Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 1065–1081

Potential Type Operators on Curves with Vorticity Points

V. Rabinovich

Abstract. We study potential type operators on certain non-Lipschitz curves Γ. The curves under consideration are locally Lyapunov except for a finite set F of singular points. The normal vector ν(y) to the curve Γ does not have a limit at the singular points and, moreover, ν(y) may be an oscillating and rotating vector function in a neighborhood of the singular points. We establish a Fredholm theory of potential type operators in the spaces L_{p,w}(Γ, ℂ^{n}) where p ∈ (1, ∞) and w is a weight satisfying the Muckenhoupt condition.

Keywords: Potential operators, Fredholmness, essential spectrum AMS subject classification: 31 A 10

1. Introduction

Let Γ be a curve which is locally Lyapunov except for a finite set F of points on Γ. We refer to the points in F as the singular points of Γ. We consider operators of the form

Bu(x) = a(x)u(x) + ∫_{Γ} \frac{(ν(y), x − y)}{|x − y|^{2}} g(x, y)u(y) dl_{y}  (x ∈ Γ)  (1)

where dl_{y} is the oriented Lebesgue measure on Γ, a and g are bounded matrix functions on Γ with elements having second kind discontinuities with respect to x and y at singular points of Γ, and ν(y) is the interior normal vector of the curve Γ at the point y ∈ Γ\F. We assume that the vector ν(y) does not approach a limit at singular points. Moreover, we allow ν(y) to be an oscillating and rotating vector function in a neighborhood of each singular point. Singular points of the latter kind will be called vorticity points. We require that the curves under consideration satisfy the well-known Carleson condition (see, for instance, [1]). The class of operators (1) includes the operators of harmonic potentials, wave potentials, and other operators on curves with vorticity points which are important in mathematical physics. The results of this paper allow us to consider interior and exterior boundary value problems in planar domains whose boundary has vorticity points.

We construct a Fredholm theory of operators (1) in the Lebesgue spaces L_{p,w}(Γ, ℂ^{n}) with p ∈ (1, ∞) and a weight w satisfying the Muckenhoupt condition [1].

V. Rabinovich: ESIME-Zacatenco, Nat. Polit. Inst., Dept Telecommun., Ed. 1, Av. IPN, Mexico 07738, D.F. MEXICO; rabinov@maya.esimez.ipn.mx; partially supported by Russian Fund of Fundamental Investigations, grant RFFI-98-01-01-023 ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

In the case where Γ is a so-called Radon curve without peaks and a and b are piecewise continuous functions, potential operators have been considered by many authors, starting with Radon’s classical paper [2]. Later these operators have been investigated by Y. B. Lopatinskiy [3], I. Daniluk [4], V. Shelepov [5], and other authors (see, for instance, the surveys [6, 7]). Potential operators on curves with peaks are considered in papers by V. Maz’ya and A. Solov’ev (see, for example, [8 - 10]). Note that the class of curves studied here is not contained in the classes of curves treated in the aforementioned papers. It should be noted that the methods of the papers [3 - 5] are based on the theory of Wiener-Hopf equations on the half-line developed in the well-known paper of I. Gohberg and M. Krein [11]. But in the case of vorticity singular points this method is not applicable. Our approach is based on the theory of Mellin pseudodifferential operators. Earlier this approach was successfully applied to singular integral operators on some Carleson curves in the author’s papers [12 - 14] and in joint work of the author with A. Böttcher and Yu. I. Karlovich [15 - 16].

The theory of potential type operators on curves with vorticity points differs essentially from the theory of such operators on “nice” curves. For instance, the operator

Au(x) = ∫_{Γ} \frac{(ν(y), x − y)}{|x − y|^{2}} u(y) dl_{y} (x ∈ Γ)

where

Γ = {x ∈ ℂ : x = t exp iδ ln t with t ∈ (0, t_{0}) and δ ∈ ℝ}

is a piece of a logarithmic spiral, is not a compact operator in the space L_{p}(Γ), which is in contrast to the case when Γ is a simple Lyapunov arc of finite length. It should also be noted that oscillations of the curve Γ and of the weight w may lead to the appearance of massive local and Fredholm spectra of the operator A : L_{p}(Γ, w) → L_{p}(Γ, w).

The paper contains five sections. In Section 2 we provide some auxiliary material on Mellin pseudodifferential operators following the papers [13, 14, 17]. In Section 3 we consider potential type operators on certain simple unbounded curves γ. We construct a local symbol at singular points, give a criterion for Fredholmness and an index formula. We also prove a criterion for the compactness of certain operators in the spaces L_{p,w}(Γ, ℂ^{n}). Section 4 is devoted to potential operators on closed Jordan curves with vorticity points. We give a description of the local and essential spectra and derive an index formula. In Section 5 we discuss applications of the results of Section 4 to the double layer potential operator for the interior Dirichlet problem.

2. Mellin pseudodifferential operators

2.1 Main properties of Mellin pseudodifferential operators. We say that a matrix function (a_{ij}(r, λ))_{i,j=1}^{n} defined on ℝ_{+} × ℝ belongs to E^{m}(n) if the functions a_{ij}(r, λ) are in C^{∞}(ℝ_{+} × ℝ) and satisfy the estimate

sup_{ℝ_{+}×ℝ} |(r∂_{r})^{α}∂_{λ}^{β}a_{ij}(r, λ)|⟨λ⟩^{−m+β} < ∞

for all α, β ∈ ℕ_{0} = \{0\} ∪ ℕ, where ⟨λ⟩ = (1 + λ^{2})^{\frac{1}{2}} . A matrix function a(r, λ) (∈ E^{0}(n)) is said to be slowly varying at the point 0 if

lim_{r→+0} sup_{λ∈ℝ} |(r∂_{r})^{α}∂_{λ}^{β}a_{ij}(r, λ)|⟨λ⟩^{β} = 0  (2)

for all α ∈ ℕ_{0} and β ∈ ℕ, and it is said to be slowly varying at the point +∞ if

lim_{r→+∞} sup_{λ∈ℝ} |(r∂_{r})^{α}∂_{λ}^{β}a_{ij}(r, λ)|⟨λ⟩^{β} = 0  (3)

for all α ∈ ℕ_{0} and β ∈ ℕ. We denote by Ẽ(n) the class of matrix functions in E^{0}(n) which are slowly varying both at the origin and at infinity. Let J_{0}(n) and J_{∞}(n) denote the set of matrix functions for which condition (2) or (3), respectively, holds for all α, β ∈ ℕ_{0}. Finally, set

J(n) = J_{0}(n) ∩ J_{∞}(n) ∩ E^{−1}(n).
Let a(r, λ) ∈ E^{m}(n). The operator
(Op(a)u)(r) = a(r, D_{r})u = ∫_{ℝ} d′λ ∫_{ℝ_{+}} a(r, λ) (rρ^{−1})^{iλ}u(ρ)ρ^{−1}dρ  (4)

where d′λ = \frac{1}{2π} dλ and u ∈ C_{0}^{∞}(ℝ_{+}, ℂ^{n}), is called the Mellin pseudodifferential operator with the symbol a(r, λ). The class of all such operators is denoted by OP E^{m}(n). The notations OP Ẽ(n), OP J_{0}(n), OP J_{∞}(n), OP J(n) have the obvious meaning.

