Kakeya Sets and Directional Maximal Operators in the Plane

TL;DR

This paper characterizes when planar directional maximal operators are bounded on L^p spaces, linking boundedness to the absence of Kakeya-type sets in the set of directions.

Contribution

It provides a complete characterization of the boundedness of directional maximal operators in the plane based on the geometric properties of the directions set.

Findings

Operators are unbounded if directions admit Kakeya sets.

Operators are bounded if directions form a generalized lacunary set.

Boundedness holds for all p > 1 when no Kakeya sets are present.

Abstract

We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.