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Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces
Author(s) -
Franců Martin
Publication year - 2017
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.201500418
Subject(s) - mathematics , carnot cycle , isoperimetric inequality , sobolev space , pure mathematics , standard probability space , bounded function , lp space , type (biology) , invariant (physics) , lorentz transformation , interpolation space , mathematical analysis , sobolev inequality , mathematical physics , functional analysis , banach space , thermodynamics , ecology , physics , classical mechanics , biology , biochemistry , chemistry , gene
A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.