Let a(r, ρ, λ) (∈ C^{∞}(ℝ_{+} × ℝ_{+} × ℝ)) be a matrix function such that

sup_{ℝ_{+}^{2}×ℝ} |(r∂_{r})^{β}(ρ∂_{ρ})^{γ}∂_{λ}^{α}a_{ij}(r, ρ, λ)|⟨λ⟩^{α} < ∞  (5)

for all α, β, γ ∈ ℕ_{0} and i, j = 1, …, n. We denote by E_{d}^{0}(n) the class of all matrix functions satisfying (5).

An operator which is defined by (4) with a(r, λ) replaced by a(r, ρ, λ) is called a Mellin pseudodifferential operator with a double symbol. Let OP E_{d}^{0}(n) stand for the class of such operators. We say that a double symbol a(r, ρ, λ) is slowly varying if

lim_{r→+0} sup_{ρ∈K,λ∈ℝ} |(r∂_{r})^{β}(ρ∂_{ρ})^{γ}∂_{λ}^{α}a_{ij}(r, rρ, λ)|⟨λ⟩^{α} = 0
lim_{r→+∞} sup_{ρ∈K,λ∈ℝ} |(r∂_{r})^{β}(ρ∂_{ρ})^{γ}∂_{λ}^{α}a_{ij}(r, rρ, λ)|⟨λ⟩^{α} = 0

for all α, β, γ ∈ ℕ_{0} such that β + γ ≠ 0 and for every compact K ⊂ ℝ_{+}. We denote by Ẽ_{d}(n) the class of slowly varying double symbols and by OP Ẽ_{d}(n) the corresponding class of operators.

Throughout what follows we suppose that 1 < p < ∞. Let L_{p}(ℝ_{+}, dμ) be the Lebesgue space on ℝ_{+} = (0, ∞) with the measure dμ(ϱ) = \frac{dϱ}{ϱ}. We denote by L_{p}^{n}(ℝ_{+}, dμ) the Banach space of all measurable complex vector functions u(r) = (u_{1}(r), …, u_{n}(r)) (r ∈ ℝ_{+}) with the norm

∥u∥_{L_{p}^{n}(ℝ_{+},dμ)} = ( ∑_{j=1}^{n} ∫_{0}^{∞} |u_{j}(r)|^{p}dμ)^{\frac{1}{p}} .

We also use the following notations: if X is a Banach space, then L(X) is the Banach algebra of all bounded linear operators on X, while K(X) is the ideal of all compact operators in L(X).

We now summarize some properties of Mellin pseudodifferential operators we will need in what follows.

Proposition 1.

(a) Every operator A ∈ OP E^{0}(n) is bounded in L_{p}^{n}(ℝ_{+}, dμ) (p ∈ (1, ∞)) and we have ∥A∥_{LL_{p}(Γ))} ≤ C_{p,s}M where

M = max_{1≤i,j≤n} ∑_{{0≤α≤l \atop 0≤β≤k}} sup_{ℝ_{+}×ℝ} |(r∂_{r})^{β}∂_{λ}^{α}a_{ij}(r, λ)| ⟨λ⟩^{α}

and k, l are independent of A.

(b) If A ∈ OP E^{0}(n) is invertible in L_{p}^{n}(ℝ_{+}, dμ) (p ∈ (1, ∞)), then A^{−1} also belongs to OP E^{0}(n).

(c) Let A, B ∈ OP Ẽ(n). Then AB ∈ OP Ẽ(n) and the symbol σ_{AB}(r, λ) of AB is given by the formula σ_{AB}(r, λ) = a(r, λ)b(r, λ) + t_{1}(r, λ) where t_{1}(r, λ) ∈ J(n).

(d) Let A (∈ OP Ẽ(n)) act on L_{p}^{n}(ℝ_{+}, dμ) (p ∈ (1, ∞)). Then A^{∗} belongs to OP E(n) and σ_{A^{∗}}(r, λ) is given by the formula σ_{A^{∗}}(r, λ) = a^{∗}(r, λ) + t_{2}(r, λ) where t_{2}(r, λ) ∈ J(n).

(e) Let A be a Mellin pseudodifferential operator with a double symbol a(r, ρ, λ) ∈ S̃_{d}(n). Then A ∈ OP Ẽ(n) and σ_{A}(r, λ) = a(r, r, λ) + t_{3}(r, λ) where t_{3}(r, λ) ∈ J(n).

(f) OP J(n) ⊂ K(L_{p}^{n}(ℝ_{+}, dμ)) for all p ∈ (1, ∞).

2.2 Mellin pseudodifferential operators with analytic symbols in weighted L_{p}-spaces. Let w ∈ C^{∞}(ℝ_{+}). We say that a vector function u belongs to L_{p}^{n}(ℝ_{+}, wdμ) if ∥u∥_{L_{p}^{n}(ℝ_{+},wdμ)} = ∥wu∥_{L_{p}^{n}(ℝ_{+},dμ)} < ∞. In what follows we consider weights w(r) = exp v(r) where the function v satisfies the estimates

|(r \frac{d}{dr} )^{k}v(r)| < ∞  (k ∈ ℕ).  (6)

Moreover, we require that there is an interval (c, d) ∋ 0 such that the function κ_{w}(r) = rv′(r) satisfies the condition

c < inf_{r∈ℝ_{+}} κ_{w}(r) ≤ sup_{r∈ℝ_{+}} κ_{w}(r) < d.  (7)

We say that a weight w is slowly varying on ℝ_{+} if conditions (6) - (7) hold, lim_{r→0} rκ′_{w}(r) = 0 and lim_{r→∞} rκ′_{w}(r) = 0. We denote by R(c, d) the class of slowly varying weights.

