no. 5
Subalgebras to a Wiener type algebra of pseudo-differential operators
[Sous-algèbres de Wiener d'opérateurs pseudo différentiels]
Toft, Joachim1
1 Blekinge Technical University, Department of Mathematics, IHN, 371-79 Karlskrona (Suède)
Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1347-1383.

Nous étudions des propriétés générales de continuité pour une famille croissante d’espaces de Banach S w p de symboles pseudo-différentiels, où S w ∞ =S w a été introduit par J. Sjöstrand en 1993. Nous montrons que les opérateurs associés à ces symboles sont des opérateurs de Schatten-von Neumann d’ordre p sur L 2 . Nous prouvons aussi que Op (S w p ) Op (S w r )⊂ Op (S w r ) et que S w p ·S w q ⊂S w r si 1/p+1/q=1/r. Si par contre 1/p+1/q=1+1/r, alors S w p w*S w q ⊂S w r . En modifiant la définition des espaces S w p , on obtient aussi des classes de symboles apparentés aux espaces S(m,g).

We study general continuity properties for an increasing family of Banach spaces S w p of classes for pseudo-differential symbols, where S w ∞ =S w was introduced by J. Sjöstrand in 1993. We prove that the operators in Op (S w p ) are Schatten-von Neumann operators of order p on L 2 . We prove also that Op (S w p ) Op (S w r )⊂ Op (S w r ) and S w p ·S w q ⊂S w r , provided 1/p+1/q=1/r. If instead 1/p+1/q=1+1/r, then S w p w*S w q ⊂S w r . By modifying the definition of the S w p -spaces, one also obtains symbol classes related to the S(m,g) spaces.

DOI : 10.5802/aif.1857
Classification : 35S05, 47B10, 47B33, 42B99, 28E99
Keywords: pseudo-differential operators, Weyl calculus, Schatten-von Neumann classes, admissible functions, Hölder's inequality, Young's inequality
Mot clés : opérateurs pseudo différentiels, calcul de Weyl, classes de Schatten-von Neumann, fonctions admissibles, inégalités de Hölder, inégalité de Young

Toft, Joachim 1

1 Blekinge Technical University, Department of Mathematics, IHN, 371-79 Karlskrona (Suède) 
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Toft, Joachim. Subalgebras to a Wiener type algebra of pseudo-differential operators. Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1347-1383. doi : 10.5802/aif.1857. https://aif.centre-mersenne.org/articles/10.5802/aif.1857/

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