Recoverability of some classes of analytic functions from their boundary values

Abstract:

The technique devised by D. J. Patil to recover the functions of the Hardy space ${H^p}(1 \leqslant p \leqslant \infty )$ from the restrictions of their boundary values to a set of positive measure on the unit circle was modified by S. E. Zarantonello in order to extend the result to ${H^p}(0 < p < 1)$. In this paper, we show that Zarantonello’s technique can be slightly modified to extend the result to a larger class of analytic functions in the unit disc. In particular, if $f(z)$ is analytic in the unit disc and satisfies \[ \lim \limits _{r \to 1} {(1 - r)^\beta }\log M(r,f) = 0\quad {\text {for}}\;{\text {some}}\;\beta \geqslant 1,\] then $f(z)$ can be recovered from the restriction of its boundary value to an open arc.