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BECKNER, WILLIAM

William Beckner

Professor
Department of Mathematics

Paul V. Montgomery Centennial Memorial Professorship in Mathematics

To examine how the interplay of symmetry, smoothness and uncertainty can characterize structure for geometric manifolds.

beckner@math.utexas.edu

Phone: 512-471-7711

Office Location
RLM 10.140

Postal Address
The University of Texas at Austin
MATHEMATICS
2515 SPEEDWAY, Stop C1200
AUSTIN, TX 78712-1202

Ph.D., Princeton University (1975)
B.S. in Physics, University of Missouri

Research 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 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), 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, 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

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, Advanced Lectures in Mathematics (in press).

Functionals for multilinear fractional embedding, Acta Math. Sinica (accepted).

On Lie groups and hyperbolic symmetry -- from Kunze-Stein 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