# A note on the Schrödinger maximal function

@article{Bourgain2016ANO, title={A note on the Schr{\"o}dinger maximal function}, author={Jean Bourgain}, journal={Journal d'Analyse Math{\'e}matique}, year={2016}, volume={130}, pages={393-396} }

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

## 69 Citations

A sharp Schrödinger maximal estimate in R[superscript 2]

- Mathematics
- 2017

We show that limt→0 eit∆f(x) = f(x) almost everywhere for all f ∈ Hs(R2) provided that s > 1/3. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

Lower bounds for estimates of the Schrödinger maximal function

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- 2019

We give new lower bounds for $L^p$ estimates of the Schr\"odinger maximal function by generalizing an example of Bourgain.

A note on pointwise convergence for the Schrödinger equation

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- 2017

Abstract We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain proved that there is initial data, in Hs(ℝn) with $s<\frac{n}{2(n+1)}$, for which the…

A Note on the Refined Strichartz Estimates and Maximal Extension Operator

- Mathematics
- 2020

There are two parts for this paper. In the first part we extend the refined Strichartz estimates to hypersurfaces with nonzero Gaussian curvature. In the second part we prove some positive results…

Note on maximal estimates of generalized Schrödinger equation

- Mathematics
- 2018

In this study we extend the recent works on the pointwise convergence for the solutions of Schr\"odinger equations based on Du, Guth, and Li and Du and Zhang to generalized Schr\"odinger equations.…

Convergence of sequences of Schr\"odinger means.

- Mathematics
- 2019

We study convergence almost everywhere of sequences of Schrodinger means. We also replace sequences by uncountable sets.

Pointwise convergence along a tangential curve for the fractional Schrödinger equation with 0 < m < 1

- MathematicsMathematical Methods in the Applied Sciences
- 2021

In this article, we study the pointwise convergence problem about solution to the fractional Schrödinger equation with 0 < m < 1 along a tangential curve and estimate the capacitary dimension of the…

A sharp Schrodinger maximal estimate in $\mathbb{R}^2$

- Mathematics
- 2016

We show that $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial…

Two Theorems on Convergence of Schrödinger Means

- MathematicsJournal of Fourier Analysis and Applications
- 2018

Localization and convergence almost everywhere of Schrödinger means are studied.

## References

SHOWING 1-6 OF 6 REFERENCES

Schrödinger maximal function estimates via the pseudoconformal transformation

- Mathematics, Computer Science
- 2016

An alternative way to recover the recent result from \cite{LR} using the pseudoconformal transformation is presented.

On the Schrödinger maximal function in higher dimension

- Mathematics
- 2012

AbstractNew estimates on the maximal function associated to the linear Schrödinger equation are established. It is shown that the almost everywhere convergence property of eitΔf for t → 0 holds for f…

On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space

- Mathematics
- 2013

A complete discretization scheme for an ill-posed Cauchy problem for abstract firstorder linear differential equations with sectorial operators in a Banach space is validated. The scheme combines a…