Boundedness of Hardy operators in the unit ball of double phase

Abstract

We establish Hardy-Sobolev inequalities in the unit ball $\mathbf{B}$ in the framework of general double phase functionals given by\[\varphi_p(x,t) = \varphi_1(t^p) + \varphi_2((b(x)t)^p),\quad x\in \mathbf{B}, t \ge 0,\]where $p>1$, $\varphi_1, \varphi_2$ are positive convex functions on $(0,\infty)$ and $b$ is a non-negative function on $\mathbf{B}$ which is radially Hölder continuous of order $\theta \in (0,1]$.