On unconditional polynomial bases inL p and bergman spaces

Abstract

In this paper we consider unconditional bases inL p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.