Joint BCAM-UPV/EHU Analysis and PDE seminar: Kakeya-type sets for Geometric Maximal Operators

LOCATION: BCAM Seminar Room and Online

Abstract
We pursue the programm of studying Maximal Operators in the plane by improving and developping further the techniques of Bateman. We establish an a priori estimate for arbitrary geometric maximal operator in the plane. Precisely we associate to any family of rectangles ℬ a geometric quantity λ[ℬ] called its analytic split and satisfying log(λ[ℬ]) ≲p ∥Mℬ∥𝑝/𝑝 for all 1 < p < ∞, where Mℬ is the Hardy-Littlewood type maximal operator associated to the family ℬ. We give then two applications in order to illustrate it. To begin with, this estimate allows us to classify the Lp(ℝ2) behavior of rarefied directional bases. As a second application, we prove that the basis ℬ generated by rectangle whose eccentricity and orientation are of the form

(𝑒𝑟, 𝑤𝑟)=(1/𝑛,sin⁡(𝑛) 𝜋/4)

for some n ∈ ℕ, yields a geometric maximal operator Mℬ which is unbounded on Lp(ℝ2) for any 1 < p < ∞.


Link to the session: 
https://us06web.zoom.us/j/99649860282?pwd=SE0vemtYMFlwbFBNTXQyOTBONG0vZz09


More info at https://sites.google.com/view/apdebilbao/home