Topics in Fourier Analysis

1.我们将对谐波分析中的各种问题进行研究。该项目将侧重于波动方程解的局部平滑特性。局部平滑的结果遵循一个重要的不等式，涉及分解的锥乘数，这是最初制定沃尔夫。这个不等式和它的几个变形有着广泛的应用，其中之一是关于平均算子沿着曲线的Sobolev正则性 和相关极大算子的有界性。确定这些不等式的范围是有意义的。该提案的其他部分涉及傅立叶限制和扩展问题，与变分Carleson定理，奇异极大函数和海森堡群上的波动方程。2.这个项目的一个主要部分是关于数学物理中一些基本线性偏微分算子的正则性。这些算子的定量结果适用于在物理科学中具有广泛应用的非线性方程问题。 他们也产生应用数学，即问题的扩大本征函数的拉普拉斯算子紧流形和规律性的平均运营商。 这些平均算子的研究是由医学成像理论中的问题激发的。

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.