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] .