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

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 H\"older 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.