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 $$Q_{0}[\phi;\Omega^{*}]\triangleq\int_{\Omega^{*}}H(x,\nabla \phi)^{p}\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(\Omega^{*}),$$ and the Finsler $p$-Dirichlet integral with a potential $$Q_{V}[\phi;\Omega]\triangleq\int_{\Omega}\left(H(x,\nabla \phi)^{p}+ V|\phi|^{p}\right)\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(\Omega),$$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 $M^{q}_{{\rm loc}}(p;\Omega)$.