Mapping properties of the Schr\"odinger maximal function on Damek--Ricci spaces

For $f \in \mathscr{S}^2(\mathcal S)_{o}$, the collection of radial $L^2$-Schwartz class functions on Damek--Ricci spaces $\mathcal S$, we consider the Schr\"odinger maximal function, \begin{equation*} S^* f(x):= \displaystyle\sup_{0<t<4/Q^2} \left|S_tf(x)\right|\:,\:\:\:\:\:\:x\in\mathcal S\:, \end{equation*} corresponding to the Laplace--Beltrami operator $\Delta$ with initial data $f$. We first obtain the complete description of the pairs $(q, \alpha) \in [1, \infty] \times [0,\infty)$ for which the estimate \begin{equation*} {\|S^*f\|}_{L^q\left(B_R\right)} \le C_R\: {\|f\|}_{H^{\alpha}(\mathcal S)}\:, \end{equation*} holds on geodesic balls $B_R$, for all $f \in \mathscr{S}^2(\mathcal S)_{o}$. Our results are sharp and agree with the Euclidean case. We also prove that for all $f \in \mathscr{S}^2(\mathcal S)_{o}$, the following global estimate \begin{equation*} {\|S^*f\|}_{L^{2,\infty}(\mathcal S)} \le C\: {\|f\|}_{H^{\alpha}(\mathcal S)},\:\:\:\:\alpha>1/2, \end{equation*} holds true.