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Journal of Applied Mathematics
Volume 2012, Article ID 969685, 10 pages
Research Article

Some Notes on the Poincaré-Bertrand Formula

Department of Mathematics, Izmir University of Economics, 35330 Izmir, Turkey

Received 18 February 2012; Revised 25 February 2012; Accepted 27 February 2012

Academic Editor: Renat Zhdanov

Copyright © 2012 Baruch Schneider. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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

1. 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:1𝜋𝑖Γ𝜏𝑑𝜏1𝜏𝑡𝜋𝑖Γ𝜏1𝑓𝜏,𝜏1𝜏1𝜏𝑑𝜏11=𝑓(𝑡,𝑡)+𝜋𝑖Γ𝜏1𝑑𝜏11𝜋𝑖Γ𝜏𝑓𝜏,𝜏1(𝜏𝜏𝑡)1𝜏𝑑𝜏,(1.1) where Γ is a smooth curve in 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].

Another important extension has been achieved by Hang and Jiang [8] on a smooth hypersurfaces in higher dimensions and for more recent references under different contexts, see for instance [914].

The Poincaré-Bertrand formula concerning two repeated integral operators of the double-layer potential on a piecewise Lyapunov curve in the plane is not generally known. Indeed, the author has been unable to find any explicit reference to such a result at all.

In this paper, we obtain the Poincaré-Bertrand formula for the double layer potential on a piecewise Lyapunov curve in the plane.

2. 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].

2.1. The Cauchy-Type Integrals

We will denote by 𝛾 a closed curve in the complex plane 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 . Suppose that a domain Ω with boundary 𝛾 is given in the complex plane . 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 by𝐾[𝑓](1𝑧)=2𝜋𝑖𝛾𝑓(𝜁)𝜁𝑧𝑑𝜁,𝑧𝛾.(2.1) We now define the singular Cauchy-type integral of 𝑓 as𝑆[𝑓](1𝑡)=𝜋𝑖𝛾𝑓(𝜁)1𝜁𝑡𝑑𝜁𝜋𝑖lim𝜀0{𝜁𝛾|𝜁𝑡|𝜀}𝑓(𝜁)𝜁𝑡𝑑𝜁,𝑡𝛾.(2.2) We say that 𝑓 is of Hölder class with exponent 𝜆, denoted by 𝐻𝜆(𝛾), where 0<𝜆1, if there exist a constant 𝑐>0 such that||𝑓𝑧1𝑧𝑓2||||𝑧𝑐1𝑧2||𝜆,𝑧1,𝑧2𝛾.(2.3)

The following theorem gives the classical Sokhotski-Plemelj formulae.

Theorem 2.1 (see [16]). Let Ω be a bounded domain in with a piecewise Lyapunov boundary, and let 𝑓𝐻𝜆(𝛾), 0<𝜆<1. Then the following limits exist: limΩ±𝑧𝑡𝛾𝐾[𝑓](𝑧)=𝐾±[𝑓](𝑡),(2.4) and moreover, the following equalities hold: 𝐾+[𝑓]1(𝑡)=2𝑆[𝑓]𝐾(𝑡)+𝑓(𝑡),𝑡𝛾,(2.5)[𝑓]1(𝑡)=2𝑆[𝑓](𝑡)𝑓(𝑡),𝑡𝛾,(2.6) here, 𝑆[𝑓](𝑡)=𝜋𝛼(𝑡)𝜋[𝑓]𝑓(𝑡)+𝑆(𝑡)(2.7) 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 Hölder condition. Thus, we have that the modified singular integral operator 𝑆 in (2.7) acting invariantly on 𝐻𝜆, 0<𝜆<1 and we have𝑆2=𝐼,(2.8) where 𝐼 is the identity operator.

