Subalgebras to a Wiener type algebra of pseudo-differential operators

Toft, Joachim

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 $S_w^{\infty }=S_w$ 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 $\mathrm{Op}\left(S_w^p\right)\mathrm{Op}\left(S_w^r\right)\subset \mathrm{Op}\left(S_w^r\right)$ et que $S_w^p·S_w^q\subset S_w^r$ si $1/p+1/q=1/r$. Si par contre $1/p+1/q=1+1/r$, alors $S_w^{p}w*S_w^q\subset S_w^r$. 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 $S_w^{\infty }=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in $\mathrm{Op}\left(S_w^p\right)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that $\mathrm{Op}\left(S_w^p\right)\mathrm{Op}\left(S_w^r\right)\subset \mathrm{Op}\left(S_w^r\right)$ and $S_w^p·S_w^q\subset S_w^r$, provided $1/p+1/q=1/r$. If instead $1/p+1/q=1+1/r$, then $S_w^{p}w*S_w^q\subset S_w^r$. By modifying the definition of the $S_w^p$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.

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