Sparse Bounds and Improving Estimates, Continuous and Discrete

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Averaging always smooths out, or improves, functions or data. In the continuous case, this phenomena has been intensively studied for decades. If we average over objects in a discrete setting like the integers, or some other high dimensional lattice, the smoothing properties of the average have only recently started to be studied. A natural object to study is the average over a discrete sphere in a five dimensional lattice. The proof that the average improves functions engages a range of deep aspects within number theory and analysis. However, resolving the natural limits for this question still seem difficult. In this project, these questions will be explored in a setting that involves both continuous and discrete phenomena. These questions are also elementary to state, which makes them amenable for mentoring programs, from undergraduate through postdoctoral levels. An improving inequality in the continuous setting has been widely studied since the 1970's. However, the corresponding questions in the discrete setting have only just attracted attention. For instance, averages over the discrete sphere in the d-dimensional integers, with d at least 5, have a rich theory of improving inequalities, paralleling the much better known continuous case. The proofs however involve complications arising from multi-frequency analysis, as well as fine estimates on Kloosterman sums. There is a richer theory, the outlines of which are appearing, giving a sharp range of inequalities for some arithmetic varieties. This subject also allows for the proof of certain sparse bounds. The latter are scale-free versions of the improving inequalities. A sparse bound immediately implies other weighted and vector valued consequences. The latter are new in this subject. These questions reveal new aspects of these averaging operators, and require new modes of investigation, deepening the connection between harmonic analysis and number theory. 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.