Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces

We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +\Psi(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:, \end{cases} \end{equation*} where $\mathcal{L}= \Delta$, the Laplace-Beltrami operator or $\tilde{\Delta}$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $\psi$ satisfies for some $a \in (0,1)$, the large frequency asymptotic: \begin{equation*} \psi(\lambda)=\lambda^a + \mathcal{O}(1)\:,\:\: \lambda \gg 1\:. \end{equation*} For almost everywhere pointwise convergence of the solution $u$ to its radial initial data $f$, we obtain the almost sharp regularity threshold $\beta>a/4$. This result is new even for $\mathbb{R}^n$ and in the special case of the fractional Schr\"odinger equations, generalizes classical Euclidean results of Walther.