The weighted Hardy constant

Let $\mathrm{\Omega }$ be a domain in $R^d$ and $d_{\mathrm{\Gamma }}$ the Euclidean distance to the boundary $\mathrm{\Gamma }$. We investigate whether the weighted Hardy inequality $$\mathrm{\parallel }{d}_{\mathrm{\Gamma }}^{\delta \mathrm{/}2-1}\phi {\mathrm{\parallel }}_2\le a_{\delta }\mathrm{\parallel }{d}_{\mathrm{\Gamma }}^{\delta \mathrm{/}2}(\mathrm{\nabla }\phi ){\mathrm{\parallel }}_2$$ is valid, with $\delta \ge 0$ and $a_{\delta }>0$, for all $\phi \in C_c^1({\mathrm{\Gamma }}_r)$ and all small $r>0$ where ${\mathrm{\Gamma }}_r=\{x\in \mathrm{\Omega }:d_{\mathrm{\Gamma }}(x)<r\}$. First we prove that if $\delta \in [0,2⟩$ then the inequality is equivalent to the weighted version of Davies’ weak Hardy inequality on $\mathrm{\Omega }$ with equality of the corresponding optimal constants. Secondly, we establish that if $\mathrm{\Omega }$ is a uniform domain with Ahlfors regular boundary then the inequality is satisfied for all $\delta \ge 0$, and all small $r$, with the exception of the value $\delta =2-(d-d_H)$ where $d_H$ is the Hausdorff dimension of $\mathrm{\Gamma }$. Moreover, the optimal constant $a_{\delta }(\mathrm{\Gamma })$ satisfies $a_{\delta }(\mathrm{\Gamma })\ge 2\mathrm{/}\mathrm{\mid }(d-d_H)+\delta -2\mathrm{\mid }$. Thirdly, if $\mathrm{\Omega }$ is a $C^{1,1}$-domain or a convex domain $a_{\delta }(\mathrm{\Gamma })=2\mathrm{/}\mathrm{\mid }\delta -1\mathrm{\mid }$ for all $\delta \ge 0$ with $\delta \mathrm{\ne }1$. The same conclusion is correct if $\mathrm{\Omega }$ is the complement of a convex domain and $\delta >1$ but if $\delta \in [0,1⟩$ then $a_{\delta }(\mathrm{\Gamma })$ can be strictly larger than $2\mathrm{/}\mathrm{\mid }\delta -1\mathrm{\mid }$. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators.