# [PIMS Hugh C. Morris Lecture] Conditional Expectation and Calderón-Zygmund Operators

In his 1983 research announcement “Some results in harmonic analysis in ℝn, for 𝙣 → ∞ ” (Bulletin of the AMS), E.M. Stein announced new results for several classical operators in harmonic analysis with operator norms that do not depend on the dimension 𝙣, and raised the following question: “Can one find an appropriate infinite-dimensional formulation of (that part of) harmonic analysis in ℝn, which displays in a natural way the above uniformity in 𝙣?” This talk describes the representation of some classical Calderón-Zygmund operators as conditional expectations of stochastic integrals. From this representation, probability gives bounds on their norms that are not only universal in terms of the geometry of the space where they are defined but in several instances are also sharp. To illustrate the techniques, we describe an application to a long-standing open problem concerning the norm of the discrete Hilbert transform. The latter is joint work with Mateusz Kwaśnicki of Wrocław University of Science and Technology, Poland.