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1997 | 122 | 2 | 167-182
Tytuł artykułu

On the maximal operator associated with the free Schrödinger equation

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EN
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.
Czasopismo
Rocznik
Tom
122
Numer
2
Strony
167-182
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-11-06
poprawiono
1996-07-23
Twórcy
autor
Bibliografia
  • [1] M. Ben-Artzi and A. Devinatz, Local smoothing and convergence properties of Schrödinger type equations, J. Funct. Anal. 101 (1991), 231-254.
  • [2] J. Bourgain, A remark on Schrödinger operators, Israel J. Math. 77 (1992), 1-16.
  • [3] L. Carleson, Some analytical problems related to statistical mechanics, in: Euclidean Harmonic Analysis, Lecture Notes in Math. 779, Springer, 1979, 5-45.
  • [4] M. Cowling, Pointwise behavior of solutions to Schrödinger equations, in: Harmonic Analysis, Lecture Notes in Math. 992, Springer, 1983, 83-90.
  • [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] C. E. Kenig, G. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.
  • [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.
  • [8] E. Prestini, Radial functions and regularity of solutions to the Schrödinger equation, Monatsh. Math. 109 (1990), 135-143.
  • [9] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976.
  • [10] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715.
  • [11] P. Sjölin, Radial functions and maximal estimates for solutions to the Schrödinger equation, J. Austral. Math. Soc. Ser. A 59 (1995), 134-142.
  • [12] P. Sjölin, Global maximal estimates for solutions to the Schrödinger equation, Studia Math. 110 (1994), 105-114.
  • [13] P. Sjölin, $L^p$ maximal estimates for solutions to the Schrödinger equation, informal notes, Aug. 1994.
  • [14] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.
  • [15] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878.
  • [16] J. Walker, Fourier Analysis, Oxford Univ. Press, 1988.
  • [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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Bibliografia
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bwmeta1.element.bwnjournal-article-smv122i2p167bwm
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