Convolution operators and variable Hardy spaces on the Heisenberg group

Let ${\mathbb{H}}^n$ be the Heisenberg group. For $0\le \alpha <Q=2n+2$ and $N\in \mathbb{N}$ we consider exponent functions $p(\cdot ):{\mathbb{H}}^n\to (0,+\mathrm{\infty })$, which satisfies Hölder conditions, such that $\frac{Q}{Q+N}<p_{-}\le p(\cdot )\le p_{+}<\frac{Q}{\alpha }$. In this article we prove the $H^{p(\cdot )}({\mathbb{H}}^n)\to L^{q(\cdot )}({\mathbb{H}}^n)$ and $H^{p(\cdot )}({\mathbb{H}}^n)\to H^{q(\cdot )}({\mathbb{H}}^n)$ boundedness of convolution operators with kernels of type $(\alpha ,N)$ on ${\mathbb{H}}^n$, where $\frac{1}{q(\cdot )}=\frac{1}{p(\cdot )}-\frac{\alpha }{Q}$. In particular, the Riesz potential on ${\mathbb{H}}^n$ satisfies such estimates.