Coherence on fractals versus pointwise convergence for the Schrödinger equation (Q521147)

The paper deals with the Schrödinger equation \(i \partial_t u + \Delta u = 0\) on \(\mathbb R^{d}\) with initial datum \(u_0\) in the Bessel/potential Sobolev space \(H^s (\mathbb R^d)\). The main result shows that if the solution \(u(t, \cdot)\) converges almost everywhere with respect to \(\alpha\)-Hausdorff measure to its initial datum \(u_0\) as time tends to zero for all data \(H^s (\mathbb R^d)\) then \(s \geq d (d+1-\alpha)/[2(d+2)]\). Such a result strengthens and generalises some results of \textit{J. Bourgain} [Isr. J. Math. 77, No. 1--2, 1--16 (1992; Zbl 0798.35131)] and \textit{B. E. J. Dahlberg} and \textit{C. E. Kenig} [Lect. Notes Math. 908, 205--209 (1982; Zbl 0519.35022)] on the same problem.