On the Schrödinger maximal function in higher dimension

New estimates on the maximal function associated to the linear Schrödinger equation are established. It is shown that the almost everywhere convergence property of $e^{it\Delta}f$ for $t\to0$ holds for $f\in H^s(\mathbb R^n)$, $s>\frac12-\frac1{4n}$, which is a new result for $n\geq3$. We also construct examples showing that $s\geq\frac12-\frac1n$ is certainly necessary when $n\geq4$. This is a further contribution to our understanding of how L. Carleson's result for $n=1$ generalizes in higher dimension. From the methodological point of view, crucial use is made of J. Bourgain and L. Guth's results and techniques that are based on the multi-linear oscillatory integral theory developed by J. Bennett, T. Carbery and T. Tao.