Banach algebras of pseudodifferential operators and their almost diagonalization

[Algèbres de Banach d’opérateurs pseudo-différentiels et leur presque diagonalisation]

Gröchenig, Karlheinz ; Rzeszotnik, Ziemowit

Annales de l'Institut Fourier, Tome 58 (2008), p. 2279-2314 / Harvested from Numdam

Résumé

Nous étudions une nouvelle classe de symboles pour les opérateurs pseudo-différentiels et leurs calculs symboliques. À chaque algèbre $\mathcal{A}$ commutative par rapport aux convolutions sur un réseau $\Lambda $ correspond une classe de symboles $M^{\infty , \mathcal{A} } $. Chaque opérateur pseudo-différentiel dans $M^{\infty , \mathcal{A} } $ est presque diagonale par rapport aux états cohérents, et sa décroissance hors de la diagonale est décrite par l’algèbre $\mathcal{A}$. Les opérateurs pseudo-différentiels avec des symboles dans $M^{\infty ,\mathcal{A} } $ sont bornés sur $L^2(\mathbb{R}^d) $ et constituent une algèbre de Banach. Si une version du lemme de Wiener s’applique à $\mathcal{A}$, 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 $S^0_{0,0}$.

Abstract

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 $\mathcal{A} $ over a lattice $\Lambda $ we associate a symbol class $M^{\infty , \mathcal{A} } $. Then every operator with a symbol in $M^{\infty ,\mathcal{A} } $ 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 $\mathcal{A} $. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on $L^2({\mathbb{R}}^d) $. If a version of Wiener’s lemma holds for $\mathcal{A} $, 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 $S^0_{0,0}$.

Détail

Publié le : 2008-01-01

MR 2498351 | Zbl 1168.35050

DOI : https://doi.org/10.5802/aif.2414
Classification:  42C40,  35S05
Mots clés: opérateur pseudodifferentiel, classe de symboles, calcul symbolique, algèbre de Banach, lemme de Wiener

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