Some Results of Topological Genericity

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