Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces

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 Schrödinger equation given by

$$\begin{aligned} {\left\{ \begin{array}{ll} i\frac{\partial u}{\partial t} =\Delta u\,,\, (x,t) \in {\mathbb {R}}^n \times {\mathbb {R}} \\ u(0,\cdot )=f\,, \text { on } {\mathbb {R}}^n \,, \end{array}\right. } \end{aligned}$$

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 Schrödinger operator u converges pointwise to f, \(\displaystyle \lim _{t \rightarrow 0+} u(x,t)=f(x)\), almost everywhere. In this article, we consider the Carleson’s problem for the Schrödinger 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.