Cork Spring 2025

Abstract: 

We characterize the function φ of minimal L^1 norm among all functions f of exponential type at most π for which f(0)=1. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros ±τ_n, n=1,2,…. We identify φ in the following way. We factor φ(z) as Φ(z)Φ(−z), and show that Φ satisfies a certain second order linear differential equation along with a functional equation, either of which characterizes Φ. Furthermore, we use these facts to establish a series expansion for the zeros and a power series expansion of the Fourier transform of φ, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of Φ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.

This is joint work with Andriy Bondarenko, Danylo Radchenko and Kristian Seip. https://arxiv.org/abs/2504.05205

Abstract:

We report on joint work with Spyros Dendrinos, Isroil Ikromov and Detlef Müller on a new class of "FIO-cone multipliers". In previous work, we studied the boundedness range of the maximal average of any smooth compact hypersurface in three-dimensional space, up to a certain "exceptional class", which is linked to the cone multiplier.  We encounter a convolution operator, being the composition of two operators: the classic cone multiplier and additionally a certain translation invariant Fourier integral operator (FIO) with non-standard phase functions with singularities near the light cone. We develop a new theory for a class of these "FIO-cone multipliers", that allows phase functions that are in a particular way adapted to the geometry of the cone. Our approach uses the recent breakthrough for the cone multiplier conjecture by Guth, Wang and Zhang.

Abstract:

The study of the boundary behaviour of holomorphic and harmonic functions is of significant importance in many areas in Analysis. In this talk I will present an overview of my research on this topic, focusing on two theorems which complement and strengthen some classical results. The first one concerns Abel's Limit Theorem, which connects the behaviour of a Taylor series as we approach the boundary from the interior with its behaviour on the boundary itself. The second one strengthens Plessner's and Spencer's theorems about the angular behaviour of holomorphic functions on the unit disc. Moreover, its harmonic analogue in higher dimensions improves classical results of Stein and Carleson. As we will see, these two theorems, which are based on a variety of tools from potential theory, find applications to certain classes of holomorphic functions with wild boundary behaviour. (Based on joint works with Stephen Gardiner, Stéphane Charpentier and Konstantinos Maronikolakis.)

Abstract:

The goal of the talk is to present a certain ray bundle representation of the Fourier extension operator in terms of the Wigner transform to investigate weighted estimates in restriction theory and their connections to time-frequency analysis and geometric combinatorics.

In joint work with Bennett, Gutierrez and Nakamura, we show how Sobolev estimates for the Wigner transform can be converted into certain tomographic bounds for the Fourier extension operator, which implies a variant of the (recently shown by H. Cairo to be false) Mizohata-Takeuchi conjecture. Together with Bez and the previous three authors, we employed of our phase-space approach in the context of Strichartz inequalities for orthonormal systems in the spirit of the work of Frank and Sabin. If time allows, we will make a further connection between our results and Flandrin's conjecture in signal processing through the study of certain singular integral operators similar to those studied by Lacey, Lie, Muscalu, Tao and Thiele.