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. ***
