Abstract
We consider the Schrödinger operator \({e^{it\Delta}}\) acting on initial data f in \({\dot{H}^s}\). We show that an affirmative answer to a question of Carleson, concerning the sharp range of s for which \({\lim_{t\to 0}e^{it\Delta}f(x)=f(x)}\) a.e. \({x\in \mathbb {R}^n}\), would imply an affirmative answer to a question of Planchon, concerning the sharp range of q and r for which \({e^{it\Delta}}\) is bounded in \({L_x^q(\mathbb {R}^n,L^r_t(\mathbb {R}))}\). When n = 2, we unconditionally improve the range for which the mixed norm estimates hold.
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Supported in part by the Spanish projects MTM2007-60952 and CCG07-UAM/ESP-1664.
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Rogers, K.M. Strichartz estimates via the Schrödinger maximal operator. Math. Ann. 343, 603–622 (2009). https://doi.org/10.1007/s00208-008-0283-7
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DOI: https://doi.org/10.1007/s00208-008-0283-7