RUI: Harmonic Analysis on Weighted Lebesgue Spaces

该项目的目标是进一步研究谐波分析的首席研究员（PI），谐波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。这项研究将在三一学院进行，三一学院是一所文理学院，专注于本科教育，但要求其教师保持积极的研究项目。这个项目将进一步发展三一学院的数学研究。本科生将有机会参与该项目。这将有助于三一学院扩大其本科研究项目，将数学纳入其中。本科研究提高了本科教育的质量，并为学生在科学、数学和工程领域的高级工作做好了更好的准备。特别是，PI（他本人是墨西哥裔美国人）希望招募妇女和代表性不足的少数群体成员参与研究项目，从而增加数学和科学的多样性。在这个项目中，PI将研究谐波分析中的加权范数不等式。目标是进一步发展PI在三个密切相关领域的最新工作：卢比奥·德·弗朗西亚外推法、尖锐常数估计和双权范数不等式。包括PI在内的许多数学家最近在Muckenhoupt权值和外推理论的尖锐常数估计方面的工作已经产生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值的外推估计。这些结果将反过来应用于退化偏微分方程的正则性估计和变量Lebesgue空间的研究。PI也将用于双权估计，特别是对于奇异积分和Riesz势的分离碰撞猜想。

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.