Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Abstract

We give sharp remainder terms of $L^p$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

23 pages; a revised version

References in corpus (6)

• On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups
• Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators
• Local Hardy and Rellich inequalities for sums of squares of vector fields
• Layer potentials, Green's formulae, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups
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Cited by in corpus (14)

• On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups
• Hardy inequalities on metric measure spaces
• Caffarelli-Kohn-Nirenberg and Sobolev type inequalities on stratified Lie groups
• $L^p$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups
• Hardy inequalities on metric measure spaces, II: The case $p>q$
• Layer potentials, Green's formulae, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups
• Uncertainty relations on nilpotent Lie groups
• Some functional inequalities for the fractional p-sub-Laplacian
• Hardy-type inequalities for the Carnot-Carathéodory distance in the Heisenberg group
• A note on stability of Hardy inequalities
• The sharp higher order Hardy--Rellich type inequalities on the homogeneous groups
• A unified approach to $L^p$ Hardy and Rellich-type inequalities in Euclidean and non-Euclidean settings
• A Bakry-Émery criterion for weighted contractivity and $L^2$-Hardy inequalities
• Finsler Hardy inequalities