Definition 2. We say that a matrix-function a(r, ρ, λ) is in E_{d}^{0}(n, (c, d)) if a(r, ρ, λ) is analytically continued with respect to λ into the strip Π = {λ ∈ ℂ : Im λ ∈ (c, d)} and

sup_{ℝ_{+}^{2}×Π} |(r∂_{r})^{β}(ρ∂_{ρ})^{γ}∂_{λ}^{α}a_{ij}(r, ρ, λ)|⟨λ⟩^{α} < ∞.

We let OP E_{d}^{0}(n, (c, d)) denote the corresponding class of Mellin pseudodifferential operators with analytic symbols.

The class OP Ẽ_{d}(n, (c, d)) of Mellin pseudodifferential operators with slowly varying double analytic symbols is introduced in evident way.

Proposition 3.

(a) Let a(r, ρ, λ) ∈ E_{d}^{0}(n, (c, d)) and let the weight w satisfy conditions (6) − (7). Then wOp(a)w^{−1} ∈ E_{d}^{0}(n) and wOp(a)w^{−1} = Op(a(r, ρ, λ + iϑ_{w}(r, ρ)) where ϑ_{w}(r, ρ) = ∫_{0}^{1} κ_{w}(r^{1−τ} ρ^{τ}) dτ (it is easy to see that condition (7) yields ϑ_{w}(r, ρ) ∈ (c, d) for all r, ρ ∈ ℝ_{+}).

(b) Let Op(a) ∈ OP Ẽ_{d}(n, (c, d)) and w ∈ R(c, d). Then wOp(a)w^{−1} ∈ OP Ẽ_{d}(n) and wOp(a)w^{−1} = Op(a(r, r, λ + iκ_{w}(r))) + q(r, λ) where q(r, λ) ∈ J(n).

Corollary 4. Let Op(a) ∈ OP E_{d}^{0}(n, (c, d)) and suppose a weight w satisfies conditions (6) − (7). Then A is bounded in L_{p}^{n}(ℝ_{+}, wdμ).

2.3 Local invertibility. First we give the following definition.

Definition 5. Let A ∈ L(L_{p}^{n}(ℝ_{+}, wdμ)). We say that A is a locally invertible operator at the point 0 if there exist R > 0 and operators B′, B′′ ∈ L(L_{p}^{n}(ℝ_{+}, wdμ)) such that B′Aχ_{R} = χ_{R} and χ_{R}AB′′ = χ_{R} where χ_{R} is the operator of multiplication by the characteristic function of the segment [0, R]. In the same way we define local invertibility at the point +∞.

Theorem 6. Let A = a(r, D_{r}) ∈ OP Ẽ(n, (c, d)) and w ∈ R(c, d). Then the operator A : L_{p}^{n}(ℝ_{+}, wdμ) → L_{p}^{n}(ℝ_{+}, wdμ) is locally invertible at the origin or at infinity if and only if

lim_{ε→0} inf_{(0,ε)×ℝ} |det a(r, λ + iκ_{w}(r))| > 0  (8)

or

lim_{R→+∞} inf_{(R,∞)×ℝ} |det a(r, λ + iκ_{w}(r))| > 0,  (9)

respectively.

2.4 Fredholmness and index. Here we formulate the following result.

Theorem 7.

(a) Let A = Op(a) ∈ OP Ẽ(n, (c, d)) and w ∈ R(c, d). Then the operator A : L_{p}^{n}(ℝ_{+}, wdμ) → L_{p}^{n}(ℝ_{+}, wdμ) is Fredholm if and only if conditions (8) − (9) are fulfilled and

lim_{R→+∞} inf_{ℝ×(R,∞)} |det a(r, λ)| > 0.

(b) If A ∈ OP Ẽ(n, (c, d)) is a Fredholm operator, then

Ind A = −\frac{1}{2π} [arg a(r, λ + iκ_{w}(r))]_{Γ(R′,R′′)}  (10)

where Γ(R′, R′′) is the positively oriented boundary of the rectangle

Π(R′, R′′) = \{(r, λ) ∈ ℝ_{+} × ℝ : \frac{1}{R′} < r < R′ and |λ| < R′′\}.

Here the numbers R′, R′′ are such that a^{−1}(r, λ + iκ_{w}(r)) exists for all points of the domain (ℝ_{+} × ℝ)\Π(R′, R′′).

3. Potential operators on simple curves

We now consider the potential type operator

Au(x) = \frac{1}{π} ∫_{γ} \frac{(ν(y), x − y)}{|x − y|^{2}} g(x, y)u(y) dl_{y}  (x ∈ γ)  (11)

on a simple unbounded curve γ. We suppose that γ has the parametrization

γ = {x ∈ ℂ : x = t exp iθ(t)} (t ∈ ℝ_{+}) (12)

where θ is a real-valued C^{∞}-function on ℝ_{+} satisfying the conditions

\begin{matrix}sup_{ℝ_{+}} |(t\frac{d}{dt})^{k}θ(t)| & < ∞ (k ∈ ℕ) \\ lim_{t→+0} (t\frac{d}{dt})^{2}θ(t) & = 0 \\ lim_{t→+∞} (t\frac{d}{dt})^{2}θ(t) & = 0. \end{matrix} \}  (13)

The simplest example of such a curve is the logarithmic spiral x = t exp(iδ ln t) (t ∈ ℝ_{+}) with the twisting coefficient δ ∈ ℝ. More complicated examples are provided by curves with a variable twisting coefficient δ(t) : x = t exp(iδ(t) ln t) (t ∈ ℝ_{+}) where δ(t) = sin(ln^{α}⟨ln⟩) with α ∈ (0, 1). It is easy to check that θ(t) = δ(t) ln t satisfies the above conditions (13).

We suppose that g(x, y) = (g_{kl}(x, y))_{k,l=1}^{n} where g_{kl} ∈ C^{∞}(γ × γ) and the functions

g̃_{k,l}(t, τ) = g_{k,l}(t exp iθ(t), τ exp iθ(τ))

are subject to the coditions

sup_{ℝ_{+}×ℝ_{+}} |(t∂_{t})^{α}(τ∂_{τ})^{β}g̃_{k,l}(t, τ)| < ∞  (α, β ∈ ℕ_{0}).  (14)

Moreover, we require that

lim_{t→0} sup_{τ} |t\frac{∂}{∂t}g̃_{k,l}(t, τ)| = lim_{t→∞} sup_{τ} |t\frac{∂}{∂t}g̃_{k,l}(t, τ)| = 0  (15)
lim_{ι→0} sup_{t} |τ \frac{∂}{∂τ} g̃_{k,l}(t, τ)| = lim_{ι→∞} sup_{t} |τ \frac{∂}{∂τ} g̃_{k,l}(t, τ)| = 0.  (16)
Let Φ : C_{0}^{∞}(γ, ℂ^{n}) → C_{0}^{∞}(ℝ_{+}, ℂ^{n}) be the mapping acting by the formula
ũ(t) = (Φu)(t) = t^{\frac{1}{p}} u(t exp iθ(t)) (t ∈ ℝ_{+}).

