Oberwolfach Reports

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Volume 5, Issue 3, 2008, pp. 1771–1850
DOI: 10.4171/OWR/2008/32

Published online: 2009-06-30

Real Analysis, Harmonic Analysis and Applications

Detlef Müller[1] and Elias M. Stein[2]

(1) Christian-Albrechts-Universität zu Kiel, Germany
(2) Princeton University, United States

This workshop, which continued the triennial series at Oberwolfach on Real and Harmonic Analysis that started in 1986, has brought together experts and young scientists working in harmonic analysis and its applications (such as to dispersive PDE's and ergodic theory) with the objective of furthering the important interactions between these fields. Three prominent experts, Elon Lindenstrauss (Princeton), Amos Nevo (Technion, Haifa), and Terence Tao (UCLA), gave survey respectively introductory lectures. Their topics included "Effective equidistribution on the torus", "Non-Euclidean lattice point counting problems, and the ergodic theory of lattice subgroups," and "The van der Corput lemma, equidistribution in nilmanifolds, and the primes." \medskip Major further areas and results represented at the workshop are: \begin{itemize} \item Application of Time Frequency analysis: this is an outgrowth of the method of "tile decomposition" which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress includes applications of these techniques and the theory of multilinear singular integral operators to ergodic theory and an extension of the celebrated Carleson-Hunt theorem to the "polynomial Carleson operator." \item Estimates for maximal functions: this includes recent progress on best weak $(1,1)$ constants for the Hardy-Littlewood maximal function on metric measure spaces, estimates for maximal functions associated to monomial polyhedra, with applications to sharp estimates for the Bergman kernel on a general class of weakly pseudoconvex domains of finite type in $\C^n,$ as well as estimates for maximal functions for the Schr\"odinger and the wave equation. \item Fourier and spectral multipliers: a breakthrough has been obtained on the characterization of radial Fourier multipliers. Contrary to a general belief that for $p\ne 1,2$ or $\infty,$ no "concrete" characterization of Fourier multipliers for $L^p(\R^d)$ would be possible, radial Fourier multipliers have been characterized for the range $1

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Müller Detlef, Stein Elias: Real Analysis, Harmonic Analysis and Applications. Oberwolfach Rep. 5 (2008), 1771-1850. doi: 10.4171/OWR/2008/32