NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES

We establish necessary conditions for domains Omega subset of R(n) which admit, the pointwise (p, beta)-Hardy inequality vertical bar u(x)vertical bar <= Cd(Omega)(x)(1-beta/p)M(2d Omega(x),q)(vertical bar del u parallel to d(Omega)(beta/p))(x), u is an element of C(0)(infinity)(Omega), where 1 < q < p, d(Omega)(x) = dist(x,partial derivative Omega), and M(R,q) is a maximal operator. In particular, the complement of such a domain must have. even locally, Hausdorff dimension strictly greater than n - p + beta.