Weights in multilinear harmonic analysis and partial differential equations

多线性调和分析和偏微分方程中的权重

• 批准号: 1201504
• 负责人: Kabe Moen
• 金额: $ 10.26万
• 依托单位: University of Alabama Tuscaloosa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201504&HistoricalAwards=false
• 关键词: Weights; multilinear; harmonic; analysis; partial

Kabe MoenThe overall goal of this project is to study the theory of weights in harmonic analysis and partial differential equations. There are two distinct main themes that share the theory of weights as a common thread. The first theme aims to further develop the theory of weighted inequalities in the multilinear setting. This theory, while still in its adolescence, is expected to have significant ramifications to several multilinear operators. For example it may be used to improve the known bounds of the bilinear Hilbert transform and related bilinear maximal function; an emphasis of study will be to develop multilinear extrapolation techniques. The second theme focuses on the regularity of solutions to degenerate p-Laplacian equations. Such equations arise naturally in the study of mappings of bounded distortion (quasiconformal mappings) and more generally mappings of finite distortion. Considering distortion functions that belong to certain classes of weights allows one to work below the natural size assumptions, yet still obtain interesting regularity results. The common thread between the two motifs is the theory of weights.This mathematics research project contributes to the long-standing synergy between the two fields of harmonic analysis and partial differential equations, in particular to the study of nonlinear quantities. In the present setting such nonlinear terms correspond to mappings of finite distortion that are relevant within the pure mathematics realm and also have far-reaching applications to other sciences, such as physics and engineering; materials sciences; hydrodynamics; and medical imaging. Moen will continue to establish fruitful collaborations both nationally and abroad, and the outcomes will be disseminated through publications of research articles, as well as presentations at professional meetings. Moen will integrate this research project with education, mentoring, and community outreach. In particular, he plans to continue his community involvement by establishing outreach programs that will connect local grade schools with the University of Alabama-Tuscaloosa.

这个项目的总体目标是学习调和分析和偏微分方程式中的权重理论。有两个截然不同的主题将权重理论作为一条共同的主线。第一个主题旨在进一步发展多线性背景下的加权不等式理论。虽然这一理论仍处于青春期，但预计将对几个多线性算子产生重大影响。例如，它可以用来改进双线性Hilbert变换和相关双线性极大函数的已知界；研究的重点将是发展多线性外推技术。第二个主题是关于退化p-拉普拉斯方程解的正则性。这类方程在研究有界偏差映射(拟共形映射)和更一般的有限偏差映射时自然产生。考虑属于某些权重类别的失真函数允许人们在自然大小假设下工作，但仍然获得有趣的规律性结果。这两个主题之间的共同主线是权重理论。这一数学研究项目有助于调和分析和偏微分方程这两个领域之间的长期协同，特别是对非线性量的研究。在目前的设置中，这些非线性项对应于在纯数学领域内相关的有限失真的映射，并且还对其他科学具有深远的应用，例如物理和工程、材料科学、流体力学和医学成像。教育部将继续在国内外建立卓有成效的合作，并将通过发表研究文章和在专业会议上发表演讲来传播成果。MOEN将把这一研究项目与教育、指导和社区推广结合起来。特别是，他计划通过建立外联计划，将当地小学与阿拉巴马大学塔斯卡卢萨分校联系起来，继续他的社区参与。