  One of the most celebrated problems in Euclidean Harmonic analysis is the
Carleson's problem: determining the optimal regularity of the initial condition
$f$ of the Schr\"odinger equation given by \begin{equation*}\begin{cases}
i\frac{\partial u}{\partial t} =\Delta u\:,\: (x,t) \in \mathbb{R}^n \times
\mathbb{R} \\ u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:,
\end{cases}\end{equation*} in terms of the index $\alpha$ such that $f$ belongs
to the inhomogeneous Sobolev space $H^\alpha(\mathbb{R}^n)$ , so that the
solution of the Schr\"odinger operator $u$ converges pointwise to $f$, $\lim_{t
\to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we consider the
Carleson's problem for the Schr\"odinger equation with radial initial data on
Damek-Ricci spaces and obtain the sharp bound up to the endpoint $\alpha \ge
1/4$, which agrees with the classical Euclidean case.
