Hardy-Sobolev Type Inequalities with Sharp Constants in Carnot-Carathéodory Spaces

Donatella Danielli; Nicola Garofalo; Nguyen Cong Phuc

doi:10.1007/s11118-010-9190-0

Hardy-Sobolev Type Inequalities with Sharp Constants in Carnot-Carathéodory Spaces

Donatella Danielli, Nicola Garofalo, Nguyen Cong Phuc

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

27 Scopus citations

Abstract

We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot-Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case p ne; 2. We also obtain a sharp inequality of Hardy-Sobolev type.

Original language	English (US)
Pages (from-to)	223-242
Number of pages	20
Journal	Potential Analysis
Volume	34
Issue number	3
DOIs	https://doi.org/10.1007/s11118-010-9190-0
State	Published - Apr 2011
Externally published	Yes

Keywords

Carnot groups
Carnot-Carathéodory spaces
Hardy type inequalities
Horizontal p-Laplacian

ASJC Scopus subject areas

Analysis

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10.1007/s11118-010-9190-0

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title = "Hardy-Sobolev Type Inequalities with Sharp Constants in Carnot-Carath{\'e}odory Spaces",

abstract = "We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot-Carath{\'e}odory spaces associated with a system of vector fields of H{\"o}rmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case p ne; 2. We also obtain a sharp inequality of Hardy-Sobolev type.",

keywords = "Carnot groups, Carnot-Carath{\'e}odory spaces, Hardy type inequalities, Horizontal p-Laplacian",

author = "Donatella Danielli and Nicola Garofalo and Phuc, {Nguyen Cong}",

note = "Funding Information: Nicola Garofalo supported in part by NSF Grant DMS-0701001 and by NSF Grant DMS-1001317. Funding Information: Nguyen Cong Phuc supported in part by NSF Grant DMS-0901083. Funding Information: Donatella Danielli supported in part by NSF CAREER Award DMS-0239771 and by NSF Grant DMS-0801090.",

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doi = "10.1007/s11118-010-9190-0",

language = "English (US)",

volume = "34",

pages = "223--242",

journal = "Potential Analysis",

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T1 - Hardy-Sobolev Type Inequalities with Sharp Constants in Carnot-Carathéodory Spaces

AU - Danielli, Donatella

AU - Garofalo, Nicola

AU - Phuc, Nguyen Cong

N1 - Funding Information: Nicola Garofalo supported in part by NSF Grant DMS-0701001 and by NSF Grant DMS-1001317. Funding Information: Nguyen Cong Phuc supported in part by NSF Grant DMS-0901083. Funding Information: Donatella Danielli supported in part by NSF CAREER Award DMS-0239771 and by NSF Grant DMS-0801090.

PY - 2011/4

Y1 - 2011/4

N2 - We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot-Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case p ne; 2. We also obtain a sharp inequality of Hardy-Sobolev type.

AB - We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot-Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case p ne; 2. We also obtain a sharp inequality of Hardy-Sobolev type.

KW - Carnot groups

KW - Carnot-Carathéodory spaces

KW - Hardy type inequalities

KW - Horizontal p-Laplacian

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DO - 10.1007/s11118-010-9190-0

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JO - Potential Analysis

JF - Potential Analysis

IS - 3

ER -

Hardy-Sobolev Type Inequalities with Sharp Constants in Carnot-Carathéodory Spaces

Abstract

Keywords

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