Clearly, Φ can be continued to an isometric isomorphism of L_{p}(γ; ℂ^{n}) on L_{p}(ℝ_{+}, ℂ^{n}, dμ).

Proposition 8. Let A be an operator of the form (11). Then A_{Φ} = ΦAΦ^{−1} is a Mellin pseudodifferential operator in the class OP Ẽ(n) with the symbol

σ_{A_{Φ}}(t, λ) = \frac{i}{2} sin \frac{2πδ_{θ}(t)(λ+i \frac{1}{p})}{1+δ_{θ}^{2}(t)} ; sinh \frac{π(λ+i \frac{1}{p})}{1+iδ_{θ}(t)} sinh \frac{π(λ+i \frac{1}{p})}{1−iδ_{θ}(t)} g̃(t, t) + q(t, λ)  (17)

where δ_{θ}(t) = tθ′(t), q(t, λ) ∈ J_{0}.

Proof. After the change of variables y = τ exp iθ(τ), the vector ν(y) assumes the form

ν(z + τ exp iθ(τ)) = \frac{(1 + iτθ′(τ)) exp i(θ(τ) − \frac{π}{2} )}{(1 + (τθ′(τ))^{2})^{\frac{1}{2}}} = d(τ) exp iθ(τ)  (18)

where

d(τ) = −i \frac{1 + iτθ′(τ)}{(1 + (τθ′(τ))^{2})^{\frac{1}{2}}} .  (19)

Further,

\begin{aligned}\frac{(ν(y), x − y)}{|x − y|^{2}} & = \frac{Re[ν(y)\overline{(x − y)}]}{|x − y|^{2}} \\ & = \frac{1}{2} \frac{ν(y)\overline{(x − y)} + \overline{ν(y)}(x − y)}{|x − y|^{2}} \\ & = \frac{1}{2} (\frac{ν(y)}{x − y} + \frac{\overline{ν(y)}}{\overline{x − y}} ). \end{aligned}  (20)

It follows from (20) that

\frac{(ν(y), x − y)}{|x − y|^{2}} = \frac{1}{2}(\frac{d(τ)}{t exp i(θ(t) − θ(τ)) − τ} + \frac{\overline{d(τ)}}{t exp −i(θ(t) − θ(τ)) − τ} ).  (21)

Taking into account (13) we obtain that d(τ) satisfies the estimates sup_{ℝ_{+}} |(t\frac{d}{dt})^{k}d(t)| < ∞ (k ∈ ℕ_{0}) and that lim_{t→+0} (t\frac{d}{dt})d(t) = 0 as well as lim_{t→∞} (t\frac{d}{dt})d(t) = 0. Let us introduce the function

m(θ)(t, τ) = \frac{θ(t) − θ(τ)}{ln t − ln τ} = ∫_{0}^{1} t^{1−r}τ^{r}θ′(t^{1−r}τ^{r}) dr.  (22)

It is easy to check that m(θ)(t, τ) satisfies conditions (14) - (16). Formula (22) yields

exp i(θ(t) − θ(τ)) = (tτ^{−1})^{im(θ)(t,τ)}.  (23)

Applying formulas (21) and (23) we see that A_{Φ} is the integral operator

(A_{Φ}v)(t) = \frac{1}{2} ∫_{ℝ_{+}} (k(t, τ) + \overline{k(t, τ)})g̃(t, τ)v(τ)\frac{dτ}{τ}

where k(t, τ) = i \frac{1+iτθ′(τ)}{(1−(tτ^{−1})^{1+im(θ)(t,τ)})}. If Re ξ ≥ 1, then

\frac{1}{πi} \frac{ξ}{1 − x^{ξ}} = \frac{1}{2π} ∫_{ℝ} coth π(\frac{λ}{ξ})x^{iλ}dλ  (x ∈ ℝ_{+})  (24)

(see [15: p. 414]). Using (24), (18) - (19) and the equality dl = (1 + (τθ′(τ))^{2})^{\frac{1}{2}} dτ we arrive at the representation

(A_{Φ}v)(t) = \frac{1}{2π} ∫_{ℝ} dλ ∫_{ℝ_{+}} a(t, τ, λ + i\frac{1}{p})(tτ^{−1})^{iλ}v(τ)\frac{dτ}{τ}

where

a(t, τ, λ) = \frac{1}{2} (a_{1}(t, τ, λ) + \overline{a_{1}(t, τ, λ)})g̃(t, τ)

and

a_{1}(t, τ, λ) = −\frac{1 + iτθ′(τ)}{1 + im (θ)(t, τ)} coth π(\frac{λ}{1 + im (θ)(t, τ)}).

From estimates (13) and (15) - (16) we infer that a(t, τ, λ) ∈ E_{d}^{0}(n). Hence, by Proposition 1/(e), A_{Φ} is a Mellin pseudodifferential operator with the symbol

a(t, t, λ) = \frac{1}{2}[ coth π(\frac{λ + i\frac{1}{p}}{1 + iτθ′(τ)}) − coth π(\frac{λ + i\frac{1}{p}}{1 − iτθ′(τ)})]g̃(t, τ) + q(t, λ)  (25)

where q(t, λ) ∈ J_{0}. Finally, the identity coth a − coth b = \frac{sinh(b−a)}{sinh a sinh b} in conjunction with (25) gives formula (17)

Let us now consider A as an operator acting in the space L_{p}^{n}(γ, w) with the norm ∥u∥_{L_{p}^{n}(γ,w)} = ∥wu∥_{L_{p}^{n}(γ)}. We suppose that the weight w satisfies the condition

w̃(t) = w(t exp iθ(t)) ∈ R(−\frac{1}{p}, 1 − \frac{1}{p}).  (26)

This condition is similar the well-known Hunt-Muckenhoupt condition (see, for instance, [1]).

Corollary 9. Let A be an operator of the form (11) and let w be a weight on γ satisfying condition (26). Then A_{Φ,w} = ΦwAw^{−1}Φ^{−1} is a Mellin pseudodifferential operator in the class OP Ẽ . Its symbol is given by the formula

σ_{A_{Φ,w}}(t, λ) = \frac{i}{2} sin \frac{2πδ_{θ}(t)(λ+i( \frac{1}{p} +κw̃(t)))}{1+δ_{θ}^{2}(t)} ; sinh \frac{π(λ+i( \frac{1}{p} +κw̃(t)))}{1+iδ_{θ}(t)} sinh \frac{π(λ+i( \frac{1}{p} +κw̃(t)))}{1−iδ_{θ}(t)} g̃(t, τ) + q(t, λ)  (27)

where q(t, λ) ∈ J_{0} and κw̃(t) = t\frac{w̃′(t)}{w̃(t)}.

