Refined general weighted $L^{p}$-Hardy and Caffarelli-Kohn-Nirenberg type inequalities and identities related to the Baouendi-Grushin operator

TL;DR

This work advances the analysis of Hardy-type and Caffarelli–Kohn–Nirenberg (CKN) inequalities in the Baouendi–Grushin framework by establishing a refined weighted L^p Hardy identity for complex-valued functions under a weight condition; it provides explicit nonnegative remainder terms and a projection along the Baouendi–Grushin distance gradient. The main result (Theorem 1) yields generalized Hardy identities with remainder and a phi-term, enabling sharp lower bounds and various refinements, including logarithmic and power-logarithmic variants, as well as HPW-type inequalities. The authors then apply this framework to derive concrete refinements on $\mathbb{R}^{n}$ and $\rho$-balls, recover classical results (e.g., D'Ambrosio, Gar) as special cases, and develop general weighted CKN inequalities with remainder terms that imply HPW-type uncertainty principles. Overall, the paper provides a cohesive, weight-based approach to sharpen and extend Hardy and CK N-type inequalities in subelliptic Baouendi–Grushin geometry with explicit constants and remainder controls, applicable to both real and complex-valued functions.

Abstract

In this paper, we present a sufficient condition on a pair of nonnegative weights $v$ and $w$ such that we have a general weighted $L^{p}$-Hardy type identity. The result, for a certain choice of weights, gives weighted $L^{p}$-Hardy type inequalities and identities with explicit remainder terms, thereby improving previously known results. Furthermore, we obtain the corresponding general weighted Caffarelli-Kohn-Nirenberg type inequality with remainder terms, which, as a result, imply Heisenberg-Pauli-Weyl type inequalities.

Theorems & Definitions (24)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4: cazacu2024hardy
• Lemma 3.5: CT24
• Lemma 3.6: CT24
• Lemma 3.7: CT24
• Corollary 4.1
• Corollary 4.2
• Corollary 4.3