Small Time Behavior and Summability for the Schrödinger Equation

Abstract

We consider the Carleson's problem regarding small time almost everywhere convergence to initial data for the Schrödinger equation, both linear and nonlinear on $\mathbb{R}$. It is shown, via the smoothing effect of the Schrödinger flow, that the (sharp) result proved by Dahlberg and Kenig for initial data in Sobolev spaces still holds when one considers the full Schrödinger equation with a certain class of potentials. As for $s<\frac{1}{4}$, the failure of $L^p$-boundedness of the (localized) maximal operator is investigated.

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