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Geometric control of oscillatory integrals

Beltran Portalés, David (2017)
Ph.D. thesis, University of Birmingham.

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Abstract

The aim of this thesis is to provide a geometric control of certain oscillatory integral operators. In particular, if \(T\) is an oscillatory Fourier multiplier, a pseudodifferential operator associated to a symbol \(\alpha\) \(\in\) \(S\) \(^m\) \({p,o}\) or a Carleson-like operator, we obtain a weighted \(L\)\(^2\) inequality of the type

\(\int\) |\(T\)\(f\)|\(^2\)w \(\leq\) C \(\int\) |\(f\)|\(^2\)\(M\)\(_T\)w

Here \(C\) is a constant independent of the weight function w, and the operator \(M\)\(_T\), which depends on the corresponding T, has an explicit geometric character. In the case of oscillatory Fourier multipliers and of Carleson-like operators we also determine auxiliary geometric operators \(g\)1 and \(g\)2 and establish a \(pointwise\) estimate of the type

\(g\)\(_1\)(\(T\)\(f\))(x) \(\leq\) C \(g\)\(_2\)(f)(x):

Finally, we include a careful study of a method developed by Bourgain and Guth in Fourier restriction theory, that allows making progress on the Fourier restriction conjecture from their conjectured multilinear counterparts. Our conjectured progress via multilinear estimates has been recently obtained by Guth.

Type of Work:Ph.D. thesis.
Supervisor(s):Bennett, Jon
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:7566
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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