Abstract

Directional maximal operators average a function over long, thin rectangles whose orientations are drawn from a prescribed set of directions. Their analytic behaviour depends in subtle ways on the geometric and combinatorial structure of that set: sufficiently sparse, or lacunary, families yield bounded operators, while richer direction sets can generate Kakeya-type phenomena and dramatically different estimates.

In this talk, I will describe a tree-based framework for measuring the complexity of a set of directions. The central invariant, called the splitting number, records the extent of persistent branching in the associated direction tree. I will explain its connection with classical notions of finite-order lacunarity and show how this perspective leads naturally to estimates for directional maximal operators. The resulting picture brings together harmonic analysis, geometric measure theory, and combinatorics, and provides a useful language for distinguishing genuinely lacunary direction sets from those capable of supporting Kakeya-type constructions.

This is joint work with Edward Kroc and Juyoung Lee.

Biography

Professor Malabika Pramanik is a professor at the University of British Columbia, and director of the Banff International Research Station (BIRS) in Canada. She obtained a PhD in 2001 from the University of California, Berkeley, supervised by Michael Christ. Since then, she has made great contributions to modern harmonic analysis and geometric measure theory, and was an invited speaker at the International Congress of Mathematicians in 2022. She is a fellow and currently a vice-president of the American Mathematical Society, contributing greatly to community outreach and supporting inclusion and diversity in mathematics.