University of Birmingham > Talks@bham > Analysis Seminar > Maximal functions and two-dimensional restriction

Maximal functions and two-dimensional restriction

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If you have a question about this talk, please contact Diogo Oliveira E Silva.

The restriction conjecture for the Fourier transform has been a very active topic in Fourier analysis for over the past 40 years, with new machinery being developed continuously to deal with its subtleties. However, its most basic manifestation, the two-dimensional case, has been solved for over 40 years, with the paper by Carleson and Sjölin giving new insights on the subject. More specifically, if 1 <= p < 4/3, they show that it make sense to talk about the restriction to the unit circle of the Fourier transform of a function in L^p.

In 2016, Müller, Ricci and Wright considered a different, stronger property: what can be actually said about the pointwise definition of the restriction operator? Is there a suitable notion of Lebesgue points of the Fourier transform over a curve? As their result would show, the answer is affirmative: in the restricted range 1 <= p < 8/7, besides having a restriction operator, they show that almost every point of the unit circle is a Lebesgue point for the Fourier transform, with respect to the (affine) arc length measure.

The aim of this talk is to extend the Müller-Ricci-Wright result to the whole range of exponents 1 <= p < 4/3. We do so by finding a way to introduce a bilinearization of our operator, along with the Carleson-Sjölin machinery, which allows us to bypass several technical details contained in their paper. We also extend their results to more general maximal functions, answering therefore questions left open by Kovač.

This talk is part of the Analysis Seminar series.

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