RUI: Harmonic Analysis on Weighted Lebesgue Spaces

这个项目的目标是进一步研究调和分析的首席研究员(PI)，调和分析是数学的一个分支，是一个活跃的研究领域，也是一个非常重要的应用于物理和工程中的各种问题的分支。这项研究将在三一学院进行，这是一所文科学院，专注于本科教育，但要求教职员工保持积极的研究项目。该项目将进一步推动三一学院数学研究的发展。本科生将有机会参与该项目。这将帮助三一扩大其本科生研究项目，将数学包括在内。本科研究提高了本科教育的质量，更好地为学生在科学、数学和工程方面的高级工作做好了准备。特别是，国际和平协会(他本人是墨西哥裔美国人)希望招募妇女和代表不足的少数群体的成员参与研究项目，从而增加数学和科学的多样性。在这个项目中，国际和平协会将研究调和分析中的加权范数不平等。其目的是促进PI在三个密切相关领域的最新工作：卢比奥·德·弗朗西亚外推、尖锐的常量估计和双权范数不等式。最近，包括PI在内的许多数学家利用Muckenoupt权重和外推理论对锐常数估计进行了研究，得到了新的结果和一些应该适用于其他问题的技术，包括有界平均振荡函数的端点估计和矩阵权重的外推估计。这些结果将反过来应用于退化偏微分方程正则性估计和可变勒贝格空间的研究。PI还将对双权估计进行工作，特别是对奇异积分和Riesz势的分离凸起猜想。

The goal of this project is to further the research of the principal investigator (PI) in harmonic analysis, a branch of mathematics that is an area of active research and also one that is very important for its applications to a wide variety of problems in physics and engineering. The research will be conducted a Trinity College, a liberal arts college that focuses on undergraduate education but requires its faculty to maintain active research programs. This project will further the development of mathematical research at Trinity. Undergraduate students will be given the opportunity to participate in the project. This will help Trinity expand its undergraduate research programs to include mathematics. Undergraduate research enhances the quality of undergraduate education and better prepares students for advanced work in science, mathematics, and engineering. In particular, the PI (himself a Mexican-American) hopes to recruit women and members of underrepresented minority groups to participate in the research project, thereby increasing diversity in mathematics and the sciences.In this project the PI will study weighted norm inequalities in harmonic analysis. The goal is to further the recent work of the PI in three closely related areas: Rubio de Francia extrapolation, sharp constant estimates, and two-weight norm inequalities. Recent work by a number of mathematicians, including the PI, on sharp constant estimates with Muckenhoupt weights and extrapolation theory has yielded new results and a number of techniques that should be applicable to additional problems, including endpoint estimates for functions of bounded mean oscillation and extrapolation estimates for matrix weights. These results in turn will yield applications to regularity estimates for degenerate partial differential equations and to the study of variable Lebesgue spaces. The PI will also work on two-weight estimates, particularly on the separated bump conjecture for singular integrals and Riesz potentials.