Convolution operators and variable Hardy spaces on the Heisenberg group

Abstract

Let \(\mathbb{H}^{n}\) be the Heisenberg group. For \(0 \leq \alpha < Q=2n+2\) and \(N \in \mathbb{N}\) we consider exponent functions \(p (\cdot) \colon \mathbb{H}^{n} \to (0, +\infty)\), which satisfy log-Hölder conditions, such that \(\frac{Q}{Q+N} < p_{-} \leq p (\cdot) \leq p_{+} < \frac{Q}{\alpha}\). 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.