Proof. This corollary follows from Proposition 3

Corollary 10. Let A be an operator of type (11) and let w be a weight on γ satisfying condition (26). Then A : L_{p}^{n}(γ, w) → L_{p}^{n}(γ, w) is a compact operator if and only if

lim_{t→0} σ_{A_{Φ,w}}(t, λ) = lim_{t→∞} σ_{A_{Φ,w}}(t, λ) = 0

uniformly with respect to λ ∈ ℝ.

Corollary 10 tells us that there are two reasons for A : L_{p}^{n}(γ, w) → L_{p}^{n}(γ, w) to be a compact operator:

1) lim_{t→0} g̃(t, t) = lim_{t→∞} g̃(t, t) = 0

2) lim_{t→0} δ_{θ}(t) = lim_{t→∞} δ_{θ}(t) = 0.

Let

Au(x) = \frac{1}{π} ∫_{γ} \frac{(ν(y), x − y)}{|x − y|^{2}} u(y) dl_{y} (x ∈ γ)

be the potential operator of the double layer. Then A is compact if and only if condition 2) holds. Let γ be a bounded curve with the parametrization γ = {x ∈ ℂ : x = t exp iθ(t)), t ∈ [0, t_{0}]} where θ satisfies condition (13)_{1} on [0, t_{0}]. Suppose also that condition (13)_{2} is fulfilled. Then A : L_{p}^{n}(γ, w) → L_{p}^{n}(γ, w) is a compact operator if and only if lim_{t→0} δ_{θ}(t).

Let us now consider the operator

Bu(x) = a(x)u(x) + \frac{1}{π} ∫_{γ} \frac{(ν(y), x − y)}{|x − y|^{2}} g(x, y)u(y) dl_{y} (x ∈ γ)

where a(x) = (a_{k,l}(x))_{k,l=1}^{n} and ã_{k,l}(t) = a_{k,l}(t exp iθ(t)) satisfy the same estimates with respect to t as g̃_{k,l}(t, τ). It follows from Corollary 9 that, up to a symbol q(t, λ) ∈ J_{0}, B_{Φ,w} is a Mellin pseudodifferential operator with the symbol

σ_{B_{Φ,w}}(t, λ) = ã(t) + σ_{A_{Φ,w}}(t, λ)

where σ_{A_{Φ,w}}(t, λ) is given by formula (27).

The next two theorems follow from Theorem 7 and Proposition 3.

Theorem 11. B : L_{p}^{n}(γ, w) → L_{p}^{n}(γ, w) is a Fredholm operator if and only if

(i) inf_{x∈γ} |deta(x)| > 0 (ii) \underline{lim}_{t→0} inf_{λ∈ℝ} |detσ_{B_{Φ,w}}(t, λ)| > 0 and \underline{lim}_{t→∞} inf_{λ∈ℝ} |detσ_{B_{Φ,w}}(t, λ)| > 0.

If conditions (i) and (ii) are fulfilled, then

Ind B = −\frac{1}{2π} [argdetσ_{B_{Φ,w}}(t, λ)]_{Γ(R′,R′′)}

where Γ(R′, R′′) is the positively oriented boundary of the rectangle

Π(R′, R′′) = \{(r, λ) ∈ ℝ_{+} × ℝ : \frac{1}{R′} < r < R′ and |λ| < R′′\}

and where the numbers R′, R′′ are chosen so that [σ_{A_{Φ,w}}(t, λ)]^{−1} exists for all points in the domain (ℝ_{+} × ℝ)\Π(R′, R′′).

Theorem 12. The operator B : L_{p}^{n}(γ, w) → L_{p}^{n}(γ, w) is locally invertible at the point 0 if and only if \underline{lim}_{t→0} inf_{λ∈ℝ} |detσ_{B_{Φ,w}}(t, λ)| > 0.

The local spectrum of an operator A : L_{p}^{n}(γ, w) → L_{p}^{n}(γ, w) at a point x ∈ γ is defined as the set of all points ς ∈ ℂ for which A − ςI is not locally invertible at the point x. We denote the local spectrum of A at the point x by Sp_{x}A.

Theorem 13. A point ς belongs to Sp_{z}A if and only if there exists a sequence t_{m} → 0 such that lim_{m→∞} inf_{λ∈ℝ} |det(σ_{A_{Φ,w}}(t_{m}, λ) − ςI)| = 0. Thus, the local spectrum of A at the singular point 0 is

Sp_{0}A_{Γ} = ⋃ {ς ∈ ℂ : det (σ̃_{A_{Φ,w}}(λ) − ςI) = 0 for some λ ∈ ℝ ∪ {∞}}

where det (σ̃_{A_{Φ,w}}(λ) − ςI) denotes the set of all partial limits of det (σ_{A_{Φ,w}}(t, λ) − ςI) as t → 0.

From the previous theorem we learn in particular that the local spectrum of A at the point 0 can be a set of positive planar Lebsegue measure and that its structure is defined by the behavior of the curve, of the coefficients, and the weight at the point zero.

4. Potential type operators on Jordan curves with vorticity points

In this section we consider potential operators on oriented closed Jordan curves Γ in the complex plane ℂ. We will suppose that Γ has a finite set F of singular points and that Γ\F is locally a Lyapunov curve.

If z ∈ F is a singular point, then there is a neighborhood U^{z} of the point z such that

Γ ∩ U_{±}^{z} = {x = z ± t exp iω_{±}^{z}(t) : t ∈ [0, s]}

where U_{+}^{z} and U_{−}^{z} is a right side and left side half-neighborhood, respectively, of the point z. We suppose that

ω_{±}^{z}(t) = θ_{0}^{z}(t) + θ_{±}^{z}(t) (t ∈ (0, s])

where the functions θ_{0}^{z} and θ_{±}^{z} satisfy estimate (13)_{1} on the interval [0, s] and equality (13)_{2}. Moreover, we suppose that

0 ≤ θ_{−}^{z}(t) ≤ M_{−} < m_{+} ≤ θ_{+}^{z}(t) ≤ M_{+} < 2π  (z ∈ F).  (28)

From (28) it follows that

lim_{t→0} t\frac{dθ_{±}^{z}(t)}{dt} = 0.

