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Caffarelli-Kohn-Nirenberg Inequalities in Weak Lebesgue Spaces

Dinghuai Wang

TL;DR

The paper addresses endpoint versions of Caffarelli–Kohn–Nirenberg inequalities in weak Lebesgue spaces, deriving weak-type forms that remain valid at critical parameter values where the classical inequalities fail. Utilizing sparse domination and two-weight theory, it obtains multiplier weak-type estimates for fractional integrals with power weights and combines them with weighted Marcinkiewicz interpolation to prove a comprehensive weak-type CKN inequality under natural scaling and weight conditions. The work also yields a weak-type Hardy inequality in the critical dimension $p=d$ and develops a flexible framework that accommodates homogeneous, non-homogeneous, and anisotropic weights, thereby unifying various endpoint cases in interpolation theory. Additional contributions include a detailed treatment of power weights, necessary and sufficient weight conditions, and a set of variants and auxiliary results that broaden the applicability of these endpoint inequalities in harmonic analysis and PDE contexts.

Abstract

By employing harmonic analysis techniques, we derive weak-type Caffarelli-Kohn-Nirenberg inequalities under natural parameter conditions. A key feature of these weak-type versions is that they remain valid even at critical parameter values where the classical inequalities fail. As an important corollary, we obtain weak-type Hardy inequalities that hold true even in the critical dimension \(d = p\). The methods developed here are sufficiently flexible to handle homogeneous, non-homogeneous and anisotropic weights, providing a unified approach to various endpoint cases in interpolation theory.

Caffarelli-Kohn-Nirenberg Inequalities in Weak Lebesgue Spaces

TL;DR

The paper addresses endpoint versions of Caffarelli–Kohn–Nirenberg inequalities in weak Lebesgue spaces, deriving weak-type forms that remain valid at critical parameter values where the classical inequalities fail. Utilizing sparse domination and two-weight theory, it obtains multiplier weak-type estimates for fractional integrals with power weights and combines them with weighted Marcinkiewicz interpolation to prove a comprehensive weak-type CKN inequality under natural scaling and weight conditions. The work also yields a weak-type Hardy inequality in the critical dimension and develops a flexible framework that accommodates homogeneous, non-homogeneous, and anisotropic weights, thereby unifying various endpoint cases in interpolation theory. Additional contributions include a detailed treatment of power weights, necessary and sufficient weight conditions, and a set of variants and auxiliary results that broaden the applicability of these endpoint inequalities in harmonic analysis and PDE contexts.

Abstract

By employing harmonic analysis techniques, we derive weak-type Caffarelli-Kohn-Nirenberg inequalities under natural parameter conditions. A key feature of these weak-type versions is that they remain valid even at critical parameter values where the classical inequalities fail. As an important corollary, we obtain weak-type Hardy inequalities that hold true even in the critical dimension . The methods developed here are sufficiently flexible to handle homogeneous, non-homogeneous and anisotropic weights, providing a unified approach to various endpoint cases in interpolation theory.
Paper Structure (14 sections, 14 theorems, 157 equations)

This paper contains 14 sections, 14 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Sketch on the proof
  4. Organization
  5. Multiplier weak-type estimates for the sparse operator
  6. The case $p > 1$ with $(\mu, w)\in A_{p,q}^{\alpha,*}$.
  7. The case $p = 1$ with $(\mu, w)\in A_{1,q}^{\alpha}$
  8. Necessity of the conditions
  9. Sufficiency of the conditions
  10. Multiplier weak-type estimates for fractional integrals with power weights
  11. Weighted Marcinkiewicz interpolation
  12. Proof of the sufficiency part of Theorem \ref{['thm:main']}
  13. Further results
  14. Technical lemmas on power weights

Key Result

Theorem 1.1

For $d\geq 1$, let $s, p,q, \gamma_{1},\gamma_{2},\gamma_{3}$ and $\theta$ satisfy CKN-1 and CKN-2. Then there exists a positive constant $C$ such that holds for all $f\in C^1_{c}(\mathbb{R}^d)$ if and only if CKN-3-CKN-5 hold.

Theorems & Definitions (34)

  • Theorem 1.1: Caffarelli, Kohn, Nirenberg CKN1982
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 2.1
  • ...and 24 more