Maximal estimates and pointwise convergence for solutions of certain dispersive equations 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 Schr\"odinger equation given by \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} -\Delta_{\mathbb{R}^n} u=0\:,\:\:\: (x,t) \in \mathbb{R}^n \times \mathbb{R}\:, \newline u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases} \end{equation*} in terms of the index $\beta$ such that $f$ belongs to the inhomogeneous Sobolev space $H^\beta(\mathbb{R}^n)$ , so that the solution of the Schr\"odinger operator $u$ converges pointwise to $f$, $\displaystyle\lim_{t \to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we address the Carleson's problem for the fractional Schr\"odinger equation, the Boussinesq equation and the Beam equation corresponding to both the Laplace-Beltrami operator $\Delta$ and the shifted Laplace-Beltrami operator $\tilde{\Delta}$, with radial initial data on Damek-Ricci spaces, by obtaining a complete description of the local (in space) mapping properties for the corresponding local (in time) maximal functions. Consequently, we obtain the sharp bound up to the endpoint $\beta \ge 1/4$, for (almost everywhere) pointwise convergence. We also establish an abstract transference principle for dispersive equations whose corresponding multipliers have comparable oscillation and also apply it in the proof of our main result.