SEMINARIO DE ANÁLISIS COMPLEJO

(COMPLEX ANALYSIS SEMINAR)

Random Carleson Measures in the Polydisc

Alberto Dayan
Saarland University, Saarbr¨ucken, Germany

Wednesday, September 25, 2024 at 15:30
Room 520, Module 17, Department of Mathematics,
Universidad Autónoma de Madrid (Autonomous University of Madrid)

Abstract:
A Carleson measure on the unit disc is a positive measure that embeds continuously
the Hardy space inside the corresponding L2 space on the unit disc. The celebrated
work of Carleson characterizes such measures in terms of a geometric condition
that has to be tested only on squares having their basis on the unit circle. Such
notions have a natural extension to the polydisc, but in this case the geometric
characterization becomes much more complicated to work with. In this talk, we will
consider atomic measures on the polydisc generated by sequences (such measures
play an important role in the theory of interpolating sequences). In particular, we
will consider a random sequence in the polydisc, and we will discuss the 0-1 law
for it to generate a Carleson measure almost surely. While in the one dimensional
case such 0-1 law can be found by using Carleson’s geometric condition, such tool
is unavailable in the multi-variable setting. We will then discuss a well known
reformulation of the problem in terms of Gram matrices, and then describe those
sequences that generates almost surely a Carleson measure for the polydisc by using
tools from the theory of random matrices.
This is a joint work with Nikolaos Chalmoukis and Giuseppe Lamberti.