# A note on local smoothing estimates for fractional Schr\"{o}dinger equations

@inproceedings{Gan2021ANO, title={A note on local smoothing estimates for fractional Schr\"\{o\}dinger equations}, author={Shengwen Gan and Changkeun Oh and Shukun Wu}, year={2021} }

. We improve local smoothing estimates for fractional Schr¨odinger equations for α ∈ (0 , 1) ∪ (1 , ∞ ).

## 4 Citations

### New bounds for Stein's square functions in higher dimensions

- Mathematics
- 2021

We improve the L(R) bounds on Stein’s square function to the best known range of the Fourier restriction problem when n ≥ 4. Applications including certain local smoothing estimates are also…

### Local smoothing estimates of fractional Schrödinger equations in $$\alpha $$-modulation spaces with some applications

- MathematicsJournal of Evolution Equations
- 2023

We show some new local smoothing estimates of the fractional Schr\"odinger equations with initial data in $\alpha$-modulation spaces via decoupling inequalities. Furthermore, our necessary conditions…

### A type of oscillatory integral operator and its applications

- MathematicsMathematische Zeitschrift
- 2022

In this paper, we consider Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…

### Sharp endpoint $L_p$ estimates of quantum Schr\"{o}dinger groups

- Mathematics
- 2023

A bstract . In this article, we establish sharp endpoint L p estimates of Schr¨odinger groups on general measure spaces which may not be equipped with good metrics but admit submarkovian semigroups…

## 20 References

### New bounds for Stein's square functions in higher dimensions

- Mathematics
- 2021

We improve the L(R) bounds on Stein’s square function to the best known range of the Fourier restriction problem when n ≥ 4. Applications including certain local smoothing estimates are also…

### A Restriction Estimate for Surfaces with Negative Gaussian Curvatures

- Mathematics, Computer SciencePeking Mathematical Journal
- 2023

We prove $L^p$ bounds for the Fourier extension operators associated to surfaces in $\mathbb{R}^3$ with negative Gaussian curvatures for every $p>3.25$.

### Endpoint maximal and smoothing estimates for Schrödinger equations

- Mathematics
- 2008

Abstract For α > 1 we consider the initial value problem for the dispersive equation i∂tu + (–Δ) α/2 u = 0. We prove an endpoint Lp inequality for the maximal function with initial values in Lp…

### A note on Fourier restriction and nested Polynomial Wolff axioms

- Mathematics
- 2020

This note records an asymptotic improvement on the known $L^p$ range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial partitioning method…

### Improved local smoothing estimates for the fractional Schrödinger operator

- MathematicsBulletin of the London Mathematical Society
- 2022

In this paper, we consider local smoothing estimates for the fractional Schrödinger operator eit(−Δ)α/2$e^{it(-\Delta )^{\alpha /2}}$ with α>1$\alpha >1$ . Using the k$k$ ‐broad ‘norm’ estimates of…

### Restriction estimates using polynomial partitioning II

- Mathematics
- 2016

We improve the estimates in the restriction problem in dimension $n \ge 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak…

### Fourier restriction for smooth hyperbolic 2-surfaces

- MathematicsMathematische Annalen
- 2022

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $$\mathbb {R}^3.$$
R
3
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### Sharp variation-norm estimates for oscillatory integrals related to Carleson’s theorem

- MathematicsAnalysis & PDE
- 2020

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates…

### Bounds on Oscillatory Integral Operators Based on Multilinear Estimates

- Mathematics
- 2010

We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved Lp estimates in the Stein…