
William Beckner
Professor
Department of Mathematics
Paul V. Montgomery Centennial Memorial Professorship in MathematicsTo examine how the interplay of symmetry, smoothness and uncertainty can characterize structure for geometric manifolds.beckner@math.utexas.edu
Phone: 5124717711
Office Location
RLM 10.140
Postal Address
The University of Texas at Austin
MATHEMATICS
2515 SPEEDWAY, Stop C1200
AUSTIN, TX 787121202

Ph.D., Princeton University (1975)
B.S. in Physics, University of MissouriResearch Interests
To examine how the interplay of symmetry, smoothness and uncertainty can characterize structure for geometric manifolds.
 Fourier Analysis
 Geometric Inequalities
 Lie Groups & Differential Geometry
 Mathematical Physics
 Partial Differential Equations
 Probability
Fourier Analysis  Sharp Inequalities and Geometric Manifolds
Geometric inequalities provide insight into the structure of manifolds. More directly, Sobolev embedding, the Fourier transform, convolution and fractional integrals are central tools for analysis on geometric manifolds. Questions concerning fractional smoothness, multilinear operators, product manifold structure and restriction phenomena on subvarieties are engaging directions for current research. The principal objective of my research is to develop a deeper understanding of the way that sharp constants for functionspace 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 nonunimodular Lie groups, including SL(2,R), hyperbolic space, and more generally, manifolds with nonpositive curvature (CartanHadamard spaces). Asymptotic arguments identify geometric invariants that characterize largescale 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 welldefined, to explain the fundamental structure of spaces and their varied geometric realizations, to calculate precise lowerorder 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, fluid dynamics, manybody 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 manybody interactions that determine the behavior of dynamical phenomena.

Selected Papers
Inequalities in Fourier analysis, Ann. Math. 102 (1975), 159182.Sobolev inequalities, the Poisson semigroup and analysis on the sphere, Proc. Nat. Acad. Sci. 89 (1992), 48164819.
Sharp Sobolev inequalities on the sphere and the MoserTrudinger inequality, Ann. Math. 138 (1993), 213242.
Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995, 3668.
Pitt's inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 18971905.
Logarithmic Sobolev inequalities and the existence of singular integrals, Forum Math. 9 (1997), 303323.
Sharp inequalities and geometric manifolds, J. Fourier Anal. Appl. 3 (1997), 825836.
Geometric proof of Nash's inequality, Int. Math. Res. Notices (1998), 6772.
Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999), 105137.
On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), 12331246.
Asymptotic estimates for GagliardoNirenberg embedding constants, Potential Analysis 17 (2002), 253266.
Estimates on Moser embedding, Potential Analysis 20 (2004), 345359.
Weighted inequalities and SteinWeiss potentials, Forum Math. 20 (2008), 587606.
Pitt's inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), 18711885.
Pitt's inequality and the fractional Laplacian: sharp error estimates, Forum Math. 24 (2012), 177209.
Multilinear embedding estimates for the fractional Laplacian, Mathematical Research Letters 19 (2012), 175189.
Multilinear embedding  convolution estimates on smooth submanifolds, Proc. Amer. Math. Soc. 142 (2014), 12171228.
Embedding estimates and fractional smoothness, Int. Math. Res. Notices (2014), 390417.
Multilinear embedding and Hardy's inequality, Advanced Lectures in Mathematics (in press).
Functionals for multilinear fractional embedding, Acta Math. Sinica (accepted).
On Lie groups and hyperbolic symmetry  from KunzeStein phenomena to Riesz potentials, Nonlinear Analysis: Theory, Methods & Applications (in preparation).

 Prix Salem 1975
 Sloan Research Fellow 1977
 ICM Invited Lecture  Helsinki, 1978
 Fellow, American Mathematical Society