The calculus of pseudo-differential operators of Fuchs type.(English)Zbl 0561.35081

Let D be the set $$\{(\xi,\eta)\in {\mathbb{R}}^ 2$$, $$\eta >0\}$$. If $$u\in L^ 2(ds/s)$$ $$(s\in {\mathbb{R}}^+_*)$$ and ($$\xi,\eta)\in D$$, put $$\sigma_{\xi,\eta}u(s)=u(\eta^ 2/s)e^{2\pi i\xi (s/\eta -\eta /s)}.$$ When $$f\in L^ 1(D)$$, the bounded operator Q(f) on $$L^ 2(ds/s)$$ defined by $$Q(f)u(s)=2\iint f(\xi,\eta)\sigma_{\xi,\eta}u(s)(1/\eta)d\xi d\eta$$ is called the operator with active symbol f. If A is any trace class operator on $$L^ 2(ds/s)$$, the passive symbol of A is the function $$h(\xi,\eta)=2Tr(A\sigma_{\xi,\eta}).$$
Let $${\mathcal S}({\mathbb{R}}^+_*)$$ (resp. $${\mathcal S}'({\mathbb{R}}^+_*))$$ be the spaces corresponding to $${\mathcal S}({\mathbb{R}})$$ (resp. $${\mathcal S}'({\mathbb{R}}))$$ via the isometry $$\Phi: L^ 2(ds/s)\to L^ 2({\mathbb{R}})$$ defined by $$\Phi u(x)=u(e^ x).$$ Let us say that $$u\in {\mathcal S}_ k({\mathbb{R}}^+_*)$$ if $$(1+s^{1/2})^ ku(s)\in {\mathcal S}({\mathbb{R}}^+_*)$$ and put $${\mathcal S}_{\infty}({\mathbb{R}}^+_*)=\cap_{k}{\mathcal S}_ k({\mathbb{R}}^+_*)$$ with the obvious topology. Finally define an order function m on D as a strictly positive function such that m(w’)/m(w) can be bounded in a suitable way then w,w’$$\in D$$, and define a symbol of order m as a $$C^{\infty}$$ function of D such that the directional derivatives of m in the directions of $$\partial_{\xi}$$ and $$\xi \partial_{\xi}+\eta \partial_{\eta}$$ can be suitably estimated in terms of m.
The author studies continuity properties (in $${\mathcal S}_{\infty}({\mathbb{R}}^+_*),L^ 2(ds/s))$$ of operators with an active symbol of order m. The passive symbol of the composition of two operators with given passive symbols is also studied. Applications are made to differential equations of Fuchs type.
Reviewer: P.Godin

MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 47Gxx Integral, integro-differential, and pseudodifferential operators
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References:

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