# Hardy inequalities, Rellich inequalities and local Dirichlet forms

Hardy inequalities, Rellich inequalities and local Dirichlet forms 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 $$\Omega \subseteq \mathbf{R}^d$$ Ω ⊆ R d . The weighting near the boundary $$\partial \Omega$$ ∂ Ω can be different from the weighting at infinity. Finally, these results are applied to weighted second-order operators on $$\mathbf{R}^d\backslash \{0\}$$ R d \ { 0 } and to a general class of operators of Grushin type. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

# Hardy inequalities, Rellich inequalities and local Dirichlet forms

, Volume 18 (3) – May 28, 2018
21 pages

/lp/springer_journal/hardy-inequalities-rellich-inequalities-and-local-dirichlet-forms-0SY3OUMx05
Publisher
Springer Journals
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
D.O.I.
10.1007/s00028-018-0454-2
Publisher site
See Article on Publisher Site

### 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 $$\Omega \subseteq \mathbf{R}^d$$ Ω ⊆ R d . The weighting near the boundary $$\partial \Omega$$ ∂ Ω can be different from the weighting at infinity. Finally, these results are applied to weighted second-order operators on $$\mathbf{R}^d\backslash \{0\}$$ R d \ { 0 } and to a general class of operators of Grushin type.

### Journal

Journal of Evolution EquationsSpringer Journals

Published: May 28, 2018

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