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

top
For d > 1, let , , where f̂ is the Fourier transform of , and its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) 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

top

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

top
  1. [1] M. Ben-Artzi and A. Devinatz, Local smoothing and convergence properties of Schrödinger type equations, J. Funct. Anal. 101 (1991), 231-254. 
  2. [2] J. Bourgain, A remark on Schrödinger operators, Israel J. Math. 77 (1992), 1-16. 
  3. [3] L. Carleson, Some analytical problems related to statistical mechanics, in: Euclidean Harmonic Analysis, Lecture Notes in Math. 779, Springer, 1979, 5-45. 
  4. [4] M. Cowling, Pointwise behavior of solutions to Schrödinger equations, in: Harmonic Analysis, Lecture Notes in Math. 992, Springer, 1983, 83-90. 
  5. [5] B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, in: Harmonic Analysis, Lecture Notes in Math. 908, Springer, 1982, 205-209. 
  6. [6] C. E. Kenig, G. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. Zbl0738.35022
  7. [7] C. E. Kenig and A. Ruiz, A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), 239-246. Zbl0525.42011
  8. [8] E. Prestini, Radial functions and regularity of solutions to the Schrödinger equation, Monatsh. Math. 109 (1990), 135-143. Zbl0777.42005
  9. [9] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976. 
  10. [10] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715. Zbl0631.42010
  11. [11] P. Sjölin, Radial functions and maximal estimates for solutions to the Schrödinger equation, J. Austral. Math. Soc. Ser. A 59 (1995), 134-142. Zbl0856.42013
  12. [12] P. Sjölin, Global maximal estimates for solutions to the Schrödinger equation, Studia Math. 110 (1994), 105-114. Zbl0829.42017
  13. [13] P. Sjölin, maximal estimates for solutions to the Schrödinger equation, informal notes, Aug. 1994. Zbl0829.42017
  14. [14] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. Zbl0232.42007
  15. [15] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. Zbl0654.42014
  16. [16] J. Walker, Fourier Analysis, Oxford Univ. Press, 1988. Zbl0669.42001
  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. 

NotesEmbed ?

top

You must be logged in to post comments.