On Hardy and Rellich inequalities

In the first part of this talk we give a brief survey of the Hardy and Rellich inequalities. Early forms of both inequalities concerned the Laplacian on $L_2(R^d\backslash\{0\})$ but in recent years they have been extended to a broad class of weighted operators on general domains $\Omega\subset R^d$. The Hardy inequality, in particular, is now well understood in many situations and has been useful in various contexts, notably for the derivation of uniqueness properties of solutions of  diffusion equations. The earliest application of this nature appears to have occurred in Leray's analysis of the Navier--Stokes equation in 1936 but subsequently there have been many applications to Schr\"odinger operators.

In the second part of this talk we describe some of my recent results which are of three types. First I will explain how the Hardy inequality can be used to derive uniqueness criteria for diffusion processes on $L_1(\Omega)$ where $\Omega$ is a domain whose boundary can be very `rough', e.g. the boundary could be a fractal of the Hutchinson--Moran type. Secondly I will describe how for many domains one can derive a strong version of the Rellich inequality from the Hardy inequality. This is significant as the existence of the  Rellich inequality is not well understood for general domains. Finally, if time allows, I will describe the derivations of optimal forms of both inequalities on domains which are the complement of convex sets.