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Caffarelli–Kohn–Nirenberg inequalities on Lie groups of polynomial growth
Author(s) -
Yacoub Chokri
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700063
Subject(s) - mathematics , mathematical proof , pure mathematics , lp space , sobolev space , lie group , carnot cycle , polynomial , sobolev inequality , banach space , mathematical analysis , physics , geometry , thermodynamics
In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli–Kohn–Nirenberg type, where the weights involved are powers of the Carnot–Caratheodory distance associated with a fixed system of vector fields which satisfy the Hörmander condition. The use of weak L p spaces is crucial in our proofs and we formulate these inequalities within the framework of L p , qLorentz spaces (a scale of (quasi)‐Banach spaces which extend the more classical L p Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy–Sobolev inequalities.