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 \[ ρ(x,t) = \left( |x|^{4} + 16 t^{2} \right)^{1/4}, \,\,\,\, (x,t) \in \mathbb{H}^{n}. \] For $0\le \alpha <Q:=2n+2$, $m\in \mathbb{N}\cap \left(1-\frac{\alpha }{Q},\infty \right)$, 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 ${\mathbb{H}}^n$ \[ T_{α, \, m}f(x,t) = \int_{\mathbb{H}^{n}} f(y,s) \prod_{j=1}^{m} ρ\left((A_j y, r_j^{-2} s)^{-1} \cdot ( x, t)\right)^{-α_j} \, dy \, ds, \] 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 $r_i^2-r_j^2\ne 0$ if $i\ne j$. In this note we study the behavior of these operators on variable Hardy spaces in ${\mathbb{H}}^n$.

arXiv comment17 pages. arXiv admin note: text overlap with arXiv:2403.11467