Averaging, spectral multipliers, sparse domination and subelliptic operators

This project focuses on harmonic analysis, a vast area within the mathematical discipline of analysis. Harmonic analysis is concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms. Its methods have found wide applications in understanding phenomena in the natural sciences and engineering.

Harmonic analysis provides efficient mathematical tools for these disciplines and contributes to the unification of seemingly unrelated areas. A main objective is to expand the current mathematical toolbox in harmonic analysis to contribute towards a deeper theoretical understanding that will ultimately be beneficial for applications. The mentoring of graduate students in research is an important educational component of the project.

The principal investigator will work on several projects in harmonic analysis. The first part of the project is concerned with the precise regularity properties of certain averages over curves and the boundedness of associated maximal operators in Lebesgue spaces. The model cases tend to be convolution operators but eventually the goal is to understand more general non-convolution averaging operators.

A second part of the project seeks to bound maximal functions with partial, possibly fractal, dilation sets, and to understand how various notions of dimensions are relevant for the Lebesgue space boundedness properties of such maximal operators. A third part of the project deals with multiplier transformations on the Heisenberg group (functions of a certain subelliptic operator).

Via a Fourier transform they are connected to wave operators, and the fine structure of wave propagation on the Heisenberg group plays an important role in the multiplier problem. A fourth part of the project is about sparse domination inequalities; one seeks to expand the scope of the theory to cover endpoint situations.

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.