Hence, the function θ_{0}^{z} describes the rotation of the curve Γ at the point z, while the functions θ_{±}^{z} characterize the oscillation of the curve at that point. In the case where z is an angular point of the curve, which means that the limits lim_{t→0} θ_{±}^{z}(t) = θ_{±}^{z}(0) exist, condition (28) says that the point z is not a peak.

Let

A_{Γ}u(x) = \frac{1}{π} ∫_{Γ} \frac{(ν(y), x − y)}{|x − y|^{2}} g(x, y)u(y) dl_{y} (x ∈ Γ)

be a potential operator on the curve Γ, assume the matrix function g(x, y) is bounded on Γ and continuous on Γ \ F, and also assume that the matrix functions

g_{++}^{z}(t, τ) = g(z + t exp iω_{+}^{z}(t), z + τ exp iω_{+}^{z}(τ))
g_{+−}^{z}(t, τ) = g(z + t exp iω_{+}^{z}(t), z − τ exp iω_{−}^{z}(τ))
g_{−+}^{z}(t, τ) = g(z − t exp iω_{−}^{z}(t), z + τ exp iω_{+}^{z}(τ))
g_{−−}^{z}(t, τ) = g(z − t exp iω_{−}^{z}(t), z − τ exp iω_{−}^{z}(τ))

have entries satisfying conditions (14) - (15) in a neighborhood of the points t = 0 and τ = 0.

We consider the operator A_{Γ} in the weighted space L_{p}^{n}(Γ, w) where 1 < p < ∞ and w is a positive continuous function on Γ\F. In a neighborhood of a singular point z the weight w is supposed to admit a representation w(x) = exp v^{z}(|x − z|) (|x − z| < s) where the function v^{z} satisfies estimates (6) on (0, s] and lim_{t→0} (t\frac{d}{dt})^{2}v^{z}(t) = 0. As above (see condition (26)) we suppose that w̃(t) = exp v^{z}(t) ∈ R(−\frac{1}{p}, 1 − \frac{1}{p}). Put

β^{z}(t) = t\frac{dv^{z}(t)}{dt}, δ^{z}(t) = t\frac{dθ_{0}^{z}(t)}{dt} and θ^{z}(t) = θ_{+}^{z}(t) − θ_{−}^{z}(t).

Example 14. Let ω_{+}^{z}(t) = δ^{z}(t) ln t + θ_{+}^{z} + \frac{ε}{2} sin ln⟨ln t⟩ and ω_{−}^{z}(t) = δ^{z}(t) ln t + θ_{−}^{z} where δ^{z}(t) = f(ln^{α}⟨ln t⟩) with 0 < α < 1, f ∈ C_{b}^{∞}(ℝ_{+}) and ε < θ_{+}^{z} − θ_{−}^{z} < 2π − ε. Then all the above conditions for Γ are fulfilled.

Let v^{z}(t) = β^{z}(t) ln t, where β^{z} satisfies the same conditions as δ^{z} and −\frac{1}{p} < inf_{(0,s]} β^{z}(t) ≤ sup_{(0,s]} β^{z}(t) < 1 − \frac{1}{p}. Then w̃(t) = exp v^{z}(t) ∈ ℝ(−\frac{1}{p}, 1 − \frac{1}{p}).

We introduce the isometric isomorphism Φ_{z} : L_{p}^{n}(Γ ∩ U_{z,}w) → L_{p}^{2n}([0, s], dμ) by the formula

Φ_{z}f(t) = (t^{\frac{1}{p}} e^{v(t)}f(z + t exp iω_{+}^{z}(t)), t^{\frac{1}{p}} e^{v(t)}f(z − t exp iω_{−}^{z}(t)))

for t ∈ (0, s].

Proposition 15. Let χ_{z}(x) be a cut-off function of the point z, that is, χ_{z}(x) ∈ C^{∞}(Γ), χ_{z}(x) = 1 in a neighborhood of the point z, and suppχ_{z}(x) ⊂ U_{z}. Then Φ_{z}χ_{z}Aχ_{z}Φ_{z}^{−1} is a Mellin pseudodifferential operator in the class OP S̃(2n). Its symbol σ^{z}(A)(t, λ) is given up to a symbol in the class J_{0} by the formula

σ^{z}(A)(t, λ) = ( \begin{matrix}a_{++}^{z}(t, λ) & a_{−+}^{z}(t, λ) \\ a_{+−}^{z}(t, λ) & a_{−−}^{z}(t, λ) \end{matrix} )

where

a_{++}^{z}(t, λ) = \frac{i}{2} sin \frac{2πδ^{z}(t)(λ+i( \frac{1}{p} +β^{z}(t)))}{1+δ^{z}(t)^{2}} ; sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1+iδ^{z}(t)} sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1−iδ^{z}(t)} g_{++}^{z}(t, t)

a_{−−}^{z}(t, λ) = −\frac{i}{2} sin \frac{2πδ^{z}(t)(λ+i( \frac{1}{p} +β^{z}(t)))}{1+δ^{z}(t)^{2}} ; sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1+iδ^{z}(t)} sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1−iδ^{z}(t)} g_{−−}^{z}(t, t)

a_{−+}^{z}(t, λ) = \frac{1}{2} [\frac{exp(θ^{z}(t) − π)\frac{λ+i( \frac{1}{p} +β^{z}(t))}{1+iδ^{z}(t)}}{sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1+iδ^{z}(t)}} − \frac{exp −(θ^{z}(t) − π)\frac{λ+i( \frac{1}{p} +β^{z}(t))}{1−iδ^{z}(t)}}{sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1−iδ^{z}(t)}} ] g_{−+}^{z}(t, t)

a_{+−}^{z}(t, λ) = −\frac{1}{2} [\frac{exp −(θ^{z}(t) − π)\frac{λ+i( \frac{1}{p} +β^{z}(t))}{1+iδ^{z}(t)}}{sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1+iδ^{z}(t)}} − \frac{exp(θ^{z}(t) − π)\frac{λ+i( \frac{1}{p} +β^{z}(t))}{1−iδ^{z}(t)}}{sinh \frac{π(λ+i( \frac{1}{p} +β^{z}(t)))}{1−iδ^{z}(t)}} ] g_{+−}^{z}(t, t).

