A quantization of the Cartan domain BD I \((q=2)\) and operators on the light cone (Q1095479)

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.