For \(f \in \mathscr {S}^2(\mathcal S)_{o}\) , the collection of radial \(L^2\) -Schwartz class functions on Damek–Ricci spaces \(\mathcal S=N\ltimes A\) , we consider the maximal function, \(\begin{aligned} S^* f(x):= \displaystyle \sup _{0<t<4/Q^2} \left| S_tf(x)\right| \,,\,\,\,\,\,\,x\in \mathcal S\,, \end{aligned}\) where \(S_tf\) is the Schrödinger propagation corresponding to the Laplace-Beltrami operator \(\Delta \) with initial data f and Q is the homogeneous dimension of Heisenberg type groups N. We first obtain the complete description of the pairs \((q, \alpha ) \in [1, \infty ] \times [0,\infty )\) for which the estimate \(\begin{aligned} {\Vert S^*f\Vert }_{L^q\left( B_R\right) } \le C_R\, {\Vert f\Vert }_{H^{\alpha }(\mathcal S)}\,, \end{aligned}\) holds on geodesic balls \(B_R\) , \(R>0\) , for constants \(C_R>0\) , depending only on R, for all \(f \in \mathscr {S}^2(\mathcal S)_{o}\) , where \(H^{\alpha }(\mathcal S)\) is the fractional \(L^2\) -Sobolev space with Sobolev index \(\alpha \) . 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 \(\begin{aligned} {\Vert S^*f\Vert }_{L^{2,\infty }(\mathcal S)} \le C\, {\Vert f\Vert }_{H^{\alpha }(\mathcal S)},\,\,\,\,\alpha >1/2, \end{aligned}\) holds true for some constant \(C>0\) .
