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.
- 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),
- 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
- Multilinear embedding and Hardy's inequality, Some topics in harmonic
analysis and applications, Advanced Lectures in Mathematics vol 34 (2015),
- Functionals for multilinear fractional embedding, Acta Math. Sinica (Engl.
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)
- A sharp Hardy-Littlewood-Sobolev inequality with weights and mixed
homogeneity (in progress)
- 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
This research program has been partially supported
by the National Science Foundation.