Proof. The expressions for a_{++}^{z}(t, λ) and for a_{−−}^{z}(t, λ) follow from Corollary 9. The formulas for a_{−+}^{z}(t, λ) and a_{+−}^{z}(t, λ) can be proved as in [14] (see the proof of Theorem 3.18 there). We remark that the proofs are based on the formula (see [14, 15])

J(λ, a, ξ) = \frac{aξ}{πi} ∫_{0}^{∞}(a − t^{ξ})^{−1}t^{−iλ−1}dt = \frac{2a exp(−i\frac{λ}{ξ} − 1) ln a}{exp(2π \frac{λ}{ξ}) − 1} (a ∈ ℂ\ℝ̄_{+})

where Im λ ∈ (0, 1) and Re ξ ≥ 1

Theorem 16. The operator A_{Γ} : L_{p}^{n}(Γ, w) → L_{p}^{n}(Γ, w) (1 < p < ∞) is bounded.

Proof. Let ∑_{j=1}^{N} χ_{j}(x) = 1 (x ∈ Γ) be a C^{∞}-partition of unity on Γ and let ψ_{j} ∈ C^{∞}(Γ) satisfy ψ_{j}(x)χ_{j}(x) = χ_{j}(x). Then

A_{Γ} = ∑_{j=1}^{N} χ_{j}A_{Γ} = ∑_{j=1}^{N} χ_{j}A_{Γ}ψ_{j} + ∑_{j=1}^{N} χ_{j}A_{Γ}(1 − ψ_{j}).

The operator ∑_{j=1}^{N} χ_{j}A_{Γ}(1 − ψ_{j}) is bounded because supp χ_{j} ∩ supp ψ_{j} = ∅. If supp χ_{j} contains a singular point, the boundedness of χ_{j}A_{Γ}ψ_{j} follows from Proposition 1/(a). In the opposite case, χ_{j}A_{Γ}ψ_{j} is bounded as a potential type operator on a Lyapunov curve

Remark 1. Let the curve Γ have zero rotation at a singular point z ∈ F, that is, assume lim_{t→0} δ^{z}(t) = 0. Then the formulas for the symbol σ^{z}(A_{Γ})(t, λ) can be simplified. Namely, in this case

σ^{z}(A_{Γ})(t, λ) = ( \begin{matrix}0 & a_{−+}^{z}(t, λ) \\ a_{+−}(t, λ) & 0 \end{matrix} )

where

a_{−+}^{z}(t, λ) = \frac{sinh [(θ^{z}(t) − π)(λ + i(\frac{1}{p} + β^{z}(t)))]}{sinh π(λ + i(\frac{1}{p} + β^{z}(t)))} g_{−+}(t, t)

a_{+−}^{z}(t, λ) = \frac{sinh [(θ^{z}(t) − π)(λ + i(\frac{1}{p} + β^{z}(t)))]}{sinh π(λ + i(\frac{1}{p} + β^{z}(t)))} g_{+−}(t, t).

Let us consider the operator

B_{Γ}u(x) = a(x)u(x) + (A_{Γ}u)(x) (x ∈ Γ)

where the matrix function a(x) has the components in C(Γ\F) and

\begin{matrix}a_{+}^{z}(t) & = a(z + t exp iω_{+}^{z}(t)) \\ a_{−}^{z}(t) & = a(z − t exp iω_{−}^{z}(t)) \end{matrix} \}

are matrix functions with components satisfying conditions (14) - (15) in a neighborhood of the point t = 0. Set

a^{z}(t) = ( \begin{matrix}a_{+}^{z}(t) & 0 \\ 0 & a_{−}^{z}(t) \end{matrix} ) .

Theorem 17. The operator B_{Γ} : L_{p}^{n}(Γ, w) → L_{p}^{n}(Γ, w) is locally invertible at the point z ∈ F if and only if its local symbol σ^{z}(B_{Γ})(t, λ) = a^{z}(t) + σ^{z}(A)(t, λ) satisfies the condition

\underline{lim}_{t→0} inf_{λ∈ℝ} |detσ^{z}(B_{Γ})(t, λ)| > 0.  (29).

Proof. The proof of this theorem follows from Proposition 15 and Theorem 6

As above, we say that ς ∈ ℂ is a point of the local spectrum of the operator A_{Γ} at the point x ∈ Γ if A_{Γ} − ςI is not locally invertible at the point x. We denote by Sp_{x}A_{Γ} the local spectrum of A_{Γ} at the point x.

Corollary 18. Let z ∈ F. Then ς ∈ Sp_{z}A_{Γ} if and only if there exists a sequence t_{m} → 0 such that lim_{m→∞} inf_{λ∈ℝ} |detσ^{z}(A_{Γ} − ςI)(t_{m}, λ)| = 0. Thus, the local spectrum of A_{Γ} at the singular point z is

Sp_{z}A_{Γ} = ⋃ {ς ∈ ℂ : det σ̃^{z}(A_{Γ} − ςI)(λ) = 0 for some λ ∈ ℝ ∪ {∞}}

where det σ̃^{z}(A_{Γ} −ςI)(λ) is the set of the partial limits of det σ^{z}(A_{Γ} −ςI)(t, λ) as t → 0.

One can see that the local spectrum of the potential operator A_{Γ} : L_{p}^{n}(Γ, w) → L_{p}^{n}(Γ, w) may be a massive set, that is, a set with a positive planar Lebesgue measure. There are three reasons for Sp_{z}A_{Γ} to be a massive set:

1) the oscillation of the curve Γ

2) the oscillation of the coefficients

3) the oscillation of the weight at the singular point z .

Theorem 19. The operator B_{Γ} : L_{p}^{n}(Γ, w) → L_{p}^{n}(Γ, w) is a Fredholm operator if and only if

1) inf_{x∈Γ} |deta(x)| > 0

2) \underline{lim}_{t→0} inf_{λ∈ℝ} |det σ^{z}(B_{Γ})(t, λ)| > 0 for any point z ∈ F.

If both conditions are fulfilled, then

Ind B_{Γ} = − ∑_{z∈F} \frac{1}{2π} [arg det (I + a^{z}(t_{0})^{−1}σ^{z}(A_{Γ})(t_{0}, λ)]_{λ=−∞}^{∞}  (30)

where t_{0} is any point close enough to 0.

Proof. Simonenko’s local principle [20] says that B_{Γ} : L_{p}^{n}(Γ, w) → L_{p}^{n}(Γ, w) is a Fredholm operator if and only if B_{Γ} is locally invertible at every point x ∈ Γ. The operator A_{Γ} is locally compact at any point x ∈ Γ\F. The operator B_{Γ} is locally equivalent (in the sense of [20]) to the operator a(x)I at a point x ∈ Γ\F. This implies that condition 1) is necessary and sufficient for the local invertibility of B_{Γ} at non-singular points. From Theorem 17 it follows that condition 2) is necessary and sufficient for the local invertibility of B_{Γ} at singular points z ∈ F. The index formula can be proved, by the standard method of separation of singularities (see, for instance, [18, 19]) and is based on formula (10)

Remark 2. The Fredholm spectrum of the operator A_{Γ} is given by the formula ess sp A_{Γ} = ∪_{z∈F} Sp_{z}A_{Γ}.

