# Real Analysis, Harmonic Analysis and Applications

### Detlef Müller

Christian-Albrechts-Universität zu Kiel, Germany### Elias M. Stein

Princeton University, United States

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## Abstract

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 $1<p<2d/(d+1),$ at least when acting on radial functions, and in sufficiently high dimension even when acting on arbitrary $L^p$- functions, in terms of Fourier localized pieces of the convolution kernels. Moreover, improvements on Wolff's inequality for the cone multiplier have been achieved. \smallskip Also, a theory of Hardy spaces on metric measure spaces with exponential growth has been developed, which allows for instance to significantly improve on a spectral multiplier theorem for Riemannian manifolds with bounded geometry by M.Taylor. For instance, the new results apply to complex powers of the Laplacian, which could not be handled before. \item Oscillatory and Fourierintegral operators: this includes endpoint $L^p - L^q$ and Sobolev inequalities for certain broad classes of highly degenerate Radon-like averaging operators. \item Applications to PDE's: this includes optimal global existence theorems by means of abstract Strichartz estimates for small amplitude nonlinear wave equations associated to certain linear wave equations involving compact perturbations of the standard Laplacian, global well-posedness and scattering in $H^1$ for defocusing nonlinear Schr\"odinger equations on hyperbolic space, and a smoothing property for the $L^2$-critical nonlinear Schr\"odinger equation. \end{itemize} \medskip The meeting took place in a lively and active atmosphere, and greatly benefited from the ideal environment at Oberwolfach. It was attended by 43 participants. The program consisted of 3 survey lecture series and 25 lectures. The organisers made an effort to include young mathematicians, and greatly appreciate the support through the joint Oberwolfach/NSF program "US Junior Oberwolfach Fellows," which allowed to invite several outstanding young scientists from the United States.