The weighted Hardy constant

Let Ω be a domain in R^d and d_Γ the Euclidean distance to the boundary Γ. We investigate whether the weighted Hardy inequality ‖d_Γ^δ/2−1φ‖₂≤a_δ‖d_Γ^δ/2(∇φ)‖₂ is valid, with δ≥0 and a_δ>0, for all φ∈C_c¹(Γ_r) and all small r>0 where Γ_r={x∈Ω:d_Γ(x)<r}. First we prove that if δ∈[0,2〉 then the inequality is equivalent to the weighted version of Davies' weak Hardy inequality on Ω with equality of the corresponding optimal constants. Secondly, we establish that if Ω is a uniform domain with a locally uniform Ahlfors regular boundary then the inequality is satisfied for all δ≥0, and all small r, with the exception of the value δ=2−(d−d_H) where d_H is the Hausdorff dimension of Γ. Moreover, the optimal constant a_δ(Γ) satisfies a_δ(Γ)≥2/|(d−d_H)+δ−2|. Thirdly, if Ω is a C^1,1-domain or a convex domain a_δ(Γ)=2/|δ−1| for all δ≥0 with δ≠1. The same conclusion is correct if Ω is the complement of a convex domain and δ>1 but if δ∈[0,1〉 then a_δ(Γ) can be strictly larger than 2/|δ−1|. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators.