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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.

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