Bilinear spherical maximal function on the Heisenberg group

TL;DR

This paper introduces bilinear spherical means on the Heisenberg group and establishes sharp $L^p$ estimates for associated maximal operators, advancing harmonic analysis on non-commutative groups.

Contribution

It develops the first $L^p$ bounds for bilinear spherical maximal functions on the Heisenberg group, including sharp results for the full maximal operator.

Findings

Established $L^{p_1} imes L^{p_2} o L^p$ bounds for bilinear averages

Proved sharp $L^p$ estimates for the full bilinear maximal operator

Developed new estimates and adapted ergodic theorems for non-commutative harmonic analysis

Abstract

We introduce the bilinear Nevo-Thangavelu spherical means on the Heisenberg group $\mathbb{H}^n,$ and derive $L^{p_1}(\mathbb{H}^n) \times L^{p_2}(\mathbb{H}^n) \to L^{p}(\mathbb{H}^n)$ estimates for the single-scale bilinear averaging operators, the (full) bilinear Nevo-Thangavelu maximal operator and finally for the bilinear lacunary maximal operator on $\mathbb{H}^n; n \geq 2$. Our result for the full maximal operator is sharp. The principal tools in our analysis include newly developed estimates for single-scale bilinear averages, Hopf's maximal ergodic theorem, and a $T^*T$ argument adapted to this setting.