Caffarelli-Kohn-Nirenberg Inequalities in Weak Lebesgue Spaces

TL;DR

This paper develops weak-type Caffarelli-Kohn-Nirenberg inequalities using harmonic analysis, which remain valid at critical parameters where classical inequalities do not, extending to Hardy inequalities and various weights.

Contribution

It introduces a unified harmonic analysis approach to derive weak-type inequalities at endpoint cases for a broad class of weights and parameters.

Findings

Weak-type inequalities valid at critical parameters

Extension to Hardy inequalities in critical dimension

Applicable to homogeneous, non-homogeneous, and anisotropic weights

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.