Bilinear Riesz means on the Heisenberg group

In this article, we investigate the 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 $L^{p}$ for $1\leq p_{1}, p_{2}\leq \infty$ and $1/p=1/p_{1}+1/p_{2}$ when $\alpha$ is large than a suitable smoothness index $\alpha (p_{1},p_{2})$. There are some essential differences between the Euclidean space and the Heisenberg group for studying the bilinear Riesz means problem. We make use of some special techniques to obtain a lower index $\alpha (p_{1},p_{2})$.