Convolution operators and variable Hardy spaces on the Heisenberg group

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