Abstract
Let be a simply connected bounded domain on the complex plane , let , and assume that is a closed rectifiable Jordan curve. Denote by the Lebesgue linear measure on . For a function analytic on and for let where is the Euclidean distance from to . It is proved that if for some (1)then has a finite nontangential boundary value for almost all , and where the integral on the left-hand side is understood as an -integral. It is also proved that under condition (1) the function is representable in by the Cauchy -integral of its nontangential boundary values on . Further, if is regular (i.e., for all and , where the constant is independent of and ), then these assertions are valid if condition (1) holds for some . The question of representability of integrals of Cauchy type by Cauchy -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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