Title: Hardy inequalities, Rellich inequalities and local Dirichlet forms,
Abstract: First the Hardy and Rellich inequalities are defined for the submarkovian
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 $\Omega\subseteq \Ri^d$. 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 $\Ri^d\backslash\{0\}$ and to
a general class of operators of Grushin type.