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A quantization of the Cartan domain BD I $$(q=2)$$ and operators on the light cone. (English) Zbl 0632.58033
Let $$\pi =SO_ 0(2,n+1)/(SO(2)\times SO(n+1))$$ be a symmetric space. For a real number $$\lambda$$ let $$H_{\lambda}$$ be the Hilbert space that consists of measurable complex-valued functions on the light cone $C=\{x=(x_ 0,x_ 1,...,x_ n)\in R^{n+1}| \quad x_ 0>0,\quad r(x)=x^ 2_ 0-x^ 2_ 1-...-x^ 2_ n>0\}$ satisfying $$\| u\|^ 2_{\lambda}=\int_{C}| u(t)|^ 2(r(t))^{-\lambda /2}dt<\infty$$. The main result of the paper is a proof of the fact that for any reasonable function f on $$\pi$$ one can associate it with a bounded linear operator op(f) on $$H_{\lambda}$$. The authors point out that this $$H_{\lambda}$$-calculus is closely related to the Weyl calculus of pseudodifferential operators and the Fuchs calculus, and in some sense the $$H_{\lambda}$$-calculus is better than the Fuchs calculus.
Reviewer: G.Tu

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 35S05 Pseudodifferential operators as generalizations of partial differential operators
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