On directional maximal operators in higher dimensions

TL;DR

This paper introduces a higher-dimensional notion of lacunarity to bound directional maximal operators in L^p spaces, characterizing direction sets that yield bounded operators and enabling differentiation of integrals along specific directions.

Contribution

It develops a new concept of lacunarity in higher dimensions and characterizes direction sets for bounded maximal operators, extending previous results.

Findings

Bounded directional maximal operators in higher dimensions for certain lacunarity classes.

Characterization of direction sets that produce bounded maximal operators.

Enabling Lebesgue differentiation using directional tubes instead of balls.

Abstract

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously considered by Nagel--Stein--Wainger, Sj\"ogren--Sj\"olin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in $L_{\text{loc}}^p(\mathbb{R}^n)$, replacing balls by tubes which point in these directions.