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An algebra of pseudodifferential operators. (English) Zbl 0840.35130
The author considers pseudo-differential operators in $$\mathbb{R}^n$$ with symbol $$a(z)$$, $$z= (x, \xi)$$, satisfying the following property: $\sup_j |(f_j a)\hat{}(\zeta)|\in L^1(\mathbb{R}^{2n}_\zeta),$ where $$j$$ runs over the lattice of the $$2n$$-tuples of integers, and $f_j(z)= f_0(z- j)\in C^\infty_0(\mathbb{R}^{2n})$ with $$\sum_j f_j= 1$$. Such symbols turn out to be bounded continuous functions in $$\mathbb{R}^{2n}$$. The author proves that the corresponding class of pseudo-differential operators: 1) is independent of the $$t$$-quantization, $$t\in \mathbb{R}$$, $\text{Op}_t(a) u(x)= (2\pi)^{- n} \int e^{i(x- y)\xi} a(tx+ (1- t) y,\xi) u(y) dy d\xi;$ 2) is stable under composition; 3) consists of $$L^2$$-bounded operators.
Reviewer: L.Rodino (Torino)

MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators
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