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

Abstract

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{equation*}\begin{cases} i\frac{\partial u}{\partial t} =Δu\:,\: (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 Schrödinger 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 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.