L1 Sobolev estimates for (pseudo)-differential operators and applications
| Autor: | Jorge Hounie, Tiago Picon |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Constant coefficients
General Mathematics 010102 general mathematics Mathematical analysis Order (ring theory) A priori estimate Characterization (mathematics) Space (mathematics) Differential operator 01 natural sciences 010101 applied mathematics Sobolev space Operator (computer programming) 0101 mathematics Mathematics |
| Zdroj: | Mathematische Nachrichten. 289:1838-1854 |
| ISSN: | 0025-584X |
| Popis: | 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∥L1 holds 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. |
| Databáze: | OpenAIRE |
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