Variable Calderón-Hardy spaces on the Heisenberg group

Let ${\mathbb{H}}^n$ be the Heisenberg group and $Q=2n+2$. For $1<q<\mathrm{\infty }$, $\gamma >0$ and an exponent function $p(\cdot )$ on ${\mathbb{H}}^n$, which satisfy log-Hölder conditions, with $0<p_{-}\le p_{+}<\mathrm{\infty }$, we introduce the variable Calderón-Hardy spaces ${\mathcal{H}}_{q,\gamma }^{p(\cdot )}({\mathbb{H}}^n)$, and show for every $f\in H^{p(\cdot )}({\mathbb{H}}^n)$ that the equation $$\mathcal{L}F=f$$ has a unique solution $F$ in ${\mathcal{H}}_{q,2}^{p(\cdot )}({\mathbb{H}}^n)$, where $\mathcal{L}$ is the sublaplacian on ${\mathbb{H}}^n$, $1<q<\frac{n+1}{n}$ and $Q(2+\frac{Q}{q}{)}^{-1}<\underset{‾}{p}$.