Maximal estimates for the bilinear Riesz means on Heisenberg groups

TL;DR

This paper establishes boundedness results for maximal bilinear Riesz means on the Heisenberg group, identifying conditions on smoothness index and auxiliary operators crucial for these bounds.

Contribution

It introduces new bounds for bilinear Riesz means on the Heisenberg group, including defining auxiliary operators and analyzing their $L^{p}$ estimates.

Findings

Boundedness of $S^{ ext{alpha}}_{*}$ for large alpha

Identification of critical smoothness index alpha(p1,p2)

Development of auxiliary operators for analysis

Abstract

In this article, we investigate the maximal 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 $2\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 $\alpha (p_{1},p_{2})$. 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.