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L 1 Sobolev estimates for (pseudo)‐differential operators and applications
Author(s) -
Hounie Jorge,
Picon Tiago
Publication year - 2016
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.201500017
Subject(s) - mathematics , sobolev space , constant coefficients , differential operator , mathematical proof , operator (biology) , elliptic operator , homogeneous , pure mathematics , a priori and a posteriori , mathematical analysis , principal part , linear map , combinatorics , biochemistry , chemistry , philosophy , geometry , epistemology , repressor , transcription factor , gene
In this work we show that if A ( x , D ) is a linear differential operator of order ν with smooth complex coefficients in Ω ⊂ R Nfrom a complex vector space E to a complex vector space F , the Sobolev a priori estimate∥ u ∥ W ν − 1 , N / ( N − 1 )≤ C∥ A ( x , D ) u ∥ L 1holds locally at any pointx 0 ∈ Ω if and only if A ( x , D ) is elliptic and the constant coefficient homogeneous operatorA ν ( x 0 , D )is canceling in the sense of Van Schaftingen for everyx 0 ∈ Ω which means that⋂ ξ ∈ R N ∖ { 0 }a ν ( x 0 , ξ ) [ E ] = { 0 } . HereA ν ( x , D )is the homogeneous part of order ν of A ( x , D ) anda ν ( x , ξ )is the principal symbol of A ( x , D ) . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.