Sparseness and Bellman Functions in Harmonic Analysis

Harmonic analysis can be broadly described as unlocking difficult operators or functions by breaking them down into smaller, simpler parts – albeit infinitely many. The type of parts depends on the nature of the object at hand. This proposal is focused on questions amenable to a dyadic approach: the simpler parts are usually some variation of a small, square signal which takes only two values.

Huge advancements have been made since Stefanie Petermichl’s discovery in 2000, that one of the most studied operators in analysis, the Hilbert transform, can be expressed in a meaningful way as an average of such simple operators. Central to this project will be the advancement of Petermichl’s work by relating it to several long-standing open questions in harmonic analysis.

An especially exciting path will be building a bridge between two “competing” methods in modern harmonic analysis, the Bellman function method and the sparse operator domination approach. This has the likelihood of becoming a new “method” in its own, as it involves new ideas from two leading methods in harmonic analysis. The project further includes organization of activities in harmonic analysis, as well as working with and training graduate students.

The starting point is an open problem on the dyadic square function which has baffled the field for quite some time. Here, weighted inequalities, specifically with Muckenhoupt weights, will be settled. These types of inequalities have recently dominated modern harmonic analysis, and the dyadic square function is one of the most important operators in the field - especially because it can be viewed as the starting point of Littlewood-Paley theory.

The question to then tackle will be the sharp power of the Muckenhoupt characteristic in the weak norm in the Hilbert space setting (parameter p=2). All other situations except the Hilbert space setting (p not equal to 2) are known, which makes it even more strange that this question is still open – usually, the Hilbert space setting is the “simple” case one extracts all other values from.

However, the nonlinear nature of the dyadic square function turns this all on its head, in a quite unique situation. The planned approach, to merge new ideas for Bellman functions with new ideas for sparse operators, has seen significant recent progress. Much broader applications of this approach are anticipated as well.

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.