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

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 with the double phase functional $\Phi (x,t) = t^{p} + (b(x)t)^{q}$ in the half-space, where $1 \lt p \lt q$ and $b(\cdot )$ is nonnegative, 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 with the double phase functional $\Phi (x,t)$ to weighted Campanato spaces. Finally, we treat the similar embedding for Sobolev functions.