Sharp weighted Young's inequalities and Moser-Trudinger inequalities on Heisenberg type groups and Grushin spaces

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Abstract

We obtain sharp weighted Moser-Trudinger inequalities for first-layer symmetric functions on groups of Heisenberg type, and for x-symmetric functions on the Grushin plane. To this end, we establish weighted Young's inequalities in the form ||K*W L||r,W ≤ ||K|| p,W||L||q,W, 1 + 1/r = 1/p + 1/q, for first-layer radial weights W on a general Carnot group script G sign and functions K, L:script G sign → ℝ with L first-layer symmetric. The proofs use some sharp estimates for hypergeometric functions.

Original languageEnglish (US)
Pages (from-to)357-384
Number of pages28
JournalPotential Analysis
Volume24
Issue number4
DOIs
StatePublished - Jun 2006

Keywords

  • Carnot group
  • Grushin plane
  • Moser-Trudinger inequality
  • Nonlinear potential theory
  • Young's inequality

ASJC Scopus subject areas

  • Analysis

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