On the lacunary spherical maximal function on the Heisenberg group

TL;DR

This paper studies the boundedness of a lacunary spherical maximal function on the Heisenberg group, establishing sparse bounds and weighted estimates using $L^p$ improving and continuity properties.

Contribution

It introduces new sparse bounds and weighted estimates for the lacunary spherical maximal function on the Heisenberg group, extending Euclidean techniques to a non-commutative setting.

Findings

Established $L^p$ boundedness of the maximal function

Derived sparse bounds leading to weighted inequalities

Proved $L^p$ improving and continuity properties of the spherical averages

Abstract

In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.