Abstract. We present a unified approach to improved L p Hardy inequalities in R N . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension 1 < k < N. In our main result, we add to the right hand side of the classical Hardy inequality a weighted L p norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted L q norms, q = p.

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets ø(Ω) parametrized by Lipschitz homeomorphisms ø defined on a fixed reference domain Ω. For two open sets ø(Ω) and \tildeø(Ω) we estimate the variation of resolvents, eigenvalues, and eigenfunctions via the Sobolev norm ||\tildeø -ø f||_{W^{1,p}(Ω)} for finite values of p, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem

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We obtain a series improvement to higher-order L p -Rellich inequalities on a Riemannian manifold M. The improvement is shown to be sharp as each new term of the series is added. Mathematics Subject Classification (2000) 35J35 (35P15, 26D10)

We obtain Sobolev inequalities for the Shcro¨dinger operator ÀD À V; where V has critical behaviour V ðxÞ ¼ ððN À 2Þ=2Þ 2 jxj À2 near the origin. We apply these inequalities to obtain point-wise estimates on the associated heat kernel, improving upon earlier results. r