Vol. 2, No. 4, 2020

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A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields

François Vigneron

Vol. 2 (2020), No. 4, 851–880
Abstract

We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.

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Keywords
Hardy inequality, Carnot group, stratified algebra, commutators, hypoelliptic vector fields, Hörmander bracket condition
Mathematical Subject Classification 2010
Primary: 43A80
Secondary: 35R03, 42B37
Milestones
Received: 14 August 2019
Accepted: 20 October 2019
Published: 10 December 2019
Authors
François Vigneron
Université Paris-Est
LAMA (UMR 8050), UPEC, UPEM, CNRS
61, avenue du Général de Gaulle
F94010 Créteil
France