Bilinear spherical maximal function on the Heisenberg group

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 ${\mathbb{H}}^n;n\ge 2$. 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 $T^{\ast }T$ argument adapted to this setting.