Maximal estimates for the bilinear Riesz means on Heisenberg groups

In this article, we investigate the maximal bilinear Riesz means $S_{\ast }^{\alpha }$ associated to the sublaplacian on the Heisenberg group. We prove that the operator $S_{\ast }^{\alpha }$ is bounded from $L^{p_1}\times L^{p_2}$ into $$ for $2\le p_1,p_2\le \mathrm{\infty }$ and $1\mathrm{/}p=1\mathrm{/}{p}_1+1\mathrm{/}{p}_2$ when $$ 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.