Bill Beckner's Research

Fourier Analysis - Sharp Inequalities: from Geometric Manifolds & Lie Groups to Dynamical Processes -- Embedded Symmetry & Encoding Information

The principal themes of my research are symmetry, inequalities, and analysis on manifolds. Geometric inequalities provide 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. The overall objective of my research is to develop a deeper understanding for 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, Cartan-Hadamard manifolds with nonpositive curvature, and more generally semisimple Lie groups and symmetric spaces extending to the family of Heisenberg groups with broken homogeneity and symplectic structure. Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation. The intrinsic character of the Heisenberg group makes it the natural playing field on which to explore the laws of symmetry and the interplay between analysis and geometry on a manifold. 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 define a new paradigm for the development of analysis on a geometric manifold. Model problems and exact calculations are a source of insight and stimulus. Vortex dynamics, the Keller-Segel model for chemotaxis, and the global Biot-Savart law suggest intriguing new applications. Fundamental identities, formulas, and inequalities are essential for developing analysis on a geometric manifold -- explicit, elegant, influential -- plus they encode information about the manifold. Formulas are the most basic computational recipe and lead always to new mathematical structure as one moves beyond the horizon to develop new computational insight.

Selected papers

Inequalities in Fourier analysis, Ann. Math. 102 (1975), 159-182.
Sobolev inequalities, the Poisson semigroup and analysis on the sphere, Proc. Nat. Acad. Sci. 89 (1992), 4816-4819.
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math. 138 (1993), 213-242.
Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995, 36-68.
Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897-1905.
Logarithmic Sobolev inequalities and the existence of singular integrals, Forum Math. 9 (1997), 303-323.
Sharp inequalities and geometric manifolds, J. Fourier Anal. Appl. 3 (1997), 825-836.
Geometric proof of Nash's inequality, Int. Math. Res. Notices (1998), 67-72.
Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999), 105-137.
On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), 1233-1246.
Asymptotic estimates for Gagliardo-Nirenberg embedding constants, Potential Analysis 17 (2002), 253-266.
Estimates on Moser embedding, Potential Analysis 20 (2004), 345-359.
Weighted inequalities and Stein-Weiss potentials, Forum Math. 20 (2008), 587-606.
Pitt's inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), 1871-1885.
Pitt's inequality and the fractional Laplacian: sharp error estimates, Forum Math. 24 (2012), 177-209.
Multilinear embedding estimates for the fractional Laplacian, Mathematical Research Letters 19 (2012), 175-189.
Multilinear embedding -- convolution estimates on smooth submanifolds, Proc. Amer. Math. Soc.142 (2014), 1217-1228.
Embedding estimates and fractional smoothness, Int. Math. Res. Notices (2014), 390-417.
Multilinear embedding and Hardy's inequality, Some topics in harmonic analysis and applications, Advanced Lectures in Mathematics vol 34 (2015), 1-26.
Functionals for multilinear fractional embedding, Acta Math. Sinica (Engl. Ser.) 31 (2015), 1-28.
On Lie groups and hyperbolic symmetry -- from Kunze-Stein phenomena to Riesz potentials, Nonlinear Analysis 126 (2015), 394-414.
Symmetry and the Heisenberg group (in progress)
Symmetry in Fourier analysis -- Heisenberg group to Stein-Weiss integrals, J. Geom. Anal. 31 (2021), 7008-7035.
Symmetry-breaking on the Heisenberg group (in progress)

Talks

Fractional Smoothness and Embedding Potentials -- Austin, January 2012
Inequalities in Harmonic Analysis -- A modern panorama on classical ideas -- Nanjing, May 2012
Fourier Analysis -- Workshop for Analysis -- Austin, March 2014
Embedding and Potentials in Fourier Analysis -- Measuring Functional Smoothness -- UCLA, May 2014
Lie groups and beyond: Kunze-Stein phenomena and Riesz potentials -- Johns Hopkins, October 2015
On Lie Groups and Beyond -- Kunze-Stein Phenomena, SL(2,R), the Heisenberg Group and the Role of Symmetry -- Hangzhou, June 2017
On Lie Groups and Beyond -- Kunze-Stein Phenomena -- SL(2,R) and the Heisenberg Group -- the Role of Symmetry -- Edinburgh, July 2017
Symmetry in Fourier Analysis -- From Heisenberg to Stein-Weiss -- July 2020

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