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Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. (English) Zbl 0776.58037
The author establishes local regularity theorems for a certain class $$I^ \mu(Z,Y;{\mathcal C})$$ of Fourier integral operators where $$Y$$ and $$Z$$ are smooth paracompact manifolds of dimensions $$n\geq 2$$ and $$n+1$$, respectively. These estimates are applied to prove versions of the Carlson-Sjölin theorem on compact two dimensional manifolds with periodic geodesic flow.
As an important application the $$L^ p\to L^ p$$ local smoothing of order $$\delta>0$$ is obtained if $$2<p<\infty$$ for the solutions of the Cauchy problem of the wave equation $$((\partial/\partial t)^ 2- \Delta)u=0$$.
The $$L^ p\to L^ p$$ local smoothing in case $$n=2$$ requires a different proof from that of the case $$n\geq 3$$. An ingenious method is devised in this part of the proof.

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 58J45 Hyperbolic equations on manifolds 42B25 Maximal functions, Littlewood-Paley theory
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