On Lipschitz and α-Bloch spaces on the unit polydisc

Abstract

In this paper we show that if (0, 1), the holomorphic Lipschitz space Lalpha;( ) on the unit polydisc is equivalent to the 1 - -Bloch space , as well as to the space consisting of all holomorphic functions f on , continuous on the closure of and satisfying the following condition where e i is an abbreviation for ( ). Moreover, we show that the corresponding norms , and , on these spaces are comparable, that is, there are positive constants C 1, C 2 and C 3 depending only on and n such that for every f holomorphic on .