Mathematical Sciences: Harmonic Analysis Related To Weight Functions

这一项目的工作重点是估算 微分算子的解的唯一性 这可以从这样的估计中看出。 特别 重点放在p-幂积分条件上， 算子的低阶系数。 在具体的 研究的目标是描述 称之为势的唯一连续性质， 微分方程 这些先验测量确定何时 光滑函数必须从非常局部的 其衍生物的消失。 将支付特别利息 到拉普拉斯算子，其中的潜力是采取在 最大的逻辑类，称为Kato类。 还将就全球环境问题开展相关工作， 唯一的延续，其中下面的电位消失 超平面就是确保它们无处不在 结果 由于C.拉普拉斯算子扰动的Febriman证实 对于具有较高电位，唯一的延续是可能的 指数. 这项研究将寻求降低条件， 所有大于1的幂的可积性。 努力将 也可以将任何新的结果应用于狄拉克算子，其中 唯一连续仅在特定情况下可验证， 函数类。 理解工作中所需的不平等的关键是 这是一个必须建立的限制现象。 基本上， 这些现象表明傅立叶的二次范数 一个球体的表面上的变换由 它的前件的加权二次积分。 重量是 微分不等式的潜力。 这项研究将 在椭圆偏微分方程领域具有重要意义 微分方程

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.