Integrability for Hardy operators of double phase

Abstract

We establish Hardy–Sobolev and Hardy–Trudinger inequalities in weighted Orlicz spaces on ${\mathbb{R}}^n$. As an application, we prove Hardy–Sobolev and Hardy–Trudinger inequalities in the framework of general double phase functionals given by

where $p>1$, ${\phi }_1,{\phi }_2$ are positive convex functions on $(0,\infty )$ and $b$ is a non-negative function on $[0,\infty )$ which is Hölder continuous of order $\theta \in (0,1]$.