Caffarelli-Kohn-Nirenberg Inequalities in Weak Lebesgue Spaces

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.