Fractional type operators on the Heisenberg group

Abstract

Let $\rho(\cdot)$ be the Koranyi norm on the Heisenberg group $\mathbb{H}^{n} \equiv (\mathbb{R}^{2n} \times \mathbb{R}, \, \cdot \, )$ defined by \[ \rho(x,t) = \left( |x|^{4} + 16 t^{2} \right)^{1/4}, \,\,\,\, (x,t) \in \mathbb{H}^{n}. \] For $0 \leq \alpha < Q:=2n+2$, $m \in \mathbb{N} \cap \left(1 - \frac{\alpha}{Q}, \infty \right)$, and $m$ positive constants $\alpha_1, ..., \alpha_m$ such that $\alpha_1 + \cdot \cdot \cdot + \alpha_m = Q - \alpha$, we consider the following generalization of the Riesz potential on $\mathbb{H}^{n}$ \[ T_{\alpha, \, m}f(x,t) = \int_{\mathbb{H}^{n}} f(y,s) \prod_{j=1}^{m} \rho\left((A_j y, r_j^{-2} s)^{-1} \cdot ( x, t)\right)^{-\alpha_j} \, dy \, ds, \] where, in the case $0 < \alpha < Q$, the $A_j$'s are matrices belonging to $Sp (2n, \mathbb{R}) \cap SO(2n)$ and $r_j = 1$ for every $j=1, ..., m$; for $\alpha = 0$, we consider $A_j = r_j^{-1} \, I_{2n \times 2n}$ for every $j=1, ..., m$, where the $r_j$'s are positive constants such that $r_{i}^{2} - r_{j}^{2} \neq 0$ if $i \neq j$. In this note we study the behavior of these operators on variable Hardy spaces in $\mathbb{H}^{n}$.