Kakeya-type sets for Geometric Maximal Operators

Anthony Gauvan (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LMO - Laboratoire de Mathématiques d'Orsay

Given a family G of rectangles, to which one associates a tree [G], one defines a natural number λ [G] called its analytic split and satisfying, for all 1 < p < ∞ log(λ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < ∞. Precisely, if Ω is an arbitrary set of angles in [0, π 4), we prove that any rarefied basis B of the directional basis R Ω yields an operator MB that has the same L p-behavior than the directional maximal operator M Ω for 1 < p < ∞.

Mots clés: Maximal operators

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Probabilités [math.PR]

Anthony Gauvan. Kakeya-type sets for Geometric Maximal Operators: Maximal operators. 2021. ⟨hal-03295901v3⟩