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

Michael Ruzhansky; Durvudkhan Suragan

doi:10.1016/j.aim.2017.07.020

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

Michael Ruzhansky
, Durvudkhan Suragan

Research output: Contribution to journal › Article › peer-review

68 Citations (Scopus)

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.

Original language	English
Pages (from-to)	799-822
Number of pages	24
Journal	Advances in Mathematics
Volume	317
DOIs	https://doi.org/10.1016/j.aim.2017.07.020
Publication status	Published - Sept 7 2017
Externally published	Yes

Funding

The authors were supported in parts by the EPSRC grants EP/K039407/1 and EP/R003025/1, and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK grant 5127/GF4. The second author was also supported by the Ministry of Science of the Russian Federation (the Agreement number 02.a03.21.0008).

Keywords

Homogeneous Lie group
L-Hardy inequality
Rellich inequality
Uncertainty principle
Weighted Hardy inequality

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2017.07.020

Cite this

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abstract = "We give sharp remainder terms of Lp 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.",

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N2 - We give sharp remainder terms of Lp 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.

AB - We give sharp remainder terms of Lp 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.

KW - Homogeneous Lie group

KW - L-Hardy inequality

KW - Rellich inequality

KW - Uncertainty principle

KW - Weighted Hardy inequality

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DO - 10.1016/j.aim.2017.07.020

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SN - 0001-8708

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EP - 822

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

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

Abstract

Funding

Keywords

ASJC Scopus subject areas

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