Bill Beckner's Research
Fourier Analysis - Sharp Inequalities and Geometric Manifolds
Geometric inequalities provide insight into the structure of manifolds.
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. Three important examples are the Moser-Trudinger
inequality where limiting Sobolev behavior for critical exponents provides
significant understanding of geometric analysis for conformal deformation
on a Riemannian manifold, the logarithmic Sobolev inequality which can
be viewed as a limiting embedding phenomena that encompasses both an
infinite-dimensional nature and Gaussian symmetry while strengthening the
uncertainty principle and in general terms is a
comparison between entropy and smoothness, and the isoperimetric inequality
which can provide a quantitative measure of the non-Euclidean character of a
manifold. Certainly the isoperimetric inequality is the cornerstone for the
analysis of geometric variational problems, and can be linked to the ideas
of entropy and the log Sobolev inequality through the Levy-Gromov functional.
This direction seems fundamental to explore the interplay between geometry
and analysis on SL(2,R), the Heisenberg group, hyperbolic space and more
generally, manifolds with nonpositive curvature. Asymptotic arguments
identify geometric invariants that characterize large-scale structure.
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
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, quantum physics,
statistical mechanics, string theory and turbulence.
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 (in press).
This research program is supported by the National Science Foundation.