Maximal Averages Over Hypersurfaces

本项目的目的是分析与凸有限型超曲面相关的调和分析和偏微分方程中出现的各种算子。该项目的主要重点是研究e.m. Stein引入的最大平均算子，以及Littman和Strichartz最初研究的平均卷积算子。Iosevich给出了极大平均算子Lp有界性的一组充分必要条件，以及平均卷积算子（Lp, Lq）有界性的一组充分必要条件。这两组条件都用距离函数对超曲面切平面的负幂的可积性来表示。Iosevich还将研究与Rn中光滑凸体相关的最大Bochner-Riesz均值的Lp有界性问题。据推测，这些算子应该具有与球相关的标准最大Bochner-Riesz意味着相同的映射属性。对谐波分析中最大平均算子和其他类似算子的研究部分是由以下有趣的问题引起的：我们能多接近于从该数据的各种平均值中恢复一组数据？这个问题具有潜在的实用价值，因为科学家经常被要求根据平均信息做出预测。例如，气象学家根据附近城镇过去几年的平均降雨量来预测特定地点的降雨量。地震学家根据周边地区的震动模式进行地震预测。粗略地说，研究这些现象所涉及的权衡如下。如果数据非常精确，那么一般来说，它可以从任何一种合理的平均值中恢复出来。如果数据不够精确，那么我们必须确保平均过程弥补了数据的不足。本课题的主要目的是研究某一类数学函数给出的数据在曲面上取平均值时的平均现象。

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.