Bilinear Riesz means on the Heisenberg group

Abstract

In this article, we investigate the bilinear Riesz means $S^{α}$ associated to the sublaplacian on the Heisenberg group. We prove that the operator $S^{α}$ 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 $ α$ is large than a suitable smoothness index $α(p_{1},p_{2})$. 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 $α(p_{1},p_{2})$.