# On the maximal operator associated with the free Schrödinger equation

Studia Mathematica (1997)

• Volume: 122, Issue: 2, page 167-182
• ISSN: 0039-3223

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## Abstract

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For d > 1, let $\left({S}_{d}f\right)\left(x,t\right)={ʃ}_{{ℝ}^{n}}{e}^{ix·\xi }{e}^{{it|\xi |}^{d}}f̂\left(\xi \right)d\xi$, $x\in {ℝ}^{n}$, where f̂ is the Fourier transform of $f\in S\left({ℝ}^{n}\right)$, and $\left({S}_{d}*f\right)\left(x\right)=su{p}_{0 its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $\left({ʃ}_{|x| holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.

## How to cite

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Wang, Sichun. "On the maximal operator associated with the free Schrödinger equation." Studia Mathematica 122.2 (1997): 167-182. <http://eudml.org/doc/216368>.

@article{Wang1997,
abstract = {For d > 1, let $(S_\{d\}f)(x,t) = ʃ_\{ℝ^n\} e^\{ix·ξ\} e^\{it|ξ|^d\} f̂(ξ)dξ$, $x ∈ ℝ^n$, where f̂ is the Fourier transform of $f ∈ S (ℝ^n)$, and $(S_\{d\}*f)(x) = sup_\{0 < t < 1\} |(S_\{d\}f)(x,t)|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $(ʃ_\{|x| < R\} |(S_\{d\}*f)(x)|^p dx)^\{1/p\} ≤ C_\{R\}∥f∥_\{H_\{1/4\}\}$ holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.},
author = {Wang, Sichun},
journal = {Studia Mathematica},
keywords = {free Schrödinger equation; maximal functions; spherical harmonics; oscillatory integrals; Schrödinger operator; -estimates},
language = {eng},
number = {2},
pages = {167-182},
title = {On the maximal operator associated with the free Schrödinger equation},
url = {http://eudml.org/doc/216368},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Wang, Sichun
TI - On the maximal operator associated with the free Schrödinger equation
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 2
SP - 167
EP - 182
AB - For d > 1, let $(S_{d}f)(x,t) = ʃ_{ℝ^n} e^{ix·ξ} e^{it|ξ|^d} f̂(ξ)dξ$, $x ∈ ℝ^n$, where f̂ is the Fourier transform of $f ∈ S (ℝ^n)$, and $(S_{d}*f)(x) = sup_{0 < t < 1} |(S_{d}f)(x,t)|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $(ʃ_{|x| < R} |(S_{d}*f)(x)|^p dx)^{1/p} ≤ C_{R}∥f∥_{H_{1/4}}$ holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.
LA - eng
KW - free Schrödinger equation; maximal functions; spherical harmonics; oscillatory integrals; Schrödinger operator; -estimates
UR - http://eudml.org/doc/216368
ER -

## References

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17. [17] S. Wang, A note on the maximal operator associated with the Schrödinger equation, Preprint series No. 7 (1993-1994), Dept. of Math. and Statistics, McMaster Univ., Canada.

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