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It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed., 16pages, 1figure

In this note we study the relation between oscillatory integral operators and convolution operators, and also the sharpness of L p L^p -estimates for singular oscillatory integral operators.

We study the semi-classical behavior of the spectral function of the Schr\"{o}dinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function., Comment: 10 pages

Isospectral theory of the Lax pairs of both 3D and 2D Euler equations of inviscid fluids is developed. Eigenfunctions are represented through an ergodic integral. The Koopman group and mean ergodic theorem are utilized. Further harmonic analysis results on the ergodic integral are introduced. The ergodic integral is a limit of the oscillatory integral of the first kind.

We obtain a simpler proof of Theorem 3.1 of The complete (L P , L P ) mapping properties of some oscillatory integrals in several dimensions, by G. Sampson and P. Szeptycki (Canad. Math. J. 53 (5) (2001), 1031-1056).

We study doubly oscillatory integrals R ne i(ξ+y|ξ|+t|ξ|α) f(ξ)dξ and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on R n .

An endpoint boundedness result is established for a class of oscillatory integral operators.