Fractional Type Operators on the Heisenberg Group

For \(0 \leqslant \alpha < Q:=2n+2\), \(m \in \mathbb{N} \cap (1 - {\alpha}/{Q}, \infty )\), and \(m\) positive constants \(\alpha_1, \dots, \alpha_m\) such that \(\alpha_1 + \cdots + \alpha_m = Q - \alpha\), we consider the following generalization of the Riesz potential on the Heisenberg group \(\mathbb{H}^{n}= \mathbb{R}^{2n} \times \mathbb{R}\): \[ T_{\alpha, m}f(x,t) = \int_{\mathbb{H}^{n}} f(y,s) \prod_{j=1}^{m} \rho\bigl((A_j y, r_j^{-2} s)^{-1} \cdot ( x, t)\bigr)^{-\alpha_j} \, dy \, ds, \] where \(\rho\) is the Koranyi norm, the \(A_j\)'s are certain matrices belonging to \({\rm Sp}(2n, \mathbb{R}) \cap {\rm SO}(2n)\) and the \(r_j\)'s are certain positive constants. In this work, we will prove that \(T_{\alpha, m}\) can be extended to a bounded operator \(H^{p(\cdot)}(\mathbb{H}^n) \to L^{q(\cdot)}(\mathbb{H}^n)\), with \(p(\cdot) \in \mathcal{P}^{\log}(\mathbb{H}^n)\), \(0 < p_{-} \leqslant p_{+} < {Q}/{\alpha}\), \(p(A_j x, r_{j}^{-2} t)=p(x,t)\) for all \((x,t) \in \mathbb{H}^n\) and all \(j=1, \dots, m\), and \({1}/{q(\cdot)} = {1}/{p(\cdot)} - {\alpha}/{Q}\).