t}, ψ_{t}), L_{t,k,2}^{#} := L^{0}(p; ψ_{t}^{(k)}, ψ_{t}^{(k)}).

Now let λ ∈ C \ sp_{ess} S_{Γ}. The curve C_{Γ}^{#} of Theorem 14.1 with T := {t_{1}, t_{2}, t_{0}} is

L_{t_{1}}^{#} (1, \frac{λ + 1}{λ − 1}) ∪ L_{t_{2}}^{#} (\frac{λ + 1}{λ − 1}, (\frac{λ + 1}{λ − 1})^{2}) ∪ L_{t_{0}}^{#} ((\frac{λ + 1}{λ − 1})^{2} , 1) , (14.11)

whereas the curve C_{Γ}^{#} of Theorem 14.2 is

L_{t_{1}}^{#}(1, −1) ∪ L_{t_{2}}^{#}(1, −1) ∪ L_{t_{0},0,2}^{#}(−1, 1) ∪ L_{t_{0},1,2}^{#}(−1, 1) (14.12)

(14.13)  = L_{t_{1}}^{#}(1, −1) ∪ L_{t_{0},0,2}^{#}(−1, 1) ∪ L_{t_{2}}^{#}(1, −1) ∪ L_{t_{0},1,2}^{#}(−1, 1).

Note that (14.13) is nothing but a rearrangement of (14.12) which indicates a possibility of traversing (14.12) along a closed path. If, for example, all occuring leaves are circular arcs or horns, it follows almost at once from (14.13) that the possible values of Ind (S_{Γ} − λI) are −2, −1, 0, 1, 2. This is less obvious from (14.11).

Finally, let η be a simple Carleson arc. Denote the endpoints of η by t_{1} and t_{2} and suppose η is oriented from t_{1} to t_{2}. By Theorem 4.4,

sp_{ess} S_{η} = (2L_{t_{1}}^{0}(p; α_{t_{1}}, β_{t_{1}}) − 1) ∪ (1 − 2L_{t_{2}}^{0}(p; α_{t_{2}}, β_{t_{2}}))
= L_{t_{1}}^{0}(−1, 1; p; α_{t_{1}}, β_{t_{1}}) ∪ L_{t_{2}}^{0}(1, −1; p; α_{t_{2}}, β_{t_{2}}),

where we used the notation (14.1). The curve of Theorem 14.2 is

C_{η}^{#} = L_{t_{1}}^{#}(−1, 1) ∪ L_{t_{2}}^{#}(1, −1)

and thus, Ind (S_{η} − λI) = −wind (C_{η}^{#}, λ) for every λ ∈ C \ sp_{ess} S_{η}. If L_{t_{1}}(η) and L_{t_{2}}(η) are circular arcs or horns, which happens necessarily if η is piecewise Lyapunov, then L_{t_{1}}^{#}(−1, 1) ∪ L_{t_{2}}^{#}(1, −1) may be chosen to be the union of two circular arcs. In this case Theorems 14.2 implies that the only possible values of Ind (S_{η} − λI) are −1, 0, +1. However, if η := {re^{iδ log r} : 0 < r < 1} (δ ∈ R), then η is (a piece of) a logarithmic spiral and thus the simplest nontrivial example of a Carleson arc. In that case t_{1} = 0, t_{2} = 1, L_{t_{1}}(η) is a logarithmic double spiral, and L_{t_{2}}(η) is a circular arc. From Theorem 14.2 we infer that for every integer k there exists a point λ ∈ C\sp_{ess}S_{η} such that Ind (S_{Γ} − λI) = k.

Notice how drastically things change when passing from Lyapunov curves to Carleson curves. Without considering spectra, such phenomena stay in the dark.

Note added in proof. In the meanwhile the book [10] was accomplished. This book is a self-contained exposition of the spectral theory of Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients. It includes an introduction to Carleson curves, Muckenhoupt weights, weighted norm inequalities, local principles, Wiener-Hopf factorization, and Banach algebras generated by idempotents.

Acknowledgements

Bishop is partially supported by NSF Grant DMS 95-00557 and an Alfred P. Sloan research fellowship. Böttcher is partially supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation. Spitkovsky is partially supported by NSF Grant DMS 94-01848. Böttcher, Karlovich, and Spitkovsky are partially supported by NATO Collaborative Grant 950332.

76  Math. Nachr. (1999)

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Mathematics Department SUNY at Stony Brook Stony Brook, NY 11794-3651 USA

bishop@math.sunysb.edu

TU Chemnitz Fakultät für Mathematik 09107 Chemnitz Germany

karlovic@mathematik.tu-chemnitz.de TU Chemnitz Fakultät für Mathematik 09107 Chemnitz Germany

aboettch@mathematik.tu-chemnitz.de

Department of Mathematics

The College of William and Mary Williamsburg, VA 23187-8795 USA

ilya@cs.wm.edu