arXiv subjects

Lorenzo D'Arca

Publications and source records attributed to Lorenzo D'Arca.

3 recordsLinked to original sources

arXiv2025

A unified approach to $L^p$ Hardy and Rellich-type inequalities in Euclidean and non-Euclidean settings

We present a unified and concise method for establishing L^p Hardy and Rellich inequalities for a broad class of subelliptic operators of divergence type. The approach, based on a fundamental algebraic identity, provides explicit control on maximizing sequences and yields sharp constants in several significant cases. It applies beyond the Euclidean framework, covering the Heisenberg and Carnot group settings, and extends to a variety of subelliptic operators such as the Heisenberg-Greiner and Baouendi-Grushin operators.

math.AP↗

arXiv2025

A unified approach to Hardy-type inequalities with Bessel pairs

In this paper, we provide suitable characterisations of pairs of weights $(V,W),$ known as Bessel pairs, that ensure the validity of weighted Hardy-type inequalities. The abstract approach adopted here makes it possible to establish such inequalities also going beyond the classical Euclidean setting and also within a more general $L^p$ framework. As a byproduct of our method, we obtain explicit expressions for the maximizing functions and, in certain specific situations, we show that the associated constants are sharp. We emphasise that our approach unifies, generalises and improves several existing results in the literature.

math.AP↗

arXiv2025

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

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.

math.AP↗