On a Lipschitz Variant of the Kakeya Maximal Function

TL;DR

This paper improves bounds for a Lipschitz variant of the Kakeya maximal function, which is associated with Lipschitz maps from the plane to the circle, and conjectures these bounds are optimal.

Contribution

It provides improved estimates for the Lipschitz Kakeya maximal function and proposes that these bounds are likely optimal.

Findings

Enhanced bounds for the Lipschitz Kakeya maximal operator

Conjecture on the optimality of the established bounds

Advancement over previous estimates in the field

Abstract

In a prior work [Hilbert transform along smooth families of lines math.CA/0310345] the authors introduced a variant of the Kakeya maximal function associated with Lipschitz maps from the plane into the unit circle. In this paper, we improve the known estimates for this maximal operator--and raise the conjecture that the bounds established are optimal.