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The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities

Huyuan Chen, Rui Chen, Daniel Hauer

Abstract

In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the order parameter \(s\in(0,1)\). We investigate its fundamental analytic and spectral properties, including its relation to the conformal logarithmic Laplacian, its spectral representation, and the explicit form of its eigenvalues and eigenfunctions. We further establish its conformal covariance law and derive the associated Yamabe-type equation, proving its equivalence to the corresponding conformal equation in \(\mathbb R^N\) through stereographic projection. Finally, we apply this framework to sharp Sobolev-type inequalities, recovering the sharp logarithmic Sobolev inequality, revealing the failure of a naive fractional--logarithmic analogue, and establishing new sharp fractional--logarithmic inequalities.

The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities

Abstract

In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the order parameter \(s\in(0,1)\). We investigate its fundamental analytic and spectral properties, including its relation to the conformal logarithmic Laplacian, its spectral representation, and the explicit form of its eigenvalues and eigenfunctions. We further establish its conformal covariance law and derive the associated Yamabe-type equation, proving its equivalence to the corresponding conformal equation in through stereographic projection. Finally, we apply this framework to sharp Sobolev-type inequalities, recovering the sharp logarithmic Sobolev inequality, revealing the failure of a naive fractional--logarithmic analogue, and establishing new sharp fractional--logarithmic inequalities.
Paper Structure (19 sections, 21 theorems, 506 equations)

This paper contains 19 sections, 21 theorems, 506 equations.

Table of Contents

  1. Introduction and Main Results
  2. Conformally Covariant Operators
  3. Conformal Fractional--Logarithmic Laplacian
  4. Sharp Sobolev-Type Inequalities
  5. The Conformal Fractional-Logarithmic Laplacian
  6. Stereographic Projection
  7. Kernel Representation and Logarithmic Dini Regularity
  8. Eigenvalues and Eigenfunctions
  9. Conformal Covariance and Yamabe-Type Problems
  10. Conformal Covariance Law
  11. Intertwining with the Euclidean Model
  12. Q-Curvature and Yamabe-Type Problems
  13. Applications to Sharp Sobolev Inequalities
  14. Sharp Logarithmic Sobolev Inequality
  15. Failure of Sharp Fractional--Logarithmic Sobolev Inequality
  16. ...and 4 more sections

Key Result

Theorem 1.1

(i). Let $s\in(0,1)$, integer $N>2s$ and $u\in C^\beta(\mathbb{S}^N)$ with $\beta>2s$. Then for every $z\in\mathbb{S}^N$, $\mathscr{P}^{s+\ln}_g u(z)$ given by eq:def-frac-log-sphere is well-defined and admits the representation where and Here $\psi$ denotes the Digamma function and $A_{N,s}$ is given in eq:constants. (ii). Assume that $u\in C^\beta(\mathbb{S}^N)$ with $\beta>2s$. For every $1\

Theorems & Definitions (42)

  • Theorem 1.1
  • Definition 1.1
  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.3
  • Theorem 1.4
  • ...and 32 more