Inversion of a theorem on action of analytic functions and multiplicative properties of some subclasses of the hardy space H^∞

Abstract

In this paper, function spaces V∩l ^p_A (w) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values of a function in a disk (for example, C_A,Lip _Aα). We denote by l ^p_A (w) the space of power series such that their Taylor coefficients are p-summable with weight w. For an analytic function Φ acting in a space of this type, we prove the following alternative: either Φ″(z)≡0, or the space is a Banach algebra with respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embeddingmult(V∩l ^p_A (w))↪multl ^p_A . An estimate for the l^p-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by Salem-Zygmund. Bibliography: 10 titles.