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Sharp fractional Sobolev and related inequalities on H-type groups

Yaojun Wang, Qiaohua Yang

Abstract

We determine the sharp constants for the fractional Sobolev inequalities associated with the conformally invariant fractional powers $\mathcal{L}_{s}(0<s<1)$ of the sublaplacian on H-type groups. From these inequalities we derive a sharp log-Sobolev inequality by considering a limiting case and a sharp Sobolev trace inequality. The later extends to this context the result of Frank, González, Monticelli and Tan (Adv. Math, 2015).

Sharp fractional Sobolev and related inequalities on H-type groups

Abstract

We determine the sharp constants for the fractional Sobolev inequalities associated with the conformally invariant fractional powers of the sublaplacian on H-type groups. From these inequalities we derive a sharp log-Sobolev inequality by considering a limiting case and a sharp Sobolev trace inequality. The later extends to this context the result of Frank, González, Monticelli and Tan (Adv. Math, 2015).

Paper Structure

This paper contains 8 sections, 14 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Sublaplacian on H-type groups
  4. Representation theory on H-type groups
  5. Green's function of $\mathcal{L}_{s}\;(0<s<n+1)$
  6. Ground state representation of $\mathcal{L}_s$ and Hardy's inequalities
  7. Proof of Theorem \ref{['th1.2']}
  8. Proof of Theorem \ref{['th1.3']} and \ref{['th1.4']}

Key Result

Theorem \oldthetheorem

Let $0<\lambda<Q$ and $p=\frac{2Q}{2Q-\lambda}$. Then for any $f,g\in L^{p}(\mathbb{H}^{n})$, with equality if and only if, up to group translations and dilations, for some $c,c'\in\mathbb{C}$.

Theorems & Definitions (22)

  • Theorem \oldthetheorem: Frank-Lieb
  • Theorem \oldthetheorem
  • Corollary 1.1
  • Remark 1.1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 12 more