Subalgebras to a Wiener type algebra of pseudo-differential operators
Annales de l'Institut Fourier, Volume 51 (2001) no. 5, p. 1347-1383
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.
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).
DOI : https://doi.org/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
@article{AIF_2001__51_5_1347_0,
     author = {Toft, Joachim},
     title = {Subalgebras to a Wiener type algebra of pseudo-differential operators},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {5},
     year = {2001},
     pages = {1347-1383},
     doi = {10.5802/aif.1857},
     mrnumber = {1860668},
     zbl = {1027.35168},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2001__51_5_1347_0}
}
Subalgebras to a Wiener type algebra of pseudo-differential operators. Annales de l'Institut Fourier, Volume 51 (2001) no. 5, pp. 1347-1383. doi : 10.5802/aif.1857. https://aif.centre-mersenne.org/item/AIF_2001__51_5_1347_0/

[B] A. Boulkhemair Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. L., Tome 4 (1997), pp. 53-67 | MR 1432810 | Zbl 0905.35103

[BL] J. Bergh; J. Löfström Interpolation Spaces. An introduction, Springer-Verlag, Berlin-Heidelberg-New York (1976) | MR 482275 | Zbl 0344.46071

[DS] M. Dimassi; J. Sjöstrand Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, Cambridge, New York, Melbourne, Madrid, London Math. Soc. Lecture Note Series, Tome vol. 268 (1999) | MR 1735654 | Zbl 0926.35002

[H] L. Hörmander The Analysis of Linear Partial Differential Operators, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo Tome vol. III (1985) | MR 404822 | Zbl 0601.35001

[L] E. H. Lieb Gaussian kernels have only Gaussian maximizers, Invent. Math., Tome 102 (1990), pp. 179-208 | Article | MR 1069246 | Zbl 0726.42005

[RS] M. Reed; B. Simon Methods of modern mathematical physics, Academic Press, London-New York (1979)

[S1] J. Sjöstrand An algebra of pseudodifferential operators, Math. Res. L., Tome 1 (1994), pp. 185-192 | MR 1266757 | Zbl 0840.35130

[S2] J. Sjöstrand Wiener type algebras of pseudodifferential operators, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, 1994, Tome n°IV (1995) | Numdam | Zbl 0880.35145

[Si] B. Simon Trace ideals and their applications I, Cambridge University Press, Cambridge-London-New York-Melbourne, London Math. Soc. Lecture Note Series, Tome vol. 35 (1979) | MR 541149 | Zbl 0423.47001

[T1] J. Toft Continuity and Positivity Problems in Pseudo-Differential Calculus (1996) (Thesis, Department of Mathematics, University of Lund, Lund)

[T2] J. Toft Regularizations, decompositions and lower bound problems in the Weyl calculus, Comm. Partial Differential Equations, Tome 7-8 (2000), pp. 1201-1234 | Zbl 0963.35215

[T3] J. Toft Continuity properties in non-commutative convolution algebras with applications in pseudo-differential calculus, Bull. Sci. Math., Tome 125 (2001) | MR 1906240 | Zbl 1002.43003