**Abstract**Let D be a connected bounded domain in R^2, S be its boundary which is closed, connected, and smooth, or S=(-∞,∞). Let Φ(z) be the function defined as Φ(z)=1/(2πi) ∫S(f(s)ds)/(s-z), where f∈L^1(S) and z=x+iy. The singular integral operator Af is defined as Af: =1/(iπ) ∫S(f(s)ds)/(s-t), where t∈S. This new definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f∈L^1(S) to be the boundary value of an analytic function in D. The Sokhotsky-Plemelj formulas are derived for f∈L^1(S). Our new definition allows one to treat singular boundary values of analytic functions.

## New definition of a singular integral operator

Annals of Communications in Mathematics 2023

, 6 (4)

, 220-224

DOI: https://doi.org/10.62072/acm.2023.060402

## Distributional Boundary Values of Analytic Functions

Annals of Communications in Mathematics 2024

, 7 (1)

, 42-46

DOI: https://doi.org/10.62072/acm.2024.070104

**Abstract**

Let D be a connected bounded domain in R^{2}, S be its boundary which is closed, connected and smooth. Let Φ(z) = 1/2πi ∫S φ(s)ds/(s-z), φ ∈ X, z = x + iy, X is a Banach space of linear bounded functionals on H^{μ}, a Banach space of distributions, and H^{μ} is the Banach space of Hoelder-continuous functions on S with the usual norm. As X one can use also the space Hoelder continuous of bounded linear functionals on the Sobolev space H^{ℓ} on S. Distributional boundary values of Φ(z) on S are studied in detail. The function Φ(t), t ∈ S, is defined in a new way. Necessary and sufficient conditions are given for φ ∈ X to be a boundary value of an analytic in D function. The Cauchy formula is generalized to the case when the boundary values of an analytic function in D are tempered distributions. The Sokhotsky-Plemelj formulas are derived for φ ∈ X.

## Dirichlet Problem With Rough Boundary Values

Annals of Communications in Mathematics 2024

, 7 (2)

, 108-113

DOI: https://doi.org/10.62072/acm.2024.070204

**Abstract**Let D be a connected bounded domain in Rn, n ≥ 2, S be its boundary, which is closed and smooth. Consider the Dirichlet problem ∆u = 0 in D, u|S = f, where f ∈ L1 (S) or f ∈ H−ℓ , where H−ℓ is the dual space to the Sobolev space Hℓ := Hℓ (S), ℓ ≥ 0 is arbitrary. The aim of this paper is to prove that the above problem has a solution for an arbitrary f ∈ L1 (S) and this solution is unique and to prove similar result for rough (distributional) boundary values. These results are new. The method of its proof, based on the potential theory, is also new. Definition of the L1 (S)-boundary value and a distributional boundary value of a harmonic in D function is given. For f ∈ L1 (S) the difficulty comes from the fact that the product of an L1 (S) function times the kernel of the potential on S is not absolutely integrable. We prove that an arbitrary f ∈ H−ℓ , ℓ > 0, can be the boundary value of a harmonic function in D.