Oscillating multipliers on the Heisenberg group (Q2773255)

Let \({\mathcal L}\) be the subLaplacian on the Heisenberg group \(\mathbb{H}^n= \mathbb{C}^n\times \mathbb{R}\) with the group law NEWLINE\[NEWLINE(z,t)(w,s)= (z+w,t+s+ \frac 12 \text{Im} z\cdot \overline w).NEWLINE\]NEWLINE The subLaplacian \({\mathcal L}\) admits the spectral decomposition \({\mathcal L}=\int^\infty_0 \lambda dE_\lambda\). Given a bounded function \(m\) defined on \((0,\infty)\) one can define the operator \(m( {\mathcal L})\) by setting \(m({\mathcal L})f= \int^\infty_0 m(\lambda)d E_\lambda f\). A result of \textit{D. Müller} and \textit{E. M. Stein} shows that the operator \({\mathcal L}^{-1/2} \sin\sqrt {\mathcal L}\) is bounded on \(L^p(\mathbb{H}^n)\) for all \(p\) satisfying \(|1/p-1/2 |<1/(2n)\) [Rev. Mat. Iberoam. 15, 297-334 (1999; Zbl 0935.43003)]. The function \(u(z,t,s)= {\mathcal L}^{-1/2} \sin s\sqrt{\mathcal L}f(z,t)\) solves the Cauchy problem for the wave equation associated with the subLaplacian, namely, \(\partial^2_su(z,t,s)= {\mathcal L}u(z,t,s)\) with initial conditions \(u(z,t,0)=0\), \(\partial_su(z,t,0)= f(z,t)\).NEWLINENEWLINENEWLINEIn this paper the result of Müller and Stein is improved when \(f(z,t)\) is band limited in the \(t\)-variables. Let \(L^p_B (\mathbb{H}^n)\) stand for those functions \(f(z,t)\) in \(L^p (\mathbb{H}^n)\) for which the partial inverse Fourier transform \(f^\lambda(z)\) in the \(t\)-variables is supported in \(|\lambda |\leq B\). The authors have the following theorem:NEWLINENEWLINENEWLINETheorem. The operators \({\mathcal L}^{-\alpha/2} J_\alpha (\sqrt{\mathcal L})\), where \(J_\alpha\) is the Bessel function of order \(\alpha\), are bounded on \(L^p_B(\mathbb{H}^n)\) for \(|1/p-1/2|< (2\alpha+1) /(4n-2)\) provided \(6\alpha\leq 4n-5\). Otherwise, they are bounded on \(L_B^p (\mathbb{H}^n)\) in the smaller range \(|1/p-1/2 |<(2 \alpha+3)/(4n+4)\).NEWLINENEWLINENEWLINECorollary. Let \(n\geq 2\). The operator \({\mathcal L}^{-1/2} \sin\sqrt {\mathcal L}\) is bounded on \(L^p_B(\mathbb{H}^n)\) for \(|1/p-1/2 |<1/(2n-1)\).