Lebesgue space estimates for spherical maximal functions on Heisenberg groups

Abstract

We prove L p → L q estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse domination for global maximal operators. We also consider lacunary variants, and extensions to Métivier groups.

Spherical maximal operators on Heisenberg groups: Restricted dilation sets

“…Note that up to endpoints we recover the corresponding sharp results for E = [1,2] in [24]. Further examples of quasi-Assouad regular sets include convex sequences, self-similar sets with β = γ (such as Cantor sets) and many more; see [23, §6].…”

“…of the local version M [1,2] was investigated by Bagchi, Hait, Roncal and Thangavelu [2], who were motivated by applications to sparse bounds and weighted estimates for the corresponding global maximal function, as well as for a lacunary variant. L p → L q results that are sharp up to endpoints, for both the single averages and full local maximal function, were proved in our previous paper [24].…”

“…The purpose of this paper is to extend recent L p -improving results for local spherical maximal functions on the Heisenberg group in [24] to the setting of restricted dilation sets. To fix notation, for n ∈ N, we let H n denote the Heisenberg group of Euclidean dimension d = 2n + 1.…”

$L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions

“…For n ě 2, L p boundedness of M H n on the optimal range was independently proved by Müller and Seeger [10], and by Narayanan and Thangavelu [12]. Furthermore, for n ě 2, Roos, Seeger and Srivastava [15] recently obtained the complete L p -L q estimate for M H n except for some endpoint cases. Also see [7] for related results.…”

“…Beltran, Guo, Hickman and Seeger [2] obtained L p boundedness of M H 1 on the Heisenberg radial functions for p ą 2. In the perspective of the results concerning the local maximal operators ( [17,18,8,15]), it is natural to consider L p -L q estimate for M H 1 . The main result of this paper is the following which completely characterizes L p improving property of M H 1 on Heisenberg radial function except for some borderline cases.…”