Weights in multilinear harmonic analysis and partial differential equations

University Of Alabama Tuscaloosa, Tuscaloosa AL

Investigators

Abstract

Kabe Moen The 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.