On the Schr\"odinger maximal function in higher dimension

J. Bourgain

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 53-66

Résumé

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.