Sobolev type inequalities for fractional maximal functions and Riesz potentials in half spaces

Abstract

In this paper, we study Sobolev type inequalities for fractional maximal functions $M_{\mathbb{H},\nu }f$ and Riesz potentials $I_{\mathbb{H},\alpha }f$ of functions in weighted Morrey spaces of the double phase functional $\Phi (x,t)=t^p+(b(x)t{)}^q$ in the half space, where $1<p<q$ and $b(\cdot )$ is non-negative, bounded and Hölder continuous of order $\theta \in (0,1]$. We also show that the Riesz potential operator $I_{\mathbb{H},\alpha }$ embeds from weighted Morrey space of the double phase functional $\Phi (x,t)$ to weighted Campanato spaces. Finally, we treat the similar embedding for Sobolev functions.

22 pages, to appear in Studia Math