Research in Harmonic Analysis and Partial Differential Equations

小行星9306387 该项目将侧重于与偏微分方程的独特连续性有关的几个数学问题。 本研究关注的问题是，某些类型的微分方程的解是否可以等于零的一个开放的集合，而不是处处为零。在这里，关注的是最近的改进经典Carleman方法是否会产生一个弱的唯一连续性的微分不等式的小弱范数的潜力。 第二条线的调查关注加权和非加权Sobolev插值不等式向量场。向量场的Poincare-Sobolev型不等式有很多，但在插值不等式的方向上还没有做太多的工作。 工作也将做加权限制结果的傅立叶变换的权重在Festhiman-Phong类和努力将获得一个尖锐的覆盖引理在n维欧几里德空间细化早期结果斯坦和斯特龙伯格。 调和分析和不等式的组合是用来提供精确的偏微分方程从业者的背景信息。 开发的不等式提供必要的先验信息的解决方案，以及重要的估计其增长特性。 ***

9306387 Lu The project will focus on several mathematical problems related to the unique continuation property of partial differential equations. This research is concerned with the question of whether solutions of certain types of differential equations can be equal to zero on an open set without being zero everywhere. Here, the concern is whether recent improvements on the classical Carleman method will produce a weak unique continuation property of differential inequalities for small weak norm on the potentials. A second line of investigation concerns weighted and nonweighted Sobolev interpolation inequalities for vector fields. Inequalities of the Poincare-Sobolev type for vector fields abound, but nothing much has been done in the direction of interpolation inequalities. Work will also be done weighted restriction results for the Fourier transform with weights in the Fefferman-Phong class and an effort will be made to obtain a sharp covering lemma in n-dimensional Euclidean space refining earlier results of Stein and Stromberg. The combination of harmonic analysis and inequalities is used to provide precise background information for practitioners in partial differential equations. The inequalities developed give essential a priori information about solutions as well as important estimates of their growth properties. ***