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

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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 \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Donatella Danielli & Nicola Garofalo

  2. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA, 70803, USA

    Nguyen Cong Phuc

Authors
  1. Donatella Danielli
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  2. Nicola Garofalo
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  3. Nguyen Cong Phuc
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Corresponding author

Correspondence to Nicola Garofalo.

Additional information

Donatella Danielli supported in part by NSF CAREER Award DMS-0239771 and by NSF Grant DMS-0801090.

Nicola Garofalo supported in part by NSF Grant DMS-0701001 and by NSF Grant DMS-1001317.

Nguyen Cong Phuc supported in part by NSF Grant DMS-0901083.

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Danielli, D., Garofalo, N. & Phuc, N.C. Hardy–Sobolev Type Inequalities with Sharp Constants in Carnot–Carathéodory Spaces. Potential Anal 34, 223–242 (2011). https://doi.org/10.1007/s11118-010-9190-0

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  • DOI: https://doi.org/10.1007/s11118-010-9190-0

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