singular integral operators on an open arc in ...

SINGULAR INTEGRAL OPERATORS ON AN OPEN ARC IN SPACES WITH WEIGHT R. Duduchava, N. Kverghelidze and M. Tsaava Abstract. We study a singular integral operator A = aI + bSΓ with the Cauchy operator SΓ (SIO) and H¨older continuous coefficients a, b in the space H0µ (Γ, ρ) of H¨older continuous functions with an exponential ”Khvedelidze” weight. The underlying curve is an open arc. It is well known, that such operator is Fredholm if and only if, along with the ellipticity condition a2 (t) − b2 (t) 6= 0, t ∈ Γ, the so called ”Gohberg-Krupnik’s arc condition” is fulfilled (see R. Duduchava [Du1]). Based on the Poincare-Beltrami formula for a composition of singular integral operators and the celebrated N. Muskhelishvili formula describing singularities of Cauchy integral, the formula for a composition of weighted singular integral operators is proved. Using the obtained composition formula and the localization, the criterion of fredholmity of the SIO is derived in a natural way, by looking for the regularizer of the operator A and equating to 0 non-compact operators. The approach is space-independent and this is demonstrated on similar results obtained for SIOs with continuous coefficients in the Lebesgue spaces with a ”Khvedelidze” weight Lp (Γ, ρ), investigated earlier by I. Gohberg and N. Krupnik [GK1, GK2] with a different approach.

Primary 47G10; Secondary 45P05, 45E05 Cauchy singular integral operator, Arc condition, Fredholm property, Poincare-Beltramy formulae

Introduction In the present paper we study a singular operator (shortcut SIO) with the Cauchy kernel Z b(t) ϕ(τ )d(τ ) Aϕ(t) = a(t)ϕ(t) + (1) πi τ −t Γ

and continuous coefficients a, b ∈ C(Γ) in the Lebesgue space of functions ϕ ∈ Lp (Γ, ρ) for which the product ρϕ is p - integrable and ρ is an exponential ”Khvedelidze” weight ρ(t) = (t − c1 )α1 (t − c2 )α2 ,

−

1 1 < Re αj < 1 − , p p

1 < p < ∞,

j = 1, 2. (2)

Integration area is a smooth open arc Γ = (c1 , c2 ) with the endpoints c1 and c2 , oriented from the point c1 to c2 .

2

Duduchava, Kverghelidze and Tsaava

We study the operator (1) also in the space of H¨older continuous functions H0µ (Γ, ρ) with an exponential weight ρ = (t − c1 )β1 (t − c2 )β2 ,

µ < βj < µ + 1,

0 < µ < 1,

j = 1, 2.

(3)

for which the product ρϕ is H¨older -continuous with the exponent µ and the product eliminates at the endpoints c1 and c2 . It is well known that the normality condition inf |a(t) ± b(t)| = 6 0

(4)

t∈Γ

is only necessary for the operator A to be Fredholm in the Lebesgue Lp (Γ, ρ) and the H¨older H0µ (Γ, ρ) spaces with weight should be endowed the following ”arc condition” of Gohberg-Krupnik to become sufficient: Aσj (cj , ξ) := a(cj ) − (−1)j b(cj )i cot π(σj + iξ) 6= 0, −∞ < ξ < ∞,

(5)

j = 1, 2,

where    1 + aj for the Lebesgue space, σj = p  β − µ, for the H¨older space, j

0 < Re σj < 1,

j = 1, 2.

(6)

The image of the function Aσj (cj , ξ) in (5) runs a ”Gohberg-Krupnik arc” Γσj j = 1, 2. in the complex plane when the parameter ξ ranges over the real axes R = (−∞, ∞). These arcs of a circle are described as follows: 1. The arc Γσj connects the points a(cj ) − b(cj ) and a(cj ) + b(cj ), j = 1, 2. so that the straight line [a(cj ) − b(cj ), a(cj ) + b(cj )] between the indicated points is visible from any point t ∈ Γσj of the arc by the angle π Re σj ∈ [0, π], i.e., arg

t − [a(cj ) − b(cj )] = π Re σj , t − [a(cj ) + b(cj )]

j = 1, 2;

