Research in Harmonic Analysis and Partial Differential Equations

本课题将重点研究与偏微分方程唯一延拓性有关的几个数学问题。研究一类微分方程的解在开集上是否等于零而不处处为零的问题。在这里，关注的是最近对经典Carleman方法的改进是否会在势上产生小弱范数的微分不等式的弱唯一延拓性质。第二项研究涉及向量场的加权和非加权Sobolev插值不等式。向量场的庞加莱-索博列夫型不等式有很多，但在插值不等式的方向上没有做太多的研究。我们还将对feffman - phong类中带有权重的傅里叶变换的加权限制结果进行研究，并努力在n维欧几里德空间中得到一个尖锐的覆盖引理，对Stein和Stromberg的早期结果进行改进。调和分析和不等式的结合被用来为偏微分方程的实践者提供精确的背景信息。所开发的不等式给出了关于解的基本先验信息以及对其增长性质的重要估计。* * *

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. ***