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1999 | 135 | 2 | 103-142
Tytuł artykułu

Boundedness of Marcinkiewicz functions.

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley's g-function, Lusin's S function, Littlewood-Paley's $g*_λ$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley's g-function. In this note, we treat counterparts $μ_{S}^{ϱ}$ and $μ_{λ}^{*,ϱ}$ to S and $g*_λ$. The definition of $μ_{S}^{ϱ}(f)$ is as follows: $μ_{S}^{ϱ}(f)(x) = (ʃ_{|y-x| < t}| 1/t^{ϱ} ʃ_{|z|≤ t} Ω(z)/(|z|^{n-ϱ}) f(y-z) dz|^2 (dydt)/(t^{n+1}) )^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and $ʃ_{S^{n-1}} Ω(x')dσ(x') = 0$. We show that if σ = Reϱ > 0, then $μ_{S}^{ϱ}$ is $L^p$ bounded for max(1,2n/(n+2σ) < p < ∞, and for 0 < ϱ ≤ n/2 and 1 ≤ p ≤ 2n/(n+2ϱ), then $L^p$ boundedness does not hold in general, in contrast to the case of the S function. Similar results hold for $μ_{λ}^{*,ϱ}$. Their boundedness in the Campanato space $ε^{α,p}$ is also considered.
Czasopismo
Rocznik
Tom
135
Numer
2
Strony
103-142
Opis fizyczny
Daty
wydano
1999
otrzymano
1995-05-04
poprawiono
1999-01-25
Twórcy
  • Department of Mathematics, Nara Women's University, Kitauoya-Nishimachi, Nara 630-8506, Japan
  • School of Science, Kwansei Gakuin University, Uegahara 1-1-155, Nishinomiya, Hyogo 662-8501, Japan, yabuta3@kwansei.ac.jp
Bibliografia
  • [1] A. Benedek, A. P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356-3365.
  • [2] S. Chanillo and R. L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), 277-294.
  • [3] Y. S. Han, On some properties of s-function and Marcinkiewicz integrals, Acta Sci. Natur. Univ. Pekinensis 5 (1987), 21-34.
  • [4] L. Hörmander, Translation invariant operators, Acta Math. 104 (1960), 93-139.
  • [5] M. Kaneko and G.-I. Sunouchi, On the Littlewood-Paley and Marcinkiewicz functions in higher dimensions, Tôhoku Math. J. (2) 37 (1985), 343-365.
  • [6] D. S. Kurtz, Littlewood-Paley operators on BMO, Proc. Amer. Math. Soc. 99 (1987), 657-666.
  • [7] S. G. Qiu, Boundedness of Littlewood-Paley operators and Marcinkiewicz integral on $ε^α,p$, J. Math. Res. Exposition 12 (1992), 41-50.
  • [8] E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466.
  • [9] E. M. Stein, Interpolation of linear operators, ibid. 83 (1956), 482-492.
  • [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
  • [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J. 1971.
  • [12] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, Calif., 1986.
  • [13] A. Torchinsky and Shilin Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235-243.
  • [14] K. Yabuta, Boundedness of Littlewood-Paley operators, Math. Japon. 43 (1996), 134-150.
  • [15] Shilin Wang, Boundedness of the Littlewood-Paley g-function on $Lip_α(ℝ^n)$ (0 < α < 1), Illinois J. Math. 33 (1989), 531-541.
  • [16] Silei Wang, Some properties of the Littlewood-Paley g-function, in: Contemp. Math. 42, Amer. Math. Soc., 1985, 191-202.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv135i2p103bwm
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