The weighted Hardy constant

Abstract

Let Ω be a domain in Rd and dΓ the Euclidean distance to the boundary Γ. We investigate whether the weighted Hardy inequality ‖dΓδ/2−1φ‖2≤aδ‖dΓδ/2(∇φ)‖2 is valid, with δ≥0 and aδ>0, for all φ∈Cc1(Γr) and all small r>0 where Γr={x∈Ω:dΓ(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−dH) where dH is the Hausdorff dimension of Γ. Moreover, the optimal constant aδ(Γ) satisfies aδ(Γ)≥2/|(d−dH)+δ−2|. Thirdly, if Ω is a C1,1-domain or a convex domain aδ(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 aδ(Γ) can be strictly larger than 2/|δ−1|. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators.