Hardy inequalities, Rellich inequalities and local Dirichlet forms

Abstract

First, the Hardy and Rellich inequalities are defined for the sub-Markovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition, the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain $$\Omega \subseteq \mathbf{R}^d$$ . The weighting near the boundary $$\partial \Omega $$ can be different from the weighting at infinity. Finally, these results are applied to weighted second-order operators on $$\mathbf{R}^d\backslash \{0\}$$ and to a general class of operators of Grushin type.