2. While moving on the interval of the straight line [a(cj ) − b(cj ), a(cj ) + b(cj )] from the point a(cj )−b(cj ) to a(cj )+b(cj ), the arc Γj is on the left-hand side if π Re σj ∈ 1 1 [0, π] < (see Fig. 1, case a.), is on the right-hand side if π Re σj ∈ [0, π] > 2 2 (see Fig. 1, case b.) and the arc coincides with the straight interval Γj = [a(cj ) − 1 b(cj ), a(cj ) + b(cj )] if π Re σj ∈ [0, π] = , j = 1, 2. 2 Let us define the spatial (3-dimensional) contour Γ∞ in the direct product of the ¯ := [−∞, ∞] compactified with two points at the infinity complex plane with the axes R + ¯ C × R as follows: Γ∞ consists of two copies of the arc Γ+ = (c+ 1 , c2 ) ”placed” in the complex plane ×{+∞} with the same orientation as the original contour Γ = (c1 , c2 ) − and Γ− = (c− 1 , c2 ) ”placed” in the complex plane ×{−∞} with the orientation counter ± to the initial orientation of Γ. The endpoints c± 1 and c1 are ”connected” pairwise with the real axes −∞ < ξ < ∞, oriented from −∞ to +∞ for the endpoints c± 1 and oriented from +∞ to −∞ for the endpoints c± (see Fig. 2). 2

Singular integral operators on an open arc

(c+ 1 , +∞)

a(cj ) − b(cj ) π σj a. σj


Fig. 2

Now we extend the definition of the function Aσj (cj , ξ) (see (5)) in the following way  a(t) + b(t), t ∈ Γ+ ,    a(c ) − b(c )i cot π(σ + iξ), t = c , 2 2 2 2 A σ (t, ξ) = −  a(t) − b(t), t ∈ Γ    a(c1 ) − b(c1 )i cot π(σ1 + iξ), t = c1 ,

−∞ < ξ < ∞,

(7)

−∞ < ξ < ∞

and call the symbol of the singular integral operator (1). Easy to ascertain, that the symbol A σ (t, ξ), is a continuous function of the argument (t, ξ) ∈ Γ∞ . If both normality (4) and arc (5) conditions are satisfied, the argument arg A σ (t, ξ) is a continuous function on the oriented ”contour” Γ∞ . Therefore, while the variables 1 (t, ξ) range through Γ∞ in the direction of orientation, the argument arg A σ (t, ξ) 2π will get an increment, which we denote by indΓ∞ Aσ , is a positive or negative integer indΓ∞ Aσ = 0, ±1, ±2, ... and is called the index of the symbol Aσ (t, ξ) 1 ind A σ := (8) [arg A σ (t, ξ)]Γ∞ . 2π Theorem 0.1. Let conditions (2) and (3) hold. For the singular integral operator (1) to be Fredholm in the weighted Lebesgue space Lp (Γ, ρ) → Lp (Γ, ρ) or in the weighted Holder space H0µ (Γ, ρ) → H0µ (Γ, ρ) it is necessary and sufficient the normality (4) and the arc conditions (5) to hold. This is reformulated equivalently as the ellipticity of the symbol of the operator (1): inf

t∈Γ ξ∈[−∞,∞]

|Aσ (t, ξ)| = 6 0

If condition (9) holds and A is a Fredholm operator, the index is: 1 Ind A = −ind Aσ := − [arg A σ (t, ξ)]Γ×[−∞,∞] . 2π

(9)

(10)

(c− 2 , −∞)

4

Duduchava, Kverghelidze and Tsaava

If the operator A : Lp (Γ, ρ) → Lp (Γ, ρ) (or the operator A : H0µ (Γ, ρ) → H0µ (Γ, ρ))) is Fredholm, then i. It is invertible from the left iff ind Aσ > 0 and dim Coker A = ind Aσ ; ii. It is invertible from the right iff ind Aσ < 0 and dim Ker A = −ind Aσ ; iii. It is inverrtible iff Ind A = 0. The proof of this theorem will be exposed in the last Section 3. Theorem 0.1 has been proved by I. Gohberg and N. Krupnik for the Lebesgue space Lp (Γ, ρ), 1 < p < ∞, in the 60-es (see [GK1, GK2] and the literature cited therein). The case of H¨older spaces with weight H0µ (Γ, ρ) has been studied in the works of one of the authors (see [Du2, Du3, Du4]). Sufficient conditions for Fredholmity of equation (1) in the class of H¨older continuous functions with singularities at knots (endpoints) [ H∗ (Γ) := H0µ (Γ, ρ) (11) 0