Weighted Hardy inequalities and the size of the boundary

We establish necessary and sufficient conditions for a domain to admit the (p, β)-Hardy inequality , where d(x) = dist(x, ∂Ω) and . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ω ^c when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ω ^c was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that we try to point out by giving various examples. The author was supported in part by the Academy of Finland.