The Fourier extension operator on large spheres and related oscillatory integrals

Details The Fourier extension operator on large spheres and related oscillatory integrals The Fourier extension operator on large spheres and related oscillatory integrals

2006-08-04

arxiv.org/abs/math/0608129v2

Proceedings of the London Mathematical Society, 98, no.1, (2009), 45-82.

Mathematics

Classical Analysis and ODEs

Scientific paper

We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal $L^p(mathbb{S}^2)\to L^q(R \mathbb{S}^2)$ estimates for the Fourier extension operator on large spheres in $\mathbb{R}^3$, which are uniform in the radius $R$. Two appendices are included, one concerning an application to Lorentz space bounds for averaging operators along curves in $R^3$, and one on bilinear estimates.

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