Weighted estimates for bilinear fractional integral operator on the Heisenberg group

TL;DR

This paper introduces a bilinear fractional integral operator on the Heisenberg group and characterizes the conditions for its boundedness between weighted Lebesgue spaces.

Contribution

It extends the theory of bilinear fractional integrals to the Heisenberg group and provides a complete characterization of boundedness conditions.

Findings

Complete characterization of exponents for boundedness

Extension of bilinear fractional integrals to Heisenberg group

Weighted inequalities established

Abstract

In this article, we introduce an analogue of Kenig and Stein's bilinear fractional integral operator on the Heisenberg group $\mathbb{H}^n$. We completely characterize exponents $\alpha, \beta$ and $\gamma$ such that the operator is bounded from $L^{p}(\mathbb{H}^n, |x|^{\alpha p})\times L^{q}(\mathbb{H}^n, |x|^{\beta q})$ to $L^{r}(\mathbb{H}^n, |x|^{-\gamma r})$.