Adding equalities (2.5) and (2.6), and subtracting them from each other, we arrive at the formulas: 𝐾+[𝑓](𝑡)𝐾[𝑓]𝐾(𝑡)=𝑓(𝑡),𝑡𝛾,+[𝑓](𝑡)+𝐾[𝑓]𝑆[𝑓](𝑡)=(𝑡),𝑡𝛾.(2.9)

Theorem 2.2 (follows from Lemma  4.3 of [10]). Let 𝛾 be a piecewise Lyapunov curve. If 𝑡,𝜁1𝛾,𝑡𝜁1, then 𝛾𝑑𝜁𝜁(𝜁𝑡)1𝜁=0.(2.10)

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 𝑡𝛾: 12𝜋𝑖𝛾𝜁𝑑𝜁1𝜁𝑡2𝜋𝑖𝛾𝜁1𝑓𝜁,𝜁1𝜁1𝜁𝑑𝜁1=12𝜋𝑖𝛾𝜁1𝑑𝜁112𝜋𝑖𝛾𝜁𝑓𝜁,𝜁1(𝜁𝜁𝑡)1𝜁𝑑𝜁+𝛼(𝑡)2𝜋2𝑓(𝑡,𝑡),(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 1𝜁(𝜁𝑡)1=1𝜁𝜁11𝑡1𝜁𝑡𝜁𝜁1,(2.12) we find that the formula (2.11) can be presented in the form: 1𝜋𝑖𝛾𝜁𝑑𝜁1𝜁𝑡𝜋𝑖𝛾𝜁1𝑓𝜁,𝜁1𝜁1𝜁𝑑𝜁1=𝜂21(𝑡)𝑓(𝑡,𝑡)+𝜋𝑖𝛾𝜁1𝑑𝜁1𝜁11𝑡𝜋𝑖𝛾𝜁𝑓𝜁,𝜁1𝜁𝑡𝑑𝜁𝛾𝜁𝑓𝜁,𝜁1𝜁𝜁1,𝑑𝜁(2.13) where 𝜂(𝑡)=𝛼(𝑡)/𝜋.

Remark 2.3 (2.6). It is possible, and indeed desirable, to consider the analogous formulas in other spaces than the Hölder space, for example, the Banach space 𝐿𝑝(𝛾),𝑝>1. If 𝑓𝐿𝑝(𝛾), 𝑝>1, then the Sokhotski-Plemelj formulas and the Poincaré-Bertrand formula are valid almost everywhere on 𝛾.

Remark 2.4 (2.7). Another class of interesting examples is rectifiable curves. The class of rectifiable curves includes as proper subclasses many other important classes of curves, in particular, smooth (Lyapunov) curves, piecewise Lyapunov curves, and Lipschitz curves. Various properties and applications of the Cauchy type integral for hyperholomorphic functions along rectifiable curves (and domains with rectifiable boundary) can be found, for instance, in [20].

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].

2.2. Simple and Double Potentials

Suppose that 𝑓𝛾 is a continuous function, and we refer to the functions 𝑢[𝑓](𝑧) and 𝑣[𝑓](𝑧), given by the formulae:𝑢[𝑓](1𝑧)=2𝜋𝛾1𝑓(𝜁)ln||||𝜁𝑧𝑑𝑠𝜁𝜐[𝑓]1,𝑧𝛾,(𝑧)=2𝜋𝛾𝜕𝑓(𝜁)1𝜕𝑛(𝜁)ln||||𝜁𝑧𝑑𝑠𝜁,𝑧𝛾,(2.14) 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:𝑤[𝑓](1𝑧)=2𝜋𝛾𝜕𝑓(𝜁)1𝜕𝜏(𝜁)ln||||𝜁𝑧𝑑𝑠𝜁,𝑧𝛾,(2.15) where 𝜏(𝜁) is a unit tangent vector.

