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