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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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