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A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities

Mengxia Dong

TL;DR

The work presents a unified Hölder–Lebesgue framework via the scale $X^p$ and its higher-order variants $X^{k,p,a}$ to extend Caffarelli–Kohn–Nirenberg inequalities beyond the classical Lebesgue setting. It introduces a two-parameter interpolation that is continuous in $(k,1/p,a)$ and bridges integrability and regularity, enabling a generalized CKN inequality on punctured domains with endpoint $p=n$ described by a localized, weighted Brezis–Wainger–Trudinger–Moser bound and a logarithmic loss. The main technical contributions include the localized weighted Hardy inequality, the generalized Sobolev embedding in the $X^p$ scale, the endpoint Trudinger–Moser inequality, and a robust real interpolation (the $K$-method) that yields a single, coherent interpolation principle across Lebesgue and Hölder regimes. Consequently, the paper unifies several classical inequalities (Hardy, Sobolev, Morrey, Trudinger–Moser) within a single weighted framework, extends CKN-type estimates across the full spectrum, and clarifies how domain geometry near the origin governs the constants. These results pave the way for higher-order and potentially fractional generalizations with broad applicability to PDE analysis on singular or punctured domains.

Abstract

We develop a unified Hölder Lebesgue scale \(X^p\) and its weighted, higher order variants \(X^{k,p,a}\) to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two parameter interpolation theorem that is continuous in the triplet \((k,1/p,a)\) and bridges integrability and regularity across the Lebesgue Hölder spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains \(Ω\subset\mathbb{R}^n\setminus\{0\}\); the dependence of the constant on \(Ω\) is characterized precisely by the (non)integrability of the weights at the origin. At the critical endpoint \(p=n\) we establish a localized, weighted Brezis Wainger type bound via Trudinger Moser together with a localized weighted Hardy lemma, yielding an endpoint CKN inequality with a logarithmic loss. Sharp constants are not pursued; rather, we prove existence of constants depending only on the structural parameters and coarse geometry of \(Ω\). Several corollaries, including a unified Hardy--Sobolev inequality, follow from the same interpolation mechanism.

A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities

TL;DR

The work presents a unified Hölder–Lebesgue framework via the scale and its higher-order variants to extend Caffarelli–Kohn–Nirenberg inequalities beyond the classical Lebesgue setting. It introduces a two-parameter interpolation that is continuous in and bridges integrability and regularity, enabling a generalized CKN inequality on punctured domains with endpoint described by a localized, weighted Brezis–Wainger–Trudinger–Moser bound and a logarithmic loss. The main technical contributions include the localized weighted Hardy inequality, the generalized Sobolev embedding in the scale, the endpoint Trudinger–Moser inequality, and a robust real interpolation (the -method) that yields a single, coherent interpolation principle across Lebesgue and Hölder regimes. Consequently, the paper unifies several classical inequalities (Hardy, Sobolev, Morrey, Trudinger–Moser) within a single weighted framework, extends CKN-type estimates across the full spectrum, and clarifies how domain geometry near the origin governs the constants. These results pave the way for higher-order and potentially fractional generalizations with broad applicability to PDE analysis on singular or punctured domains.

Abstract

We develop a unified Hölder Lebesgue scale and its weighted, higher order variants to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two parameter interpolation theorem that is continuous in the triplet \((k,1/p,a)\) and bridges integrability and regularity across the Lebesgue Hölder spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains ; the dependence of the constant on is characterized precisely by the (non)integrability of the weights at the origin. At the critical endpoint we establish a localized, weighted Brezis Wainger type bound via Trudinger Moser together with a localized weighted Hardy lemma, yielding an endpoint CKN inequality with a logarithmic loss. Sharp constants are not pursued; rather, we prove existence of constants depending only on the structural parameters and coarse geometry of . Several corollaries, including a unified Hardy--Sobolev inequality, follow from the same interpolation mechanism.

Paper Structure

This paper contains 15 sections, 9 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A localized weighted Hardy inequality
  4. Generalized Sobolev inequality
  5. The Trudinger--Moser inequality
  6. Interpolation theory and the $K$–method
  7. Interpolation inequality
  8. Endpoint Interpolation I: the $q=\infty$ case
  9. Endpoint Interpolation II: the $q<0$ case
  10. Proof of the interpolation theorem
  11. Proof of the generalized Hardy--Sobolev inequality
  12. Generalized Caffarelli--Kohn--Nirenberg inequality
  13. Proof of the generalized Caffarelli--Kohn--Nirenberg inequality
  14. Endpoint ($p=n$) logarithmic variant
  15. Outlook and further directions

Key Result

Theorem 1.1

Let $n\ge1$ and exponents satisfy Let $\Omega\subset\mathbb{R}^n\setminus\{0\}$ be a bounded open set. For any $u\in C_c^\infty(\Omega)$ with and any $\lambda\in(0,1)$, define the interpolated parameters Then $u\in X^{0,q,b}(\mathbb{R}^n)$ and there exists a constant $C>0$ such that Here $C$ depends only on the structural parameters $(n,p,r,q,a,b,c,\lambda)$ and on the domain $\Omega$, and is

Theorems & Definitions (21)

  • Theorem 1.1: Interpolation Theorem
  • Theorem 1.2: Generalized Hardy-Sobolev type inequality
  • Remark 1.3
  • Theorem 1.4: Generalized Caffarelli-Kohn-Nirenberg inequality
  • Remark 1.5: Consequences and specializations
  • Lemma 2.1: Localized weighted Hardy
  • proof
  • Corollary 2.2: Weighted Hardy in the $X^p$ notation
  • proof
  • Remark 2.3
  • ...and 11 more