Some Notes on the Poincaré-Bertrand Formula

The aim of this present paper is to establish the Poincare-Bertrand formula for the double-layer potential on piecewise Lyapunov curve of integration.


Introduction
For repeated singular integrals, the celebrated Poincaré-Bertrand formula of Hardy 1 and Poincaré 2 plays a fundamentally important role in the theory of one-dimensional singular integral equations: where Γ is a smooth curve in R 2 , t is a fixed point on Γ, and f lies on some appropriate function space.
There have been different extensions of this Poincaré-Bertrand formula for problems with different backgrounds.For example, Mitelman and Shapiro 3 established a Poincaré-Bertrand formula for quaternion singular integrals of Cauchy type over a smooth Lyapunov surface, Kytmanov 4 has an extension for the Bochner-Martinelli integral over a smooth manifolds.
Generalizations of the Poincaré-Bertrand theorem has been the subject of research in a number of papers see 5, 6 .Important applications of the Poincaré-Bertrand theorem to nuclear physics, transport theory, condensed matter physics have been established by Davies et al. 6,7 .

Preliminary Material
In this section, we provide some well-known facts from classical complex analysis to be used in this paper.For more information, we refer the reader to 15, 16 .

The Cauchy-Type Integrals
We will denote by γ a closed curve in the complex plane C which contains a finite number of conical points.If the complement in γ of the union of conical points is a Lyapunov curve, then we shall refer to γ as piecewise Lyapunov curve in C. Suppose that a domain Ω with boundary γ is given in the complex plane C. Let Ω and Ω − be, respectively, the interior and exterior domains bounded by γ.Suppose that f is a continuous complex-valued function on γ.
The Cauchy-type integral of f will be denoted by K f and defined by

2.1
We now define the singular Cauchy-type integral of f as We say that f is of H ölder class with exponent λ, denoted by H λ γ , where 0 < λ ≤ 1, if there exist a constant c > 0 such that The following theorem gives the classical Sokhotski-Plemelj formulae.
Theorem 2.1 see 16 .Let Ω be a bounded domain in C with a piecewise Lyapunov boundary, and let f ∈ H λ γ , 0 < λ < 1.Then the following limits exist: and moreover, the following equalities hold: is the modified Cauchy singular integral, and α t is the angle between the one-sided tangents at point t.
As was proved in Privalov's book, page 199 17 and in the article by Alekseev 18 and also in 19 , the limit values K ± f t of the Cauchy-type integral satisfy a H ölder condition.Thus, we have that the modified singular integral operator S in 2.7 acting invariantly on H λ , 0 < λ < 1 and we have where I is the identity operator.Adding equalities 2.5 and 2.6 , and subtracting them from each other, we arrive at the formulas:
In two repeated Cauchy's principal integrals over piecewise Lyapunov curves, the order of integration can be changed according to the following Poincaré-Bertrand formula see, e.g., 12 for all t ∈ γ:

2.11
where the integrals being understood in the sense of the Cauchy principal value, α t is the angle between the one-sided tangents at the point t, and f lies on some appropriate function space.Noting that 12 we find that the formula 2.11 can be presented in the form: Our purpose is to study the Poincaré-Bertrand formula associated with double-layer potential for piecewise Lyapunov curve.Before introducing the main results, we need a few standard facts from potential theory.For a detailed exposition, we refer the reader to for example, 15, 21 .

Simple and Double Potentials
Suppose that f : γ → C is a continuous function, and we refer to the functions u f z and v f z , given by the formulae: as the simple-and double-layer potentials, respectively.Here ∂/∂ n ζ denotes partial differentiation with respect to the outward directed normal unit vector to the curve γ at a point ζ, and ds denotes arc-length on γ.Clearly, the simple-layer potential u f and double layer potential v f are holomorphic in the interior of Ω for any integrable f.Another option see, e.g., 16 is to use the simple layer potential of the form: where τ ζ is a unit tangent vector.For z t ∈ γ define 2.16

Elementary Observations
It is easy to verify that if f is real-valued then where

3.2
We have already noted that for real-valued function f we have where the complex conjugation, denoted by "Z .Thus, for f 1 , f 2 real we have

3.4
Then, we may write

Main Results
This section is devoted to the main results of the paper.
Theorem 4.1 Sokhotski-Plemelj Formulas .Let γ be a piecewise Lyapunov curve and f a complexvalued function defined on γ which belongs to H λ γ , 0 < λ < 1.Then the following limits exist: and moreover, the Sokhotski-Plemelj formulas hold: is the modified singular double layer potential, and α t is the angle between the one-sided tangents at point t, and the integral exists as an improper integral.
Proof.For a function f f 1 if 2 ∈ H λ γ, C with f 1 , f 2 real-valued, we can write: Note that v f 1 is the real part of K f 1 , and v f 2 is the real part of K f 2 .Now the conclusion follows directly from Section 2 and the Sokhotski-Plemelj formulas 2.5 and 2.6 .
Suppose that density f in 2.2 is real-valued and belongs to H μ γ × γ, R .By formula 2.13 we have for each t ∈ γ:

4.6
Multiplying both sides of 2.10 by f ζ 1 dζ 1 , integrating over γ with respect to ζ 1 and separate complex coordinates, the following equalities can be easily obtained:

4.7
Assume that a function f depends on ζ 1 only, then using 4.5 -4.9 we have Now, in terms of double layer potential V and simple-layer potential W we can easily represent the very important properties: 4.9 Remark 4.2.When γ is a smooth Lyapunov curve, the properties 4.9 coincide with properties given in 22, Section 4.1. .The proof of following lemma is straightforward.
Lemma 4.3.Assume that γ is a piecewise Lyapunov curve.Then for t ∈ γ Thus, we remark also that 4.7 implies that The following theorem is provided by using 4.11 .Its proof is virtually identical to that of 10, Theorem 4.4 , and is omitted.

4.12
If we now take a C-valued function f as f 1 if 2 (see Section 3), then comparing Theorem 4. 4 Then for all t ∈ γ:

4.13
where the integrals exist in the sense of the Cauchy principal value.
An easy consequence of Theorem 4.5 is the following corollary.
Corollary 4.6.Let Ω be a bounded domain in R 2 with piecewise Lyapunov curve.Suppose that Then for all t ∈ γ: Note that in the previous theorems, we assumed that Ω was a bounded region in R 2 .Let now γ R and we consider a function f on R, of the class L p , p > 1.So, we have to understand K f as the Lebesgue integral.In fact, the proof of the Poincaré-Bertrand formula is essentially local, and is valid almost everywhere.Thus, the following theorems hold.

4.15
where the integrals exist in the sense of the Cauchy principal value.

Theorem 4 . 7 .
If p > 1, f ∈ L p R × R, C , then, for almost all t: