Let $\mathbb{ H}^{ n}$ be the Heisenberg group and $Q = 2n+2$. For $1 < q < \infty$, $\gamma > 0$ and an exponent function $p(\cdot)$ on $\mathbb{ H}^n$, which satisfy log-H\"older conditions, with $0 < p_{ -} \leq p_{ +} < \infty$, we introduce the variable Calder\'on-Hardy spaces $\mathcal{ H}^{ p(\cdot)}_{ q, \gamma}(\mathbb{ H}^{ n})$, 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 $\mathcal{ H}^{ p(\cdot)}_{ q, 2}(\mathbb{ H}^{ n})$, where $\mathcal{ L}$ is the sublaplacian on $\mathbb{ H}^{ n}$, $1 < q < \frac{ n+1}{ n}$ and $Q (2 + \frac{ Q}{ q})^{ -1} < \underline{ p}$.
Categories: math.CA