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

Fix an integer $n\ge 2$, an exponent $1<p<\mathrm{\infty }$, and a domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$. Let ${\mathrm{\Omega }}^{\ast }\triangleq \mathrm{\Omega }\setminus \{\widehat{x}\}$ where $\widehat{x}\in \mathrm{\Omega }$. Under some further conditions, we construct optimal Hardy-weights for the Finsler $p$-Dirichlet integral $$Q_0[\phi ;{\mathrm{\Omega }}^{\ast }]\triangleq {\int }_{{\mathrm{\Omega }}^{\ast }}H(x,\mathrm{\nabla }\phi {)}^p\mathrm{d}xon{C}_c^{\mathrm{\infty }}({\mathrm{\Omega }}^{\ast }),$$ and the Finsler $p$-Dirichlet integral with a potential $$Q_V[\phi ;\mathrm{\Omega }]\triangleq {\int }_{\mathrm{\Omega }}(H(x,\mathrm{\nabla }\phi {)}^p+V\mathrm{\mid }\phi {\mathrm{\mid }}^p)\mathrm{d}xon{C}_c^{\mathrm{\infty }}(\mathrm{\Omega }),$$where $H(x,\cdot )$ is a family of norms on ${\mathbb{R}}^n$ parameterized by $x\in {\mathrm{\Omega }}^{\ast }$ or $x\in \mathrm{\Omega }$, respectively, and the potential $V$ lies in a subspace ${\widehat{M}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^q(p;\mathrm{\Omega })$ of a local Morrey space $M_{\mathrm{l}\mathrm{o}\mathrm{c}}^q(p;\mathrm{\Omega })$.