Banach algebras of pseudodifferential operators and their almost diagonalization

Gröchenig, Karlheinz; Rzeszotnik, Ziemowit

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2279-2314.

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

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

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