Rodrigo Banuelos AMS Invited One-Hour Address - 01/04/23

The interplay between probability and harmonic analysis has a long history that has greatly enriched both fields. The purpose of this talk is to discuss some of this history and illustrate this interplay with some recent applications of probability to new sharp inequalities for classical operators in harmonic analysis in the discrete setting.

The discrete Hilbert transform was introduced by David Hilbert in the first decade of the 20th century who also proved its boundedness on the space of square summable sequences. His proof first appeared in H. Weyl's doctoral dissertation in 1908. In 1925, M. Riesz solved a problem of considerable interest to the analysis community at the time by showing that the continuous version of the Hilbert transform defined on the real line is bounded on the Lebesgue space of p-integrable functions and from this concluded the same for the discrete Hilbert transform on the space of p-summable sequences and showed that in fact the operators have comparable norms. Proving that they have the same norms has been a long–standing open problem motivated in part by an erroneous proof by E. C. Titchmarsh in 1926. Minimizing technicalities and recalling history, we will discuss a solution to this problem based on probabilistic tools that have had many applications in the computation of norms of singular integrals and Fourier multipliers in the continuous setting. Similar problems in the higher dimensional lattice will be mentioned.

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