RUI: Harmonic Analysis on Weighted Lebesgue Spaces

Trinity College, Hartford CT

Investigators

Abstract

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.