For 𝑧=𝑡𝛾 define 𝒱[𝑓]1(𝑡)=𝜋𝛾||||𝑓(𝜁)𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋lim𝜀0||||{𝜁𝛾𝜁𝑡𝜀}||||𝑓(𝜁)𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁,𝒲[𝑓]1(𝑡)=𝜋𝛾||||𝑓(𝜁)𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋lim𝜀0||||{𝜁𝛾𝜁𝑡𝜀}||||𝑓(𝜁)𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁.(2.16)

3. Elementary Observations

It is easy to verify that if 𝑓 is real-valued then𝑆[𝑓][𝑓][𝑓](𝑡)=𝒱(𝑡)+𝑖𝒲(𝑡),(3.1) where𝒱[𝑓]1(𝑡)=Re𝜋𝑖𝛾𝑓(𝜁),𝒲[𝑓]1𝜁𝑡𝑑𝜁(𝑡)=Im𝜋𝑖𝛾𝑓(𝜁).𝜁𝑡𝑑𝜁(3.2) We have already noted that for real-valued function 𝑓 we have𝒱[𝑓][𝑓][𝑓][𝑓],=2Re𝐾=𝐾+𝑍𝐾(3.3) where the complex conjugation, denoted by 𝑍.

Thus, for 𝑓1, 𝑓2 real we have𝒱𝑓1+𝑖𝑓2𝑓=𝒱1𝑓+𝑖𝒱2𝑓=𝐾1𝑓+𝑍𝐾1𝐾𝑓+𝑖2𝑓+𝑍𝐾2=𝑓(𝐾+𝑍𝐾𝑍)1+𝑖𝑓2.(3.4) Then, we may write𝒱=𝐾+𝑍𝐾𝑍.(3.5)

4. 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 complex-valued function defined on 𝛾 which belongs to 𝐻𝜆(𝛾), 0<𝜆<1. Then the following limits exist: limΩ±𝑧𝑡𝛾𝑣[𝑓](𝑧)=𝑣±[𝑓](𝑡),(4.1) and moreover, the Sokhotski-Plemelj formulas hold: 𝑣±[𝑓]1(𝑡)=2𝑉[𝑓]±𝑓(𝑡)+(𝑡),𝑡𝛾,(4.2) where 𝑉[𝑓](𝑡)=𝜋𝛼(𝑡)𝜋[𝑓]𝑓(𝑡)+𝑉(𝑡)(4.3) 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=𝑓1+𝑖𝑓2𝐻𝜆(𝛾,) with 𝑓1, 𝑓2 real-valued, we can write: 𝐾𝑓1+𝑖𝑓2𝑓(𝑧)=𝐾1𝑓(𝑧)+𝑖𝐾2(𝑧).(4.4) Note that 𝑣[𝑓1] is the real part of 𝐾[𝑓1], and 𝑣[𝑓2] is the real part of 𝐾[𝑓2]. 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 𝐻𝜇(𝛾×𝛾,). By formula (2.13) we have for each 𝑡𝛾:1𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝜏1𝑑𝑠𝜁1=𝜂2(1𝑡)𝑓(𝑡,𝑡)+𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝑛1𝑑𝑠𝜁1×1𝜋𝛾𝜁𝑓𝜁,𝜁1||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝜏1𝑑𝑠𝜁11𝜋𝛾𝜁𝑓𝜁,𝜁1||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕𝜏(𝜁)𝑑𝑠𝜁,1(4.5)𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕n(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝜏1𝑑𝑠𝜁1+1𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜕n𝜁1𝑑𝑠𝜁1=1𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜕n𝜁1𝑑𝑠𝜁11𝜋𝛾𝜁𝑓𝜁,𝜁1||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕𝜏(𝜁)𝑑𝑠𝜁+1𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝜏1𝑑𝑠𝜁11𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕n(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕n(𝜁)𝑑𝑠𝜁.(4.6)

Multiplying both sides of (2.10) by 𝑓(𝜁1)𝑑𝜁1, integrating over 𝛾 with respect to 𝜁1 and separate complex coordinates, the following equalities can be easily obtained:1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁||𝜕ln𝜁𝜁1||𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝜏1𝑑s𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁||𝜕ln𝜁𝜁1||𝜕𝜏(𝜁)𝑑𝑠𝜁1=0,𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁||𝜕ln𝜁𝜁1||𝜕𝜏(𝜁)𝑑𝑠𝜁+1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝜏1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁||𝜕ln𝜁𝜁1||𝜕𝑛(𝜁)𝑑𝑠𝜁=0.(4.7)

