Maximal Averages Over Hypersurfaces

超曲面上的最大平均值

• 批准号: 9706825
• 负责人: Alex Iosevich
• 金额: $ 5.15万
• 依托单位: Wright State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 1999-06-04
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706825&HistoricalAwards=false
• 关键词: Maximal; Averages; Over; Hypersurfaces

ABSTRACT Iosevich The purpose of this project is to analyze various operators that arise in harmonic analysis and partial differential equations associated with the convex finite type hypersurfaces. The main focus of the project is the study of the maximal averaging operators introduced by E. M. Stein, and the averaging convolution operators originally studied by Littman and Strichartz. Iosevich proposes a set of necessary and sufficient conditions for the Lp boundedness of maximal averaging operators, and a set of necessary and sufficient conditions for the (Lp, Lq) boundedness of the averaging convolution operators. Both sets of conditions are expressed in terms of the integrability of the negative powers of the distance functions to the tangent planes to the hypersurface. Iosevich will also work on the problem of Lp boundedness of the maximal Bochner-Riesz means associated with the smooth convex bodies in Rn. It has been conjectured that these operators should have the same mapping properties as the standard maximal Bochner-Riesz means associated with the ball. The study of the maximal averaging operators and other similar operators in harmonic analysis is partially motivated by the following interesting question: How close can we come to recovering a set of data from the various kinds of averages of that data? The question is of potential practical value since scientists are often called upon to make predictions based on average information. For example, meteorologists make predictions about the rainfall in the particular location based on the average rainfall in the years past in the nearby towns. Seismologists make earthquake predictions based on the pattern of shocks in the surrounding area. The tradeoff involved in the study of these phenomena is, roughly speaking, the following. If the data is very precise, then it can, generally speaking, be recovered from any kind of a reasonable average. If the data is less pre cise, then we have to make sure that the averaging process compensates the deficiencies of the data. The main thrust of this project is to study the averaging phenomenon when the data is given by a certain kind of a mathematical function, and the average is taken over a curved surface.

摘要 约舍维奇 这个项目的目的是分析各种运营商， 出现在调和分析和偏微分方程中 与凸有限型超曲面相关。主要的重点 该项目的主要内容是研究最大平均算子 由E. M. Stein和平均卷积算子 最初是由Littman和Litthartz研究的。Iosevich提出了一个 一组极大平均算子Lp有界的充要条件，一组极大平均算子Lp有界的充要条件， 平均卷积的（Lp，Lq）有界性条件 运营商这两组条件都表示为： 距离函数的负幂的可积性， 超曲面的切平面Iosevich还将研究最大Bochner-Riesz平均的Lp有界性问题 与Rn中的光滑凸体相关。已经 说明这些运算符应具有相同的映射 性质作为标准的最大Bochner-Riesz平均相关联 用球。 最大平均算子及其它类似算子的研究 算子在调和分析中的部分动机是 以下有趣的问题：我们距离复苏还有多远 从这些数据的各种平均值中得到一组数据？的 这个问题具有潜在的实用价值，因为科学家们经常 根据平均信息进行预测。为 例如，气象学家预测 根据过去几年的平均降雨量计算的特定位置 在附近的城镇。地震学家预测地震的依据是 周围地区的地震模式的权衡 参与研究这些现象的是，粗略地说， 以下.如果数据非常精确，那么它通常可以 也就是说，从任何一种合理的平均值中恢复。如果 数据不太精确，那么我们必须确保平均值 这个过程弥补了数据的不足。主旨 这个项目的目的是研究平均现象时，数据 是由某种数学函数给出的 在曲面上取平均值。