Stein-Weiss inequality revisit on Heisenberg group

TL;DR

This paper revisits the Stein-Weiss inequality on the Heisenberg group by analyzing fractional integral operators with Zygmund dilation, establishing a two-weight norm inequality characterization, and deriving a new inequality in this setting.

Contribution

It provides a new characterization of two-weight inequalities for fractional integrals on the Heisenberg group, extending Stein-Weiss inequalities to this non-commutative setting.

Findings

Established a two-weight norm inequality characterization.

Derived a Stein-Weiss inequality on the Heisenberg group.

Analyzed kernels satisfying Zygmund dilation.

Abstract

We study a family of fractional integral operators defined on Heisenberg group whose kernels satisfy Zygmund dilation. We give a characterization between a two-weight norm inequality and the necessary constraints by considering the weights to be suitable powers. As a result, we obtain a Stein-Weiss inequality on Heisenberg group.