Title: PROBABILISITIC TOOLS IN DISCRETE HARMONIC ANALYSIS

RODRIGO BAÑUELOS,  PURDUE UNIVERSITY

Abstract. The discrete Hilbert transform was introduced by David Hilbert at the beginning

of the 20th century as an example of a singular quadratic form. Its boundedness on the

space of square summable sequences appeared in H. Wyel’s doctoral dissertation (directed

by Hilbert) in 1908. In 1925, Marcel Riesz solved a problem of great interest to the analysis

community at the time by showing that the continuous version of the Hilbert transform on

the real line is bounded on the Lebesgue Lp–spaces. From this he concluded the same for

the discrete Hilbert transform on the space of p–summable sequences. In 1926, inspired

by Riesz’s proof, E. C. Titchmarsh showed that in fact the operators have the same norm.

Unfortunately, soon after the publication he found a gap in the proof. The question of

equality has been open ever since.

In this talk the speaker will discuss a probabilistic construction that leads to sharp norm

bounds for discrete singular integrals on the d-dimensional lattice, where d is greater than

or equal to one. When d is one, it resolves the problem for the Hilbert transform. The case

d greater than one raises similar questions and conjectures for other Calderón-Zygmund

singular integrals.

This lecture is designed for a general audience. The talk will contain many historical

remarks. Technicalities will be kept to a minimum.