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The limiting case of the fractional Caffarelli-Kohn-Nirenberg inequality in dimension one

Maria del Mar Gonzalez, Ali Hyder, Mariel Saez

Abstract

In this paper we study the fractional Caffarelli-Kohn-Nirenberg inequality (CKN) in one dimension when the parameter $γ$ converges (from the left) to its critical value $1/2$, obtaining Onofri's inequality in the unit disk as the limit. A difficulty that we encounter is the lack of an explicit expression for the extremal function at which the CKN inequality is attained, which we address by studying solutions of the weighted Liouville equation for the half-Laplacian in dimension one.

The limiting case of the fractional Caffarelli-Kohn-Nirenberg inequality in dimension one

Abstract

In this paper we study the fractional Caffarelli-Kohn-Nirenberg inequality (CKN) in one dimension when the parameter converges (from the left) to its critical value , obtaining Onofri's inequality in the unit disk as the limit. A difficulty that we encounter is the lack of an explicit expression for the extremal function at which the CKN inequality is attained, which we address by studying solutions of the weighted Liouville equation for the half-Laplacian in dimension one.

Paper Structure

This paper contains 11 sections, 19 theorems, 227 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Preliminaries
  3. Function spaces
  4. An equivalent formulation
  5. Operation rules
  6. Regularity of minimizers
  7. Technical lemmas
  8. Proof of Theorem \ref{['regularity']}
  9. Compactness results
  10. Proof of Theorem \ref{['main-theorem']}
  11. Extending the inequality

Key Result

Theorem 1.1

Assume that $0<\gamma<1$, $0\leq p(\beta-\alpha)< 2 \gamma<n$ and $c_{\text{CKN}}<c_{\text{H}}$. Then:

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 28 more