Non-Standard Sparse Estimates and Weighted Inequalities

Amherst College, Amherst MA

Investigators

Abstract

Mathematical analysis of models arising in fields such as physics and engineering often requires an understanding of functions both from a quantitative (size, rate of growth) and a qualitative (boundedness) point of view. Harmonic analysis provides tools for answering these questions through decomposing a function in a convenient way, analyzing the components, and then reassembling the information about the components to provide an overall understanding. This project investigates and further develops a new, surprisingly powerful approach in harmonic analysis known as sparse domination principles. In broad terms, sparse domination gives a way of studying general functions by comparing them to positive, localized, easy-to-understand versions, thereby obtaining results of the same strength and sharpness as with much more involved techniques. One of the main advantages of this new approach is its versatility: it can be employed not only in the classical setting, but also, after a potential redefinition of the objects involved, in a variety of more general scenarios. This project intends to explore the full strength and versatility of sparse domination techniques. The project comprises three research directions aimed at a deeper analysis of the concept of sparse bounds. One direction concerns matrix weighted estimates on vector-valued function spaces, an area that has enjoyed renewed interest, partially due to its connections to elliptic partial differential equations. Some recent sparse domination results, which reinterpret the traditional definition of function averages, suggest that progress on resolving the so-called A2 conjecture may be within reach. The principal investigator plans to continue research into the matrix A2 conjecture, seeking both further evidence in support of the result and potential counterexamples. A second direction, the theory of discrete operators, relates analytic number theory and harmonic analysis. Until recently, no weighted bounds had been established in the discrete setting, but the use of sparse domination has yielded new results and problems, such as the question of weighted estimates for the discrete oscillatory Hilbert transform with a general polynomial phase and the problem of weighted bounds for discrete fractional singular integrals. The principal investigator intends to study such functions with the goal of providing a broader weighted theory for arithmetic operators. A third direction concerns a different interpretation of sparse collections, previously considered only with respect to cubes. The principal investigator is interested in developing methods with cubes replaced by rectangles to answer questions regarding Bochner-Riesz operators and boundedness results involving averages associated to the Kakeya maximal function. 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.