Boundary values of analytic functions

Abstract

Let $D$ be a connected bounded domain in ${\mathbb{R}}^2$, $S$ be its boundary which is closed, connected and smooth. Let $\Phi (z)=\frac{1}{2\pi i}{\int }_S\frac{f(s)ds}{s-z}$, $f\in L^1(S)$, $z=x+iy$. Boundary values of $\Phi (z)$ on $S$ are studied. The function $\Phi (t)$, $t\in S$, is defined in a new way. Necessary and sufficient conditions are given for $f\in L^1(S)$ to be boundary value of an analytic in $D$ function. The Sokhotsky-Plemelj formulas are derived for $f\in L^1(S)$.