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Fractional Caffarelli-Kohn-Nirenberg type inequalities on the Heisenberg group

Rama Rawat, Haripada Roy, Prosenjit Roy

Abstract

The aim of this work is to establish some cases of the Caffarelli-Kohn-Nirenberg inequalities on the Heisenberg group for the fractional Sobolev spaces. Here we work with the fractional Sobolev spaces as given by Adimurthi and Mallick in [1]. Our inequalities also give an improvement on the range of indices for the Hardy type inequality established in [1].

Fractional Caffarelli-Kohn-Nirenberg type inequalities on the Heisenberg group

Abstract

The aim of this work is to establish some cases of the Caffarelli-Kohn-Nirenberg inequalities on the Heisenberg group for the fractional Sobolev spaces. Here we work with the fractional Sobolev spaces as given by Adimurthi and Mallick in [1]. Our inequalities also give an improvement on the range of indices for the Hardy type inequality established in [1].
Paper Structure (5 sections, 20 theorems, 181 equations)

This paper contains 5 sections, 20 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Some basic properties of fractional Sobolev spaces
  3. Proof of Theorem \ref{['frac CKN in H^n']}
  4. Proof of Theorem \ref{['frac CKN in H^n b']}
  5. Proof of the limiting case

Key Result

Theorem 1

frac-CKN Let $n,p,s,q,\tau,a, \beta,\gamma,\eta_1,\eta_2$ and $\eta$ be as above satisfying the conditions condition 1, condition 2 and condition 3. (i) If $\frac{1}{\tau}+\frac{\gamma}{n}>0$, then we have (ii) If $\frac{1}{\tau}+\frac{\gamma}{n}<0$, then we have

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Proposition 2.1
  • proof
  • ...and 29 more