Fractional Poincaré inequalities for general measures

Clément Mouhot; Emmanuel Russ; Yannick Sire

doi:10.1016/j.matpur.2010.10.003

Fractional Poincaré inequalities for general measures

Clément Mouhot, Emmanuel Russ, Yannick Sire

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Abstract

We prove a fractional version of Poincaré inequalities in the context of Rn endowed with a fairly general measure. Namely we prove a control of an L2 norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures. © 2010 Elsevier Masson SAS.

Original language	English (US)
Pages (from-to)	72-84
Number of pages	13
Journal	Journal de Mathématiques Pures et Appliquées
Volume	95
Issue number	1
DOIs	https://doi.org/10.1016/j.matpur.2010.10.003
State	Published - Jan 2011
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: The first author would like to thank the Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST) for the funding provided in Cambridge University.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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@article{c4f02668059f42508f4d150be019afc8,

title = "Fractional Poincar{\'e} inequalities for general measures",

abstract = "We prove a fractional version of Poincar{\'e} inequalities in the context of Rn endowed with a fairly general measure. Namely we prove a control of an L2 norm by a non-local quantity, which plays the role of the gradient in the standard Poincar{\'e} inequality. The assumption on the measure is the fact that it satisfies the classical Poincar{\'e} inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincar{\'e} inequality for measures more general than L{\'e}vy measures. {\textcopyright} 2010 Elsevier Masson SAS.",

author = "Cl{\'e}ment Mouhot and Emmanuel Russ and Yannick Sire",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: The first author would like to thank the Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST) for the funding provided in Cambridge University. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2011",

month = jan,

doi = "10.1016/j.matpur.2010.10.003",

language = "English (US)",

volume = "95",

pages = "72--84",

journal = "Journal de Math{\'e}matiques Pures et Appliqu{\'e}es",

issn = "0021-7824",

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T1 - Fractional Poincaré inequalities for general measures

AU - Mouhot, Clément

AU - Russ, Emmanuel

AU - Sire, Yannick

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: The first author would like to thank the Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST) for the funding provided in Cambridge University. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2011/1

Y1 - 2011/1

N2 - We prove a fractional version of Poincaré inequalities in the context of Rn endowed with a fairly general measure. Namely we prove a control of an L2 norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures. © 2010 Elsevier Masson SAS.

AB - We prove a fractional version of Poincaré inequalities in the context of Rn endowed with a fairly general measure. Namely we prove a control of an L2 norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures. © 2010 Elsevier Masson SAS.

UR - http://hdl.handle.net/10754/598363

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021782410001200

UR - http://www.scopus.com/inward/record.url?scp=78650232560&partnerID=8YFLogxK

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DO - 10.1016/j.matpur.2010.10.003

M3 - Article

SN - 0021-7824

VL - 95

SP - 72

EP - 84

JO - Journal de Mathématiques Pures et Appliquées

JF - Journal de Mathématiques Pures et Appliquées

IS - 1

ER -

Fractional Poincaré inequalities for general measures

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