Seminar: Journé’s Lemma, Rey’s Conjecture and Zygmund’s Conjecture

Abstract: Journé’s covering lemma is a basic tool for controlling overlapping rectangles in product spaces. We discuss refinements of such rectangular packing estimates, including new probabilistic identities and a method for passing from L¹ control to exponential integrability. These estimates lead to a conjecture of G. Rey concerning overlap functions of sparse incomparable families of rectangles, and finally to Zygmund’s conjecture. For the full m-parameter strong maximal function, the weak-type endpoint estimate involves the logarithmic power m-1. Zygmund’s conjecture predicts that making one side length depend monotonically on the others saves one logarithm, reducing this power to m-2. We explain how these ideas fit together and what is now known.