The circular maximal operator on Heisenberg radial functions

TL;DR

This paper establishes Lebesgue space bounds for a circular maximal operator on Heisenberg radial functions, reducing the problem to a Euclidean maximal operator with unique geometric features.

Contribution

It introduces new Lebesgue space estimates for the Heisenberg group restricted to radial functions, linking it to a Euclidean maximal operator with non-standard curvature properties.

Findings

Lebesgue space estimates for the Heisenberg radial maximal function

Reduction to a Euclidean maximal operator with non-smooth curve distribution

Failure of standard curvature conditions in the associated Euclidean operator

Abstract

Lebesgue space estimates are obtained for the circular maximal function on the Heisenberg group $\mathbb{H}^1$ restricted to a class of Heisenberg radial functions. Under this assumption, the problem reduces to studying a maximal operator on the Euclidean plane. This operator has a number of interesting features: it is associated to a non-smooth curve distribution and, furthermore, fails both the usual rotational curvature and cinematic curvature conditions.