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 $$ \varphi\_p(x,t) = \varphi\_1(t^p) + \varphi\_2((b(x)t)^p), \quad x\in \mathbb{R}^n,, t \ge 0, $$ where $p>1$, $\varphi\_1, \varphi\_2$ are positive convex functions on $(0,\infty)$ and $b$ is a non-negative function on $\[0,\infty)$ which is Hölder continuous of order $\theta \in (0,1]$.