Maximal estimates for the bilinear Riesz means on Heisenberg groups

In this article, we investigate the maximal bilinear Riesz means $S^{\alpha }_{*}$ associated to the sublaplacian on the Heisenberg group. We prove that the operator $S^{\alpha }_{*}$ is bounded from $L^{p_{1}}\times L^{p_{2}}$ into $$ for $2\leq p_{1}, p_{2}\leq \infty$ and $1/p=1/p_{1}+1/p_{2}$ when $$ is large than a suitable smoothness index $\alpha (p_{1},p_{2})$. For obtaining a lower index $\alpha (p_{1},p_{2})$, we define two important auxiliary operators and investigate their $L^{p}$ estimates,which play a key role in our proof.