Mathematical Sciences: Harmonic Analysis Related To Weight Functions

本项目将开展的工作集中于对微分算子的估计以及由此可以证明的解的唯一性。特别强调关于算子低阶系数的p次方积分条件。这项研究的具体目标之一是刻画被称为微分方程式中的位势的唯一连续性质。这些先验度量决定了光滑函数何时必须从其导数的非常局部消失中相同地消失。特别的利息将支付给拉普拉斯算符，其中势在最大的逻辑类中进行，称为加藤类。还将就全球独特的延续问题开展相关工作，在这一问题中，消除超平面以下的潜力就是确保它们在任何地方都消失。C.Fefferman关于拉普拉斯微扰的结果证实，对于高指数位势，唯一延拓是可能的。这项研究将寻求将所有大于统一的幂的条件降低到可积性。还将努力将任何新结果应用于狄拉克运算符，其中唯一连续只能在特定情况下验证，而不是在函数类中。理解工作中所需的不平等的关键是必须建立的限制现象。基本上，这些现象表明，限制在球面上的傅里叶变换的二次范数是由其前件的加权二次积分控制的。权是微分不等式的势。这一研究将对椭圆型偏微分方程组的研究具有重要意义。

Work to be carried out on this project focuses on estimates of differential operators and resulting uniqueness of solutions which can be shown to follow from such estimates. Particular emphasis is to be placed on p-power integral conditions on the lower order coefficients of the operators. Among the specific goals of the research is one of characterizing what has been termed the unique continuation properties of potentials in differential equalities. These a priori measures determine when a smooth function must vanish identically from very local vanishing of its derivatives. Particular interest will be paid to the Laplace operator in which the potential is taken in the largest logical class, known as the Kato class. Related work will also be carried out on questions of global unique continuation in which vanishing of potentials below hyperplanes is to ensure that they vanish everywhere. Results due to C. Fefferman for perturbations of the Laplacian confirm that unique continuation is possible for potentials with high index. This research will seek to lower the condition to integrability for all powers greater than unity. Efforts will also be made to apply any new result to the Dirac operator where unique continuation is only verifiable in specific cases rather than function classes. They key to understanding inequalities needed in the work is a restriction phenomenon which must be established. Basically, these phenomena say that the quadratic norm of a Fourier transform confined to the surface of a sphere is dominated by the weighted quadratic integral of its antecedent. The weight is the potential of the differential inequality. This research will have fundamental implications in the field of elliptic partial differential equations.