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Inequalities of Hardy–Sobolev Type in Carnot–Carathéodory Spaces

  • Donatella Danielli
  • Nicola Garofalo
  • Nguyen Cong Phuc
Part of the International Mathematical Series book series (IMAT, volume 8)

We consider various types of Hardy-Sobolev inequalities on a Carnot-Caratheodory space (Ω, d) associated to a system of smooth vector fields Χ = {Χ 1,Χ 2,…,Χm} on Rn satisfying the Hormander finite rank condition rank Lie[Χ 1,…, Ωm] ≡ n. One of our main concerns is the trace inequality \(\int\limits_\Omega ^{} {|\ell \left( x \right)} |^p V\left( x \right)dx \le C\int\limits_\Omega ^{} {|X\ell |^P dx},\ell \in C_0^\infty \left( \Omega \right),\) where V is a general weight, i.e., a nonnegative locally integrable function on Ω, and 1 < p < +∞. Under sharp geometric assumptions on the domain Ω C Rn that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.

Keywords

Local Parameter Heisenberg Group Sobolev Inequality Hardy Inequality Carnot Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Donatella Danielli
    • 1
  • Nicola Garofalo
    • 1
  • Nguyen Cong Phuc
    • 1
  1. 1.Purdue UniversityWest LafayetteUSA

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