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Fourier integrals in classical analysis. (English) Zbl 0783.35001
Cambridge Tracts in Mathematics. 105. Cambridge: Cambridge University Press. x, 237 p. (1993).
The author gives sharp estimates for maximal operators using tools from microlocal analysis. Systematic use of restriction theorems is one of the hallmarks of this book. The analysis centers on local and global estimates for solutions of variable coefficient wave equations and estimates for a self-adjoint operator \(A\) of the spectral projections \(E(\lambda)\) and asymptotic estimates of the number, \(N(\lambda)\), of eigenvalues less than or equal to \(\lambda\), as \(\lambda\) goes to \(\infty\). The basic tools needed from Fourier analysis (wave front sets, restriction theorems, pseudodifferential operators and Fourier integral operators) are introduced and explained. Stationary phase estimates are developed and applied to prove sharp restriction theorems of the form \[ \left( \int_ S | \widehat{f(\xi)}|^ r d \mu(\xi) \right) ^{1/r} \leq C \| f \|_ s, \] where \(r={n+1 \over n-1}s'\), \(1 \leq s \leq{2(n+1) \over n+3}\), if \(n \geq 3\), and \(1 \leq s<4/3\) if \(n=2\). Here \(d \mu=\beta d \sigma\), where \(\beta \in C^ \infty_ 0\) and \(d \sigma\) is the Lebesgue measure on the surface \(S\). It is assumed that \(S\) has everywhere nonvanishing Gaussian curvature. These estimates are applied to the study of Riesz means, \(S^ \delta_ \lambda\), associated with a surface defined by \(q(\xi)\), (i.e., \(\{\xi\mid q(\xi)=1\}\)), where \[ S^ \delta_ \lambda f(x)=(2\pi)^{-n} \int e^{i\langle x,\xi \rangle} \bigl(1-q(\xi/ \lambda)\bigr)^ \delta_ +\widehat{f(\xi)} d \xi. \] The pseudodifferential operator calculus is introduced and used to show how to reduce the study of the spectrum of an elliptic self-adjoint pseudodifferential operator of arbitrary order to the first order case. The calculus and methods of stationary phase are used to construct local solutions of the associated differential operator (the half-wave operator) and to prove asymptotic estimates for \(N(\lambda)\). The explicit local solution is used to estimate the spectral projections associated with the first order self-adjoint pseudodifferential operator. A discrete restriction theorem associated to a first order pseudodifferential operator \(P(x,D)\) with positive principal symbol \(p\) is proved and used to estimate associated Riesz means.
Finally, Fourier integral operators are introduced and \(L^ 2\) and \(L^ p\) estimates are proved for them, along with maximal estimates that can be used to prove Stein’s spherical maximal theorem. He considers a special class of Fourier integral operators which contain the solution operators associated with variable coefficient wave equations and proves local smoothing estimates that can be used to improve the spherical maximal estimates to include the case of averages over geodesic circles, and gives some results for maximal operators with averages taken over thin “geodesic rectangles”.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35S30 Fourier integral operators applied to PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
35L25 Higher-order hyperbolic equations
42B25 Maximal functions, Littlewood-Paley theory
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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