Weighted Poincaré inequality and Hardy improvements related to some degenerate elliptic differential operators

2024/07/15 by Lorenzo D'Arca, D'Arca, Lorenzo · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2407.10840

openalex publication_date 2024/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we characterize the sharp constant and maximizing functions for weighted Poincaré inequalities. These results lead to refinements of Hardy's inequality obtained by adding remainder terms involving \(L^p\) norms. We use techniques that avoid symmetric rearrangement argument, simplifying the analysis of these inequalities in both Euclidean and non-Euclidean contexts. Specifically, this method applies to a variety of settings, such as the Heisenberg group, various Carnot groups and operators expressed as sums of squares of vector fields. Significant examples include the Heisenberg-Greiner operator and the Baouendi-Grushin operator.

Cited by

• Sharp remainder terms and stability of weighted Hardy-Poincaré and Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator
• Sharp remainder of the L^p-Poincaré inequality for Baouendi-Grushin vector fields

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