Continuity and differentiability of Nemytskii operators on the Hardy space $\mathcal{H}^{1,1} (T^1 )$

Abstract

Let $\mathcal{H}^{1,1} (T^1 )$ denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space $\mathcal{H}^1 (T^1 )$ . It is shown that when the weak derivative of a locally Lipschitz continuous function f has bounded variation on compact sets the Nemytskii operator F, defined by F(u)=f·u, maps $\mathcal{H}^{1,1} (T^1 )$ continuously into itself. A further condition sufficient for the continuous Fréchet differentiability of F is then added.