Composition operators in generalized Morrey spaces

Abstract

Let $\Omega$ be an open subset of ${\mathbb{R}}^n$. Let $f$ be a Borel measurable function from $\mathbb{R}$ to $\mathbb{R}$. We prove necessary and sufficient conditions on $f$ in order that the composite function $T_f[g]=f\circ g$ belongs to a generalized Morrey space ${\mathcal{M}}_p^w(\Omega )$ whenever $g$ belongs to ${\mathcal{M}}_p^w(\Omega )$. Then we prove necessary conditions and sufficient conditions on $f$ in order that the composition operator $T_f[\cdot ]$ be continuous, uniformly continuous, Hölder continuous and Lipschitz continuous in ${\mathcal{M}}_p^w(\Omega )$. We also consider its ‘vanishing’ generalized Morrey subspace ${\mathcal{M}}_p^{w,0}(\Omega )$ and prove the related results for the composition operator as operator acting from ${\mathcal{M}}_p^{w,0}(\Omega )$ to ${\mathcal{M}}_p^w(\Omega )$ and also between the spaces ${\mathcal{M}}_p^{w,0}(\Omega )$. For the uniform, Hölder and Lipschitz continuity, we also have conditions that are both necessary and sufficient. We also have both necessary and sufficient conditions for the continuity under certain additional natural assumptions. We also consider the most commonly used Morrey classes that are related to power-type weights in the context of a discussion of some of the conditions that we impose on the weights.