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


35S05 Pseudodifferential operators as generalizations of partial differential operators
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[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
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