A note on the Schrödinger maximal function

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It is shown that control of the Schrödinger maximal function sup0 <t<1 ǀeitΔfǀ for f ∈ Hs(Rn) requires s ≥ n/2(n + 1).

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A sharp Schrödinger maximal estimate in R2 (2017, background): “In general dimension n ≥ 2, Bourgain [2] showed the convergence for s > 1/2 − 1/(4n), using multilinear methods.”
Sharp L2 estimates of the Schrödinger maximal function in higher dimensions (2018, background): “In the narrow case, we use the l2 decoupling theorem of Bourgain-Demeter [5] in a lower dimension and perform induction on scales.”
POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES (2018, background): “[2] J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math., 196 (2006), 261-302 [3] J. Bourgain, On the Schrödinger maximal function in higher dimension, Proceedings of the Steklov Institute of Math.”
A note on pointwise convergence for the Schrödinger equation (2017, background): “It was thought that s ≥ 1/4 might also be sufficient in higher dimensions (see for example [11] or [19]), however Bourgain recently proved that s ≥ n 2(n+1) is necessary [4].”