Potential Analysis

, Volume 34, Issue 3, pp 223–242 | Cite as

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

  • Donatella Danielli
  • Nicola Garofalo
  • Nguyen Cong Phuc
Article

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.

Keywords

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

Mathematics Subject Classifications (2010)

35H20 35H30 

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Donatella Danielli
    • 1
  • Nicola Garofalo
    • 1
  • Nguyen Cong Phuc
    • 2
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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