Annals of Communications in Mathematics

AbstractLet 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.

AbstractLet 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.