Some Results of Topological Genericity

Abstract

We prove that a generic function from \(\bigcap \nolimits _{p<1}H^p\) has unbounded Taylor coefficients, this applies also to the Taylor coefficients of its derivatives. Results of similar nature are valid when the space \(\bigcap \nolimits _{p<1}H^p\) is replaced by \(H^p\)(\(0<p<1\)) and by localized versions of such spaces. Moreover, we prove that a generic function from \(A(\mathbb {D})\) has Taylor coefficients outside of \(\ell ^1\), this applies also to the Taylor coefficients of its derivatives. Lastly, we prove that a generic function from \(\bigcap \nolimits _{p<1}h^p\) has a harmonic conjugate that does not belong to any \(h^q(q>0)\).