Regularity and pointwise convergence of solutions of the Schr\"odinger operator with radial initial data on Damek-Ricci spaces

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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 Schr\"odinger equation given by \begin{equation*}\begin{cases} 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{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 Schr\"odinger 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 Schr\"odinger 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.