The $A$-integral and boundary values of analytic functions

T. S. Salimov

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 23-39

Résumé

Let $G$ be a simply connected bounded domain on the complex plane $\mathbf C$, let $\gamma=\partial G$, and assume that $\gamma$ is a closed rectifiable Jordan curve. Denote by $m$ the Lebesgue linear measure on $\gamma$. For a function $F$ analytic on $G$ and for $\alpha>1$ let $F_\alpha^*(t)=\sup\{|F(z)|:z\in G,\ |z-t|<\alpha\rho(z,\gamma)\}$, $t\in\gamma$, where $\rho(z,\gamma)$ is the Euclidean distance from $z$ to $\gamma$. It is proved that if for some $\alpha>2$ \begin{equation} m\{t\in\gamma:F^*_\alpha(t)>\lambda\}=o(\lambda^{-1}),\qquad\lambda\to+\infty, \end{equation} then $F$ has a finite nontangential boundary value $F(t)$ for almost all $t\in\gamma$, and $$ (A)\int_\gamma F(t)\,dt=0, $$ where the integral on the left-hand side is understood as an $A$-integral. It is also proved that under condition (1) the function $F$ is representable in $G$ by the Cauchy $A$-integral of its nontangential boundary values on $\gamma$. Further, if $\gamma$ is regular (i.e., $m\{t\in\gamma:|t-z|\leqslant r\}\leqslant Cr$ for all $z\in\mathbf C$ and $r>0$, where the constant $C$ is independent of $z$ and $r$), then these assertions are valid if condition (1) holds for some $\alpha>1$. The question of representability of integrals of Cauchy type by Cauchy $A$-integrals is studied. In particular, well-known results of Ul'yanov on this question are carried over to the case of domains with a regular boundary. It is proved that the condition of regularity of the boundary cannot be weakened here. Bibliography: 18 titles.

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