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Classes de Schatten d’opérateurs pseudo-différentiels. (French) Zbl 0574.47032
The author considers the correspondence (which goes back to H. Weyl) between a symbol $$a\in {\mathcal S}({\mathbb{R}}^{2n})$$ and the operator $$0p_{1/2}(a)$$ defined on $${\mathcal S}({\mathbb{R}}^ n)$$ as follows $0p_{1/2}(a)u(x)=\iint a(\frac{x+y}{2},\xi)e^{2i\pi <x-y,\xi >}u(y)dy d\xi.$ The main result gives a sufficient condition on a general symbol a for the operator $$0p_{1/2}(a)$$ to be in the Schatten class $$S_ p$$ as an operator acting on $$L^ 2({\mathbb{R}}^ n).$$
The condition involves a field $$\| \|_ X$$ of symplectic norms on $${\mathbb{R}}^{2n}$$ satisfying certain technical assumptions (which we do not state here) involving a number $$m_ 0>0$$. For $$V\in {\mathbb{R}}^{2n}$$ let $$V(D)=\sum^{n}_{1}V_ j\frac{\partial}{\partial x_ j}+V_{j+n}\frac{\partial}{\partial \xi_ j}.$$
Let $$a\in {\mathcal S}({\mathbb{R}}^{2n})$$, let $| a|_{q,X}=Sup\{| V^ 1(D).....V^ q(D)a(X)| | \| V^ 1\|_ X\leq 1,...,\| V^ q\|_ X\leq 1\}$ and let $$\| a\|_{L_ k^ p}=\sum_{q\leq k}(\int | a|^ p_{q,X}dX)^{1/p}$$, $$1\leq p<\infty.$$
The space $$L^ p_ k$$ (relative to the norm field) is defined as the completion of $${\mathcal S}({\mathbb{R}}^{2n})$$ for the norm $$\| \|_{L^ p_ k}$$. Then the main result asserts that if k is the smallest even integer such that $$k>2n((2/p)-1)$$ for $$p\in [1,2]$$ and $$k>(2m_ 0+n)(1-2/p)$$ for $$p\in [2,\infty]$$, then the mapping $$a\to 0p_{1/2}(a)$$ is bounded from $$L^ p_ k$$ into $$S_ p$$. A similar result holds for $$p=\infty.$$
In the second part of the paper, the author studies the converse to this result in the particular case of a constant field and for certain regularizations of the symbol a.
Reviewer: G.Pisier

##### MSC:
 47Gxx Integral, integro-differential, and pseudodifferential operators 35S05 Pseudodifferential operators as generalizations of partial differential operators 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
symbol; Schatten class; symplectic norms; norm field
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##### References:
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