Bilinear Riesz means on the Heisenberg group

TL;DR

This paper studies the boundedness of bilinear Riesz means operators on the Heisenberg group, establishing conditions under which they map products of Lebesgue spaces into another Lebesgue space, highlighting differences from Euclidean space.

Contribution

It proves boundedness of bilinear Riesz means on the Heisenberg group for large smoothness indices, using novel techniques to handle differences from Euclidean space.

Findings

Boundedness of $S^{eta}$ for large $eta$ on $L^{p_1} imes L^{p_2} o L^{p}$.

Identification of the smoothness threshold $eta(p_1,p_2)$ for boundedness.

Highlighting key differences between Euclidean and Heisenberg group cases.

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\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 $\alpha (p_{1},p_{2})$.