Annals of Communications in Mathematics

Let D be a connected bounded domain in R², 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.