Collaborative Research: Non-homogeneous harmonic analysis: matrix weights, Banach space valued Calderon--Zygmund operators, and singular integrals on graphs with cycles

About This Grant

Harmonic analysis studies the behavior of various kinds of waves. This study is needed in a wide variety of applications including probability theory, physics, engineering, and medicine. Waves are complicated objects, and the mathematical language used to study them involves the so-called singular integral operators. The word singular shows that those operators are quite intricate. At the present moment singular integral operators are well researched in relatively smooth environments. However, nature is not smooth, and it's necessary to look at singular integrals in non-homogeneous environments. This is the principal aim of this award. Besides answering fundamental mathematical questions, the research will serve as a training ground for graduate students and young researchers. The PIs start their investigation with singular integrals with matrix weights, which arise in the study of the regularity of vector stationary stochastic processes. We continue with singular integrals on graphs that have cycles -- this is a completely new area developed for multi-parameter singular integrals. These, in turn, are needed for understanding the regularity of non-linear partial differential equations. The study of Banach space valued singular integrals on a hypercube (a graph with cycles) is related (very surprisingly) to open questions in theoretical computer science. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.