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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
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