# Small Time Behavior and Summability for the Schr\"odinger Equation

## TL;DR

This paper investigates the small time convergence and summability properties of the Schr"odinger equation, extending known results to include certain potentials and analyzing the failure of boundedness for low regularity data.

## Contribution

It extends Dahlberg and Kenig's convergence results to Schr"odinger equations with potentials and examines the limitations of maximal operator boundedness for low regularity.

## Key findings

- Convergence results hold for Schr"odinger equations with specific potentials.
- Failure of $L^p$-boundedness for the maximal operator when regularity $s<1/4$.
- Smoothing effects help establish these convergence properties.

## Abstract

We consider the Carleson's problem regarding small time almost everywhere convergence to initial data for the Schr\"odinger equation, both linear and nonlinear on $\mathbb{R}$. It is shown, via the smoothing effect of the Schr\"odinger flow, that the (sharp) result proved by Dahlberg and Kenig for initial data in Sobolev spaces still holds when one considers the full Schr\"odinger 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.