On the maximal operator associated with the free Schrödinger equation

Sichun Wang

Studia Mathematica (1997)

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

Abstract

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For d > 1, let ( S d f ) ( x , t ) = ʃ ℝ n e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , x ∈ ℝ n , where f̂ is the Fourier transform of f ∈ S ( ℝ n ) , and ( S d * f ) ( x ) = s u p 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 d x ) 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.

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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  15. [15] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. Zbl0654.42014
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