×

zbMATH — the first resource for mathematics

Une algèbre maximale d’opérateurs pseudo-différentiels. (A maximal algebra of pseudo differential operators). (French) Zbl 0659.35115
The author considers pseudo differential operators P in the class \(L^ 0_{1,1}\), i.e. \(P=p(x,D)\) where the symbol p(x,\(\xi)\) satisfies \[ | D^{\alpha}_ x D^{\beta}_{\xi} p(x,\xi)| \leq C_{\alpha \beta}(1+| \xi |)^{| \alpha | -| \beta |}. \] It is well known that P is not \(L^ 2\) continuous, in general. A characterization is given of the largest self-adjoint sub-algebra \({\mathcal A}\) of \({\mathcal L}(L^ 2)\) consisting of pseudo differential oprators in \(L^ 0_{1,1}\); precisely, the author proves that \({\mathcal A}=L^ 0_{1,1}\cap (L^ 0_{1,1})^*\), i.e. \(P\in {\mathcal A}\) if and only if the adjoint \(P^*\) is also well defined as operator in \(L^ 0_{1,1}\).
Reviewer: L.Rodino

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bourdaud G, Publ,Math.Univ.Paris 7 (1987)
[2] Coifman R, Astérisque S.M.F 57 (1978)
[3] DOI: 10.2307/2006946 · Zbl 0567.47025 · doi:10.2307/2006946
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.