Nous étudions une nouvelle classe de symboles pour les opérateurs pseudo-différentiels et leurs calculs symboliques. À chaque algèbre commutative par rapport aux convolutions sur un réseau correspond une classe de symboles . Chaque opérateur pseudo-différentiel dans est presque diagonale par rapport aux états cohérents, et sa décroissance hors de la diagonale est décrite par l’algèbre . Les opérateurs pseudo-différentiels avec des symboles dans sont bornés sur et constituent une algèbre de Banach. Si une version du lemme de Wiener s’applique à , alors l’algèbre d’opérateurs pseudo-différentiels est fermée par rapport à l’inversion des opérateurs. La théorie contient comme un cas spécial la théorie de J. Sjöstrand et fournit une nouvelle démonstration d’un théorème de Beals sur les symboles de Hörmander dans .
We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra over a lattice we associate a symbol class . Then every operator with a symbol in is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra . Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on . If a version of Wiener’s lemma holds for , then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sjöstrand’s class and yields a new proof of a theorem of Beals about the Hörmander class .
@article{AIF_2008__58_7_2279_0, author = {Gr\"ochenig, Karlheinz and Rzeszotnik, Ziemowit}, title = {Banach algebras of pseudodifferential operators and their almost diagonalization}, journal = {Annales de l'Institut Fourier}, volume = {58}, year = {2008}, pages = {2279-2314}, doi = {10.5802/aif.2414}, zbl = {1168.35050}, mrnumber = {2498351}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_2008__58_7_2279_0} }
Gröchenig, Karlheinz; Rzeszotnik, Ziemowit. Banach algebras of pseudodifferential operators and their almost diagonalization. Annales de l'Institut Fourier, Tome 58 (2008) pp. 2279-2314. doi : 10.5802/aif.2414. http://gdmltest.u-ga.fr/item/AIF_2008__58_7_2279_0/
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