Bilinear Riesz means on the Heisenberg group

Abstract

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 $1\le p_1,p_2\le \infty$ 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 $\alpha (p_1,p_2)$.