Variable Calder\'on-Hardy spaces on the Heisenberg group

Abstract

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 $Q (2 + \frac{Q}{q})^{-1} < \underline{p}$.