Oscillating multipliers on noncompact Riemannian symmetric space \(\mathrm{SL}(3,\mathbb{H})/\mathrm{Sp}(3)\) (Q1389916)

Let \(M\) be the Riemannian symmetric space \(\mathrm{SL}(2,{\mathbb H})/\mathrm{Sp}(3)\). The author studies the convolution operator \(T_{a,b}\) on \(M\) associated with the radial multipliers \(m_{a,b}\), defined by \[ m_{a,b}(\lambda)=(\|\lambda\|^{2}+\|\rho\|^{2})^{-b/2} e^{i(\|\lambda\|^{2}+\|\rho\|^{2})^{a/2}},\quad \Re b\geq 0,\quad a>0. \] The main results of the paper are conditions for the operator \(T_{a,b}\) to be bounded on \(L^{p}(M)\). These results are an extension of results of \textit{S. Giulini} and \textit{S. Meda} [J. Reine Angew. Math. 409, 93--105 (1990; Zbl 0696.43007)] for noncompact Riemannian symmetric spaces of rank one.