Some results of topological genericity

Abstract

We show topological genericity for the set of functions in the space X, where X denotes the intersection of the Hardy spaces H^p with p<1, on the open unit disc such that the sequence of Taylor coefficients of the function and of all derivatives of the function are unbounded. Results of similar nature are valid when the space X is replaced by H^p(0 < p < 1) and by localized versions of such spaces. Looking at the smaller space A(D) \subseteq H^{\infty} we show topological genericity for the set of functions in A(D) and of all derivatives such that the sequence of Taylor coefficients of the function are outside of (\el)^1. We also show topological genericity for the set of functions in the space Y, where Y denotes the intersection of the harmonic Hardy spaces h^p with p<1, whose harmonic conjugate does not belong in any h^q (q > 0)