5. Integral operator of the Dirichlet problem

Let

A_{Γ}u(x) = \frac{1}{π} ∫_{Γ} \frac{(ν(y), x − y)}{|x − y|^{2}}u(y) dl_{y} (x ∈ Γ)

be the double layer potential on a Jordan curve Γ satisfying the conditions of the previous section. Moreover, suppose that the singular points z ∈ F are points with zero rotation, that is, lim_{t→0} δ^{z}(t) = 0 for all z ∈ F. We consider A_{Γ} as an operator acting in L_{p}(Γ, w) where the weight w satisfies the above conditions. Let

r_{z}(A_{Γ}) = sup {|ζ| : ζ ∈ ℂ and A_{Γ} − ζI not locally invertible in L_{p}(Γ, w)}

be the local spectral radius A_{Γ}.

Theorem 20. Let the above given conditions for the curve Γ hold. Then

r_{z}(A_{Γ}) = \overline{lim}_{t→0} \frac{sin |π − θ^{z}(t)|(\frac{1}{p} + β^{z}(t))}{sin π(\frac{1}{p} + β^{z}(t))}.

Proof. We have σ^{z}(A_{Γ} − ζI)(t, λ)

= ( \begin{matrix}−ζ & \frac{sinh[(θ^{z}(t)−π)(λ+i( \frac{1}{p} +β^{z}(t)))]}{sinh π(λ+i( \frac{1}{p} +β^{z}(t)))} \\ \frac{sinh[(θ^{z}(t)−π)(λ+i( \frac{1}{p} +β^{z}(t)))]}{sinh π(λ+i( \frac{1}{p} +β^{z}(t)))} & −ζ \end{matrix} )

From Theorem 17 we deduce that

r_{z}(A_{Γ}) = \overline{lim}_{t→0} sup_{λ∈ℝ} |\frac{sinh [(θ^{z}(t) − π)(λ + i(\frac{1}{p} + β^{z}(t)))]}{sinh π(λ + i(\frac{1}{p} + β^{z}(t)))}| .  (31)

It is well-known (see, for instance, [5]) that

sup_{λ∈ℝ} |\frac{sinh[ξ(λ + iγ]}{sinh π(λ + iγ)}| = \frac{sin ξγ}{sin πγ}  (0 < ξ < π, γ ∈ (0, 1)).  (32)

Hence formulas (31) and (32) yield (30)

Let us now consider the integral operator B_{Γ} = I +A_{Γ} corresponding to the interior Dirichlet problem for harmonic functions.

Theorem 21. The operator B_{Γ} : L_{p}(Γ, w) → L_{p}(Γ, w) is locally invertible at a singular point z ∈ F if and only if \frac{1}{p} ∉ [a_{−}^{z}, a_{+}^{z}] where

\begin{matrix}a_{−}^{z} & = \underline{lim}_{t→0}(\frac{π}{π + |θ^{z}(t) − π|} − β^{z}(t)) \\ a_{+}^{z} & = \overline{lim}_{t→0}(\frac{π}{π + |θ^{z}(t) − π|} − β^{z}(t)). \end{matrix} \}

Proof. We have

detσ^{z}(B_{Γ})(t, λ) = 1 − [\frac{sinh [(θ^{z}(t) − π)(λ + i(\frac{1}{p} + β^{z}(t)))]}{sinh π(λ + i(\frac{1}{p} + β^{z}(t)))}]^{2}.

The equation 1 − \frac{sinh^{2} |π−θ|ζ}{sinh^{2} πζ} = 0 (ζ = λ + iη ∈ ℝ + i(0, 1)) has a unique solution ζ = i \frac{π}{π+|θ−π|}. Hence, if \frac{1}{p} ∉ [a_{−}^{z}, a_{+}^{z}], then condition (29) of Theorem 16 is fulfilled. This yields the local invertibility of B_{Γ} at the point z. If \frac{1}{p} ∈ [a_{−}^{z}, a_{+}^{z}], then there is a sequence t_{m} → 0 such that lim_{m→∞} \frac{π}{π+|θ^{z}(t_{m})−π|} − β^{z}(t_{m}) = \frac{1}{p}. Thus in this case condition (29) is not fulfilled, which shows that B_{Γ} is not a locally invertible operator at the point z

Remark 3. The oscillations of the curve and the weight at singular points define a segment [a_{−}^{z}, a_{+}^{z}] of prohibited values of \frac{1}{p} for the local invertibility of the operator B_{Γ}. Note that in the case of an angular singular point and a power weight there is exactly one prohibited value of \frac{1}{p}.

Theorem 22. The operator B_{Γ} : L_{p}(Γ, w) → L_{p}(Γ, w) is of Fredholm type if and only if the condition of Theorem 21 is satisfied for all singular points z ∈ F. In this case

Ind B_{Γ} = ∑_{z∈F} κ_{z}(p)  where κ_{z}(p) = \{ \begin{matrix}1 & if \frac{1}{p} > a_{+}^{z} \\ 0 & if \frac{1}{p} < a_{−}^{z}. \end{matrix}  (33)

Proof. The Fredholm criterion for B_{Γ} follows from Simonenko’s local principle [20] and Theorem 21. The function Δ(ζ) = 1 − [\frac{sinh[|θ−π|ζ]}{sinh π(ζ)}]^{2} (ζ = λ + iη) is analytic in the strip Imζ ∈ (0, 1), it extends continuously onto the real axis, we have lim_{λ→∞} Δ(λ + iη) = 1, and Δ(ζ) has only one zero ζ = \frac{iπ}{π+|θ−π|} in this strip. It is easy to check that [argΔ(λ)]_{λ=−∞}^{∞} = 0 because Δ(λ) (λ ∈ ℝ ∪ {∞}) is a real-valued function. Thus, the argument principle gives

−\frac{1}{2π} [Δ(λ + iη)]_{λ=−∞}^{∞} = \{ \begin{matrix}1 & if η > \frac{π}{π+|θ−π|} \\ 0 & if η < \frac{π}{π+|θ−π|}. \end{matrix}  (34)

Now formulas (34) and (30) yield (33)

Acknowledgement. The author is very indebted to Prof. A. Böttcher who improved the English.

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Received 16.02.1999