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Proceedings of the London Mathematical Society Advance Access originally published online on May 9, 2008
Proceedings of the London Mathematical Society 2009 98(1):45-82; doi:10.1112/plms/pdn022
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© 2008 London Mathematical Society

The Fourier extension operator on large spheres and related oscillatory integrals

Jonathan Bennett

School of Mathematics
University of Birmingham
The Watson Building
Edgbaston
Birmingham
B15 2TT
United Kingdom

Andreas Seeger

Department of Mathematics
University of Wisconsin
Madison
WI 53706-1388
USA
seeger@math.wisc.edu

Received 17 January 2007. Revision received 11 March 2008.

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 Lp(S2)->Lq(RS2) estimates for the Fourier extension operator on large spheres in R3, 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 R3, and one on bilinear estimates.

2000 Mathematics Subject Classification 42B10.

The first author is supported by EPSRC grants GR/S27009/02 and EP/E022340/1 and the second author by the National Science Foundation grant DMS 0200186.


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