Bill Beckner's Research

Fourier Analysis - Sharp Inequalities: from Geometric Manifolds & Lie Groups to Dynamical Processes

Geometric inequalities provide basic insight into the structure of manifolds. More directly, the Fourier transform, convolution, Riesz potentials and Sobolev embedding are central tools for analysis on geometric manifolds. Questions concerning fractional smoothness, multilinear operators, product manifold structure, Riesz potentials and Kunze-Stein estimates on Lie groups, and restriction phenomena on subvarieties are essential directions for current research. The principal objective of my research is to develop a deeper understanding of the way that sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. This direction seems fundamental to explore the interplay between geometry and analysis on locally compact non-unimodular Lie groups, including SL(2,R), SL(2,C), Lorentz groups, hyperbolic space, and more generally, manifolds with nonpositive curvature (Cartan-Hadamard spaces). Asymptotic arguments identify geometric invariants that characterize large-scale structure. Weighted inequalities provide quantitative information to characterize integrability for differential and integral operators and reflect the dilation character of the manifold. Sharp estimates constitute a critical tool to determine existence and regularity for solutions to pde's, to demonstrate that operators and functionals are well-defined, to explain the fundamental structure of spaces and their varied geometric realizations, to calculate precise lower-order effects and to suggest new directions for the development of analysis on a geometric manifold. Model problems and exact calculations in differential geometry and mathematical physics are a source of insight and stimulus, particularly conformal deformation, density functional theory, fluid dynamics, many-body dynamics, quantum physics, statistical mechanics, stellar dynamics, string theory and turbulence. Functional forms that characterize smoothness lie at the heart of understanding and rigorously describing the many-body interactions that determine the behavior of dynamical phenomena.

Selected papers


This research program has been partially supported by the National Science Foundation.