Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces

For $f\in {\mathcal{S}}^2(\mathcal{S}{)}_o$, the collection of radial $L^2$-Schwartz class functions on Damek–Ricci spaces $\mathcal{S}$, we consider the Schrödinger 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 $\mathrm{\Delta }$ with initial data $f$. We first obtain the complete description of the pairs $(q,\alpha )\in [1,\mathrm{\infty }]\times [0,\mathrm{\infty })$ for which the estimate \begin{equation*} {|S^f|}{L^q\left(B_R\right)} \le C_R: {|f|}{H^α(\mathcal S)}:, \end{equation} holds on geodesic balls $B_R$, for all $f\in {\mathcal{S}}^2(\mathcal{S}{)}_o$. Our results are sharp and agree with the Euclidean case. We also prove that for all $f\in {\mathcal{S}}^2(\mathcal{S}{)}_o$, the following global estimate \begin{equation*} {|S^f|}{L^{2,\infty}(\mathcal S)} \le C: {|f|}{H^α(\mathcal S)},::::α>1/2, \end{equation} holds true.