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} +Ψ(\sqrt{-$\mathcal{L}$} )u=0:,: (x,t) \in S \times $\mathbb{R}$ :, \ u(0,\cdot)=f:,: $\text{ on }$ S :, \end{cases} \end{equation*} where $\mathcal{L}=\mathrm{\Delta }$, the Laplace-Beltrami operator or $\tilde{\mathrm{\Delta }}$, 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*} ψ(λ)=λ^a + $\mathcal{O}$(1):,:: λ\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 $\beta >a\mathrm{/}4$. This result is new even for ${\mathbb{R}}^n$ and in the special case of the fractional Schrödinger equations, generalizes classical Euclidean results of Walther.