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

Abstract

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 $\mathcal{L}=\hat{I}”$, the Laplace-Beltrami operator or $\tilde{\hat{I}}”$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $\ddot{I}ˆ$ 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 ${\hat{I}}^2>a/4$. 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.