Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

在这个项目中，首席研究员将继续他的 加权范数不等式有关问题的研究 积分和微分算子 具体来说，他将重点 关于Poisson积分的加权范数不等式，分数阶 积分，希尔伯特变换，椭圆和抛物线 运营商 这一理论工作适用于（和动机 通过）以下问题：尖锐估计的推导 对于退化抛物方程的基本解， 各种特征值分布的估计 算子，并阐明了连续性的性质， 积分和微分方程的解。 数学家所谓的“分析”， 推导和应用“先验”估计的解决方案， 积分和微分方程 先验估计 本质上为数学家提供了关于 方程本身所包含的解， 推导不需要对假定的 溶液作为说明，某些类型的聚合物的解决方案可以是： 椭圆型方程边值问题满足 以下类型的先验估计;即，解决方案是 最大的，最小的，最小的，最大的，最小 边界值。 在这个项目中，首席研究员将 使用所谓的“范数不等式”来推导出尖锐的先验 各种积分和微分方程解的估计 方程

In this project the principal investigator will continue his study of questions related to weighted norm inequalities for integral and differential operators. Specifically he will focus upon weighted norm inequalities for Poisson integrals, fractional integrals, Hilbert transforms, and elliptic and parabolic operators. This theoretical work is applicable to (and motivated by) the following problems: the derivation of sharp estimates for the fundamental solutions of degenerate parabolic equations, the estimation of the distribution of eigenvalues of various operators, and the elucidation of the continuity properties of solutions of integral and differential equations. Much of what mathematicians call "analysis" involves deriving and applying "a'priori" estimates on the solutions of integral and differential equations. An a'priori estimate esentially provides a mathematician with information about a solution that is contained in the equation itself, and its derivation does not require a detailed knowledge of the putative solution. As an illustration, solutions of certain types of boundary value problems for elliptic equations satisfy an a'priori estimate of the following type; namely, the solution is bounded from above (below) by the largest (smallest) value of the boundary values. In this project the principal investigator will use what are called "norm inequalities" to derive sharp a'priori estimates on the solutions of various integral and differential equations.