L1 Sobolev estimates for (pseudo)-differential operators and applications

• Published in: Mathematische Nachrichten 2016-10, Vol.289 (14-15), p.1838-1854
• Main Authors: Hounie, Jorge, Picon, Tiago
• Format: Article
• Language: English
• Subjects: 35F05; 35F35; 35J48; 35N10; 35S05; 35S05; Secondary: 35B45; cancelling operators; Elliptic complexes; L1 estimates; Primary: 46E35; pseudo-complex; Secondary: 35B45
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.201500017

In this work we show that if A(x,D) is a linear differential operator of order ν with smooth complex coefficients in Ω⊂RN from a complex vector space E to a complex vector space F, the Sobolev a priori estimate ∥u∥Wν−1,N/(N−1)≤C∥A(x,D)u∥L1holds locally at any point x0∈Ω if and only if A(x,D) is elliptic and the constant coefficient homogeneous operator Aν(x0,D) is canceling in the sense of Van Schaftingen for every x0∈Ω which means that ⋂ξ∈RN∖{0}aν(x0,ξ)[E]={0}.Here Aν(x,D) is the homogeneous part of order ν of A(x,D) and aν(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.