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 $\mathbb{ H}^{ n}$.
Categories: math.CA