RUI: Harmonic Analysis on Weighted Lebesgue Spaces

该项目的目标是进一步研究的主要研究者（PI）在谐波分析，数学的一个分支，这是一个活跃的研究领域，也是一个非常重要的应用到各种各样的问题，在物理和工程。 这项研究将在Trinity College进行，这是一所文科学院，专注于本科教育，但要求其教师保持积极的研究计划。 这个项目将进一步发展数学研究在Trinity。本科生将有机会参与该项目。这将有助于Trinity扩大其本科研究计划，包括数学。 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准备。 特别是，PI（他本人是墨西哥裔美国人）希望招募女性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将研究调和分析中的加权范数不等式。 我们的目标是进一步的PI在三个密切相关的领域：鲁比奥德弗朗西亚外推，尖锐的常数估计，和两个权重范数不等式的最新工作。 最近的工作由一些数学家，包括PI，尖锐的常数估计与Muckenhoupt权重和外推理论已经产生了新的结果和一些技术，应适用于其他问题，包括端点估计的功能有界平均振荡和外推估计矩阵权重。 这些结果反过来将产生应用程序的正则性估计退化偏微分方程和变量勒贝格空间的研究。 PI还将研究两个权重估计，特别是奇异积分和Riesz势的分离凸点猜想。

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.