• Corpus ID: 232404792

# Counterexamples for high-degree generalizations of the Schr\"odinger maximal operator

@inproceedings{An2021CounterexamplesFH,
title={Counterexamples for high-degree generalizations of the Schr\"odinger maximal operator},
author={Chen An and Rena Chu and L. B. Pierce},
year={2021}
}
• Published 27 March 2021
• Mathematics
. In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space H s ( R n ) that implies pointwise convergence for the solution of the linear Schr¨odinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schr¨odinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a suﬃcient condition. Analogues of Carleson’s…
5 Citations
Decoupling and Schr\"odinger maximal estimates for finite type phases in higher dimensions
• Mathematics
• 2022
In this article, we establish an l decoupling inequality for the hypersurface
The Frisch--Parisi formalism for fluctuations of the Schr\"odinger equation
• Mathematics
• 2022
We consider the solution of the Schrödinger equation u in R when the initial datum tends to the Dirac comb. Let hp,δ(t) be the fluctuations in time of ∫ |x||u(x, t)| dx, for 0 < δ < 1, after removing
Pointwise Convergence over Fractals for Dispersive Equations with Homogeneous Symbol
• Mathematics
Journal of Mathematical Analysis and Applications
• 2022
Counterexamples for the fractal Schr\"odinger convergence problem with an intermediate space trick
• Mathematics
• 2021
We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain’s counterexample and the intermediate space trick of Du–Kim–Wang–Zhang. We
Bounds on the Norms of Maximal Operators on Weyl Sums
• Mathematics
• 2021
We obtain new estimates on the maximal operator applied to the Weyl sums. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are

## References

SHOWING 1-10 OF 98 REFERENCES
Schrödinger equations: pointwise convergence to the initial data
Soit u(x,t) la solution de l'equation de Schrodinger avec des donnees initiales f dans l'espace de Sobolev H s (R n ) avec s>1/2. La convergence presque partout de u(x,t) vers f(x) se deduit d'une
A note on the Schrödinger maximal function
It is shown that control of the Schrödinger maximal function sup0 <t<1 ǀeitΔfǀ for f ∈ Hs(Rn) requires s ≥ n/2(n + 1).
Duke Math
• J., 55(3):699–715,
• 1987
Local smoothing properties of dispersive equations
• Mathematics
• 1988
Is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data? For the linear wave equation, for instance, the answer is
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.
Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces
AbstractA new moment inequality is obtained for the eigenfunctions ϕ of the flat tours Πn. More specifically, it is known that the $p = \frac{{2n}} {{n - 1}}$ is almost uniformaly bounded.
On the Schrödinger maximal function in higher dimension
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
Stein (Princeton
• NJ, 1991), volume 42 of Princeton Math. Ser., pages 83–112. Princeton Univ. Press, Princeton, NJ,
• 1995
On Bourgain’s Counterexample for the Schrödinger Maximal Function
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrodinger equation, for