1. We shall conduct research on various problems in harmonic analysis. The project will focus on local smoothing properties of solutions for wave equations. Results on local smoothing follow from a crucial inequality involving decompositions of cone multipliers which was originally formulated by Wolff. This inequality and several variants of it have a wide range of applications; one of them concerns the Sobolev regularity for averaging operators along curves and boundedness of associated maximal operators. It is of interest to determine the range of these inequalities. Other parts of the proposal deal with Fourier restriction and extension problems, with variational Carleson theorems, with singular maximal functions and with the wave equation on the Heisenberg group.
2. A major part of this project is concerned with the regularity properties of some basic linear partial differential operators of mathematical physics. Quantitative results for these operators are applicable to problems in nonlinear equations with a wide range of applications in the physical sciences. They also yield applications within mathematics, namely to problems on expansions in eigenfunctions of the Laplacian on compact manifolds and the regularity of averaging operators. The study of these averaging operators is motivated by problems in the theory of medical imaging.
The work supported on this grant resulted in a number of new results in Fourier analysis and allied topics. Most of the research was conducted in collaboration with other mathematicians. Characterizations of radial Fourier multipliers and new local space time estimates for solutions of the wave equations (both in the Euclidean case and on manifolds) are key outcomes of this work. An endpoint version of Sogge's conjecture was proved for suitable ranges of Lebesgue spaces in four and higher dimensions. Other significant results are variation norm inequalities for the partial sum operators of Fourier series and new endpoint bounds for the Fourier restriction operator associated with nondegenerate curves as well as related bounds for certain general classes curves equipped with affine arclength measure. Among other achievements, the PI and collaborators contributed new versions of Wolff inequalities for decompositions of cone and spherical multipliers, boundedness and entropy results for embeddings of low regularity spaces, various space time and maximal estimates for solutions of Schroedinger equations, an improved Calderon-Zygmund regularity estimate for generalized Radon transforms and Fourier integral operators, Stein-Tomas type Fourier restriction estimates for measures, results on lacunary maximal functions, and improved bounds for a square-function associated with Bochner-Riesz multipliers. The grant also provided support for the research and training of graduate students. The students proved new theorems on lattice point problems, generalized Fourier restriction estimates, and new differentiation results for the basis of convex sets.
