Maximal functions and two-dimensional restriction.
A talk in the Oberseminar Analysis series by
João Pedro Ramos from Bonn
| Abstract: | 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, as the paper by Carleson and Sjölin sheds light into why the phenomenon occurs. More specifically, if $1 \le p < \frac{4}{3},$ they show that it make sense to talk about the restriction to the unit circle of the Fourier transform of a function in $L^p$.
Nevertheless, in 2016, Müller, Ricci and Wright have considered a different, yet related, 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 $1 \le p \le \frac{8}{7},$ 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 $1 \le p \le \frac{4}{3}.$ We do so by finding a clever 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. Time allowing, we shall also discuss some interesting open problems and high-dimensional generalizations. |