Weights in multilinear harmonic analysis and partial differential equations

这个项目的总体目标是研究谐波分析和偏微分方程中的权重理论。有两个截然不同的主题共享权重理论作为一个共同的线索。第一个主题旨在进一步发展多线性环境下的加权不等式理论。这一理论虽然仍处于发展阶段，但有望对几种多线性算子产生重大影响。例如，它可用于改进双线性希尔伯特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。第二个主题着重于退化p-拉普拉斯方程解的正则性。在研究有界畸变的映射（拟共形映射）和更一般的有限畸变的映射时，自然会出现这样的方程。考虑属于某些权重类别的失真函数，可以在低于自然大小假设的情况下工作，但仍然可以获得有趣的规律性结果。这两个主题之间的共同点是权重理论。这个数学研究项目有助于调和分析和偏微分方程两个领域之间的长期协同，特别是对非线性量的研究。在目前的情况下，这些非线性术语对应于与纯数学领域相关的有限失真的映射，并且对其他科学（如物理和工程）也有深远的应用；材料科学;流体动力学;还有医学成像。Moen将继续在国内外建立卓有成效的合作关系，其成果将通过发表研究文章以及在专业会议上发表报告的方式进行传播。莫恩将把这个研究项目与教育、指导和社区外展结合起来。特别是，他计划通过建立将当地小学与阿拉巴马大学塔斯卡卢萨分校联系起来的外展项目，继续他的社区参与。

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.