Existence of quasiconformal mappings in a given Hardy space

Abstract

Let Ω \Omega be a simply connected domain in C \mathbb {C} and let 0 > p > ∞ 0>p>\infty . We show that there is a quasiconformal mapping f f from the unit disk D \mathbb {D} onto Ω \Omega which is in the Hardy space H p H^p .

We furthermore show that either all quasiconformal mappings from D \mathbb {D} onto Ω \Omega are in  H p H^p for every p p , or for every 0 > p > ∞ 0>p>\infty there is a quasiconformal mapping f : D → Ω f\colon \mathbb {D}\to \Omega with f ∉ H p f\notin H^p .