Boundedness of Marcinkiewicz functions.

Sakamoto, Minako ; Yabuta, Kôzô

Studia Mathematica, Tome 133 (1999), p. 103-142 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ʃ_{S^{n-1}} Ω(x^{\prime })dσ(x^{\prime }) = 0$. 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.

Détail

Publié le : 1999-01-01

Zbl 0930.42009

EUDML-ID : urn:eudml:doc:216646

Bibliographie

[00000] [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. | Zbl 0103.33402

[00001] [2] S. Chanillo and R. L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), 277-294. | Zbl 0598.34019

[00002] [3] Y. S. Han, On some properties of s-function and Marcinkiewicz integrals, Acta Sci. Natur. Univ. Pekinensis 5 (1987), 21-34. | Zbl 0648.35012

[00003] [4] L. Hörmander, Translation invariant operators, Acta Math. 104 (1960), 93-139.

[00004] [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. | Zbl 0579.42011

[00005] [6] D. S. Kurtz, Littlewood-Paley operators on BMO, Proc. Amer. Math. Soc. 99 (1987), 657-666.

[00006] [7] S. G. Qiu, Boundedness of Littlewood-Paley operators and Marcinkiewicz integral on $ε^α,p$, J. Math. Res. Exposition 12 (1992), 41-50. | Zbl 0773.42014

[00007] [8] E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. | Zbl 0105.05104

[00008] [9] E. M. Stein, Interpolation of linear operators, ibid. 83 (1956), 482-492. | Zbl 0072.32402

[00009] [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. | Zbl 0207.13501

[00010] [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J. 1971. | Zbl 0232.42007

[00011] [12] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, Calif., 1986. | Zbl 0621.42001

[00012] [13] A. Torchinsky and Shilin Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235-243.

[00013] [14] K. Yabuta, Boundedness of Littlewood-Paley operators, Math. Japon. 43 (1996), 134-150. | Zbl 0843.42008

[00014] [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. | Zbl 0671.42018

[00015] [16] Silei Wang, Some properties of the Littlewood-Paley g-function, in: Contemp. Math. 42, Amer. Math. Soc., 1985, 191-202.