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

Abstract

Fix an integer $n\ge 2$, an exponent $1<p<\infty$, and a domain $\Omega \subseteq {\mathbb{R}}^n$. Let ${\Omega }^{\ast }\triangleq \Omega \setminus \{\hat{x}\}$ where $\hat{x}\in \Omega$. Under some further conditions, we construct optimal Hardy-weights for the Finsler $p$-Dirichlet integral $$Q_{0}[ϕ;Ω^{*}]\triangleq\int_{Ω^{*}}H(x,\nabla ϕ)^{p}\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(Ω^{*}),$$ 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 $x\in {\Omega }^{\ast }$ or $x\in \Omega$, respectively, and the potential $V$ lies in a subspace ${\hat{M}}_{\mathrm{loc}}^q(p;\Omega )$ of a local Morrey space $M_{\mathrm{loc}}^q(p;\Omega )$.

44 pages