On unconditional polynomial bases in L(p) and Bergman spaces

In this paper we consider unconditional bases in L(p)(T), 1 <p <infinity, p not equal 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 in H-p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.