Jennifer Duncan (at ICMAT until 30 November 2024, currently at the Universidad Autónoma de Madrid), David Beltrán (University of Valencia) and Jonathan Hickman (University of Edinburgh) sign “Off-diagonal estimates for the helical maximal function”, published in Proceedings of the London Mathematical Society.

Harmonic analysis has had, since the introduction of the spherical maximal function by Stein in the ‘70s, an abiding interest in variants on the Hardy-Littlewood maximal function that involve taking averages along lower-dimensional sets, in no small part due to their delicate nature and their connection with dispersive PDE. Interestingly, the spherical maximal function turns out to be much harder to prove in two dimensions (wherein it is referred to as the ‘circular’ maximal function) than in higher dimensions, and many new unexpected geometric features start to play a role in its behaviour. Lebesgue estimates on the circular maximal function were eventually proved by Bougain in the ‘80s, and in following decades the helical maximal function has come to be viewed as the appropriate higher dimensional analogue of the circular maximal function problem, in that it lifts the geometric subtleties of the problem into higher-dimensions in a way that the spherical maximal function does not.

The optimal diagonal Lebesgue estimates for the helical maximal function in three dimensions were proved independently by Beltran-Guo-Hickman-Seeger and Ko-Lee-Oh via methods that centred around distinct features of the problem. By developing a methodology that incorporates both geometries, David Beltran (University of Valencia), Jennifer Duncan (Autonomous University of Madrid) and Jonathan Hickman (University of Edinburgh) were able to establish the full conjectured off-diagonal range except for certain endpoints. Both the on- and off-diagonal problems remain open in higher dimensions.

Reference: Beltran D, Duncan J, Hickman J. Off-diagonal estimates for the helical maximal function. Proc Lond Math Soc. 2024;128(4):e12594. doi:10.1112/plms.12594.