Mathematical Sciences: Weighted Norm Inequalities and Differential Equations

数学科学：加权范数不等式和微分方程

• 批准号: 9104195
• 负责人: Richard Wheeden
• 金额: $ 7.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-06-01 至 1994-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9104195&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Weighted; Norm; Inequalities

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.

在这个项目中，首席研究员将继续研究与积分和微分算子的加权范数不等有关的问题。具体地说，他将专注于泊松积分、分数次积分、希尔伯特变换以及椭圆和抛物算子的加权范数不等式。这项理论工作适用于下列问题：退化抛物方程基本解的精确估计的推导，各种算子本征值分布的估计，以及积分和微分方程解的连续性性质的阐明。许多数学家所说的“分析”涉及对积分方程解和微分方程解的“先验”估计的推导和应用。先验估计必然会为数学家提供关于方程本身所包含的解的信息，并且其推导不需要对假定解的详细了解。作为例子，某些类型的椭圆型方程边值问题的解满足如下类型的先验估计：即解从上(下)由边值的最大(最小)值有界。在这个项目中，主要研究人员将使用所谓的“范数不等”来推导各种积分和微分方程解的精确先验估计。