Optimal Hardy-weights for the Finsler p-Dirichlet integral with a potential

Abstract

Fix an integer \(n\geq 2\), an exponent \(1<p<\infty\), and a domain \(\Omega\subseteq\mathbb{R}^{n}\). Let \(\Omega^{*}\triangleq\Omega\setminus\{\hat{x}\}\), where \(\hat{x}\in\Omega\). Under some further conditions, we construct optimal Hardy-weights for the Finsler \(p\)-Dirichlet integral \[ Q_{0}[\phi;\Omega^{*}]\triangleq\int_{\Omega^{*}}H(x,\nabla \phi)^{p}\,\mathrm{d}x\quad \mbox{on}\ \, C^{\infty}_{c}(\Omega^{*}), \] and the Finsler \(p\)-Dirichlet integral with a potential \[ Q_{V}[\phi;\Omega]\triangleq\int_{\Omega}\left(H(x,\nabla \phi)^{p}+ V|\phi|^{p}\right)\,\mathrm{d}x\quad \mbox{on}\ \, C^{\infty}_{c}(\Omega), \] where \(H(x,\cdot)\) is a family of norms on \(\mathbb{R}^{n}\) parameterized by \(x\in\Omega^{*}\) or \(x\in\Omega\), respectively, and the potential \(V\) lies in a subspace \(\widehat{M}^{q}_{{\rm loc}}(p;\Omega)\) of a local Morrey space \(M^{q}_{{\rm loc}}(p;\Omega)\).