A restriction theorem for the product Heisenberg group (Q1906622)

Let \(H\) be the Heisenberg group with multiplication given by \[ (z_1, t_1) (z_1', t_1') = \left( z_1 + z_1', t_1 + t_1' - {1 \over 2} \text{Re} z_1 z_1' \right) \] for \(z_1, z_1' \in \mathbb{C}\), \(t_1, t_1' \in \mathbb{R}\). Let \[ \pi_\lambda (z,t) = \pi_{\lambda_1} (z_1, t_1) \otimes \cdots \otimes \pi_{\lambda_n} (z_n, t_n) \] be the tensor product of Schrödinger representations of \(H\) in \(L_2 (H)\) given by \[ \pi_{\lambda_j} (z_j, t_j) (\varphi) (w) = \varphi (x + w) \exp \left( - \lambda_j \left( t_j + y_jw + {1 \over 2} x_jy_j \right) \right) \] for \(0 \neq \lambda_j \in \mathbb{R}\), \(z_j = x_j + iy_j\). Let \(P_\alpha (\lambda) = P_{\alpha_1} (\lambda_1) \otimes \cdots \otimes P_{\alpha_n} (\lambda_n)\), where \(P_{\alpha_j} (\lambda_j)\) are the projections on the subspace spanned by \(\{|\lambda_j |^{{1 \over 4}} h_{\alpha_j} (|\lambda_j |^{{1 \over 2}} x)\}\) for the Hermite functions \(h_{\alpha_j}\) and \(\alpha_j \in \mathbb{Z}_+\). Let \[ \bigl |R(f, \lambda) \bigr |^2_2 = \sum_{\alpha \in \mathbb{Z}^n_+} \bigl |\pi_{\lambda \bullet \nu} (f) P_\alpha (\lambda \bullet \nu) \bigr |^2_{HS} |\nu |^2 |\lambda_1 \dots \lambda_n |, \] where \(\nu = \nu (\alpha) = ({1 \over 2 \alpha_1 + 1}, \dots, {1 \over 2 \alpha_{n + 1}})\), \(|\nu |= \prod^n_{j = 1} {1 \over 2 \alpha_j + 1}\) and \(\lambda \bullet \nu = ({\lambda_1 \over 2 \alpha_1 + 1}, \dots, {\lambda_n \over 2 \alpha_n + 1})\). The author presents the following result: Let \(1 \leq p \leq {6 \over 5}\) for \(n = 2\) and \(1 \leq p \leq {4 \over 3}\) for \(n \geq 3\). Then there exists a constant \(C\) such that for all \(f \in L_p (H^n)\), we have \(\int_{\mathbb{R}^n} |R (f, \lambda) |^2_2 d \sigma \leq C |f |_p\), where \(d \sigma\) denotes the surface measure of the unit sphere of \(\mathbb{R}^n\).