Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces

Abstract

For $f\in {\mathcal{S}}^2(\mathcal{S}{)}_o$, the collection of radial $L^2$-Schwartz class functions on Damek--Ricci spaces $\mathcal{S}$, we consider the Schrödinger maximal function, \begin{equation*} S^* f(x):= \displaystyle\sup_{0<t<4/Q^2} \left|S_tf(x)\right|\:,\:\:\:\:\:\:x\in\mathcal S\:, \end{equation*} corresponding to the Laplace--Beltrami operator $\hat{I}”$ with initial data $f$. We first obtain the complete description of the pairs $(q,\hat{I}\pm )\in [1,\infty ]\times [0,\infty )$ for which the estimate \begin{equation*} {\|S^*f\|}_{L^q\left(B_R\right)} \le C_R\: {\|f\|}_{H^α(\mathcal S)}\:, \end{equation*} holds on geodesic balls $B_R$, for all $f\in {\mathcal{S}}^2(\mathcal{S}{)}_o$. Our results are sharp and agree with the Euclidean case. We also prove that for all $f\in {\mathcal{S}}^2(\mathcal{S}{)}_o$, the following global estimate \begin{equation*} {\|S^*f\|}_{L^{2,\infty}(\mathcal S)} \le C\: {\|f\|}_{H^α(\mathcal S)},\:\:\:\:α>1/2, \end{equation*} holds true.

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