Assume that a function 𝑓 depends on 𝜁1 only, then using (4.5)–(4.9) we have1𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝜏1𝑑𝑠𝜁1=𝜂2(1𝑡)𝑓(𝑡,𝑡),𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝜏1𝑑𝑠𝜁1+1𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁1=0.(4.8)

Now, in terms of double layer potential 𝒱 and simple-layer potential 𝒲 we can easily represent the very important properties:𝒱2𝒲2=𝜂2𝐼,𝒱𝒲+𝒲𝒱=0.(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 𝑡𝛾1𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁||𝜕ln𝜁𝜁1||𝜕𝑛(𝜁)𝑑𝑠𝜁=0.(4.10)

Thus, we remark also that (4.7) implies that1𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝜏1𝑑𝑠𝜁11𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁||𝜕ln𝜁𝜁1||𝜕𝜏(𝜁)𝑑𝑠𝜁=0.(4.11) The following theorem is provided by using (4.11). Its proof is virtually identical to that of [10, Theorem  4.4], and is omitted.

Theorem 4.4. Let 𝛾 be a piecewise Lyapunov curve. Then for 𝑡𝛾: 1𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝜏1𝑑𝑠𝜁1=1𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝜏1𝑑𝑠𝜁11𝜋𝛾𝜁𝑓𝜁,𝜁1||||𝜕ln𝜁𝑡𝜕𝜏(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕𝜏(𝜁)𝑑𝑠𝜁.(4.12) If we now take a -valued function 𝑓 as 𝑓1+𝑖𝑓2 (see Section 3), then comparing Theorem 4.4 and formula (4.5) we have one of our main results, analog of the Poincaré-Bertrand formula for double layer potential.

Theorem 4.5 (Poincaré-Bertrand formula). Let Ω be a bounded domain in 2 with piecewise Lyapunov curve. Assume that 𝑓𝐻𝜆(𝛾×𝛾,),0<𝜆1. Then for all 𝑡𝛾: 1𝜋𝛾𝜁||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁1=𝜂21(𝑡)𝑓(𝑡,𝑡)+𝜋𝛾𝜁1||𝜁𝜕ln1||𝑡𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋𝛾𝜁𝑓𝜁,𝜁1||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕𝑛(𝜁)𝑑𝑠𝜁,(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 2 with piecewise Lyapunov curve. Suppose that 𝑓(𝜁,𝜁1)=𝑓(𝜁)𝐻𝜆(𝛾×𝛾,), 0<𝜆1. Then for all 𝑡𝛾: 1𝜋𝛾𝜁||||𝑓(𝜁)𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋𝛾𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁1=𝜂2(𝑡)𝑓(𝑡).(4.14)

Proof. Suppose that 𝑓(𝜁,𝜁1)=𝑓(𝜁)𝐻𝜆(𝛾×𝛾,), 0<𝜆1, then by using formula (4.10) we obtain (4.14).

Note that in the previous theorems, we assumed that Ω was a bounded region in 2. Let now 𝛾= and we consider a function 𝑓 on , of the class 𝐿𝑝,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.

Theorem 4.7. If 𝑝>1, 𝑓𝐿𝑝(×,), then, for almost all 𝑡: 1𝜋+||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋+𝑓𝜁,𝜁1||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁11=𝑓(𝑡,𝑡)+𝜋+||𝜁𝜕ln1||𝑡𝜁𝜕𝑛1𝑑𝑠𝜁11𝜋+𝑓𝜁,𝜁1||||𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋+𝑓𝜁,𝜁1||𝜕ln𝜁𝜁1||𝜕𝑛(𝜁)𝑑𝑠𝜁,(4.15) where the integrals exist in the sense of the Cauchy principal value.

Corollary 4.8. Suppose that 𝑓(𝜁,𝜁1)=𝑓(𝜁)𝐿𝑝(×,), 𝑝>1. Then for almost all 𝑡: 1𝜋+||||𝑓(𝜁)𝜕ln𝜁𝑡𝜕𝑛(𝜁)𝑑𝑠𝜁1𝜋+||𝜁𝜕ln1||𝜁𝜁𝜕𝑛1𝑑𝑠𝜁1=𝑓(𝑡).(4.16)


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