Dr. Geraldo de Souza
Department of Mathematics and Statistics at Auburn University, USA
Talk is at 11:00 AM Central Time (calculating your local time…)
Abstract
This talk presents a historical overview of several important developments in harmonic analysis, emphasizing how foundational questions led to major advances in modern mathematics. Beginning with classical Hardy spaces, Lebesgue spaces, and the Hilbert transform, we review the role of the M. Riesz theorem and the correspondence between analytic and real-variable function spaces. We then discuss the emergence of BMO and the celebrated duality between H1 and BMO through the work of Coifman and Fefferman.
The presentation highlights special atom spaces introduced by De Souza, their structural properties, and their connection to real and analytic characterizations of function spaces. We explain how questions surrounding dyadic atoms and the Haar system helped inspire the birth of wavelet theory, particularly through the pioneering work of Yves Meyer.
A second theme concerns the long-standing problem of almost everywhere convergence of Fourier series. We trace the path from Luzin’s conjecture to Calderon’s operator-theoretic reformulation, culminating in the Carleson–Hunt theorem. Finally, we describe later Carleson-type results for the spaces Bp and L(p, 1), together with atomic descriptions extending the classical H1 theory. The talk combines historical narrative with mathematical insight to illustrate the lasting influence of harmonic analysis across Pure and Applied Mathematics.