Geometric control of oscillatory integrals

Beltran Portalés, David (2017). Geometric control of oscillatory integrals. University of Birmingham. Ph.D.

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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:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):