Abstract
We consider Carleson’s problem regarding convergence for the Schrödinger equation in dimensions \({d\ge 2}\). We show that if the solution converges almost everywhere with respect to \({\alpha}\)-Hausdorff measure to its initial datum as time tends to zero, for all data \({H^{s}(\mathbb{R}^{d})}\), then \({s\ge \frac{d}{2(d+2)}(d+1-\alpha)}\). This strengthens and generalises results of Bourgain and Dahlberg–Kenig.
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Lucà, R., Rogers, K.M. Coherence on Fractals Versus Pointwise Convergence for the Schrödinger Equation. Commun. Math. Phys. 351, 341–359 (2017). https://doi.org/10.1007/s00220-016-2722-8
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DOI: https://doi.org/10.1007/s00220-016-2722-8