# Boundedness of Marcinkiewicz functions.

Studia Mathematica (1999)

• Volume: 135, Issue: 2, page 103-142
• ISSN: 0039-3223

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## Abstract

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The ${L}^{p}$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g{*}_{\lambda }$-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 ${\mu }_{S}^{\varrho }$ and ${\mu }_{\lambda }^{*,\varrho }$ to S and $g{*}_{\lambda }$. The definition of ${\mu }_{S}^{\varrho }\left(f\right)$ is as follows: ${\mu }_{S}^{\varrho }\left(f\right)\left(x\right)=\left({ʃ}_{|y-x|, 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 ${ʃ}_{{S}^{n-1}}\Omega \left({x}^{\text{'}}\right)d\sigma \left({x}^{\text{'}}\right)=0$. We show that if σ = Reϱ > 0, then ${\mu }_{S}^{\varrho }$ 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 ${\mu }_{\lambda }^{*,\varrho }$. Their boundedness in the Campanato space ${\epsilon }^{\alpha ,p}$ is also considered.

## How to cite

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Sakamoto, Minako, and Yabuta, Kôzô. "Boundedness of Marcinkiewicz functions.." Studia Mathematica 135.2 (1999): 103-142. <http://eudml.org/doc/216646>.

@article{Sakamoto1999,
abstract = {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.},
author = {Sakamoto, Minako, Yabuta, Kôzô},
journal = {Studia Mathematica},
keywords = {Marcinkiewicz function; Littlewood-Paley function; area function; boundedness; Littlewood-Paley’s -function; Campanato space},
language = {eng},
number = {2},
pages = {103-142},
title = {Boundedness of Marcinkiewicz functions.},
url = {http://eudml.org/doc/216646},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Sakamoto, Minako
AU - Yabuta, Kôzô
TI - Boundedness of Marcinkiewicz functions.
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 2
SP - 103
EP - 142
AB - 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.
LA - eng
KW - Marcinkiewicz function; Littlewood-Paley function; area function; boundedness; Littlewood-Paley’s -function; Campanato space
UR - http://eudml.org/doc/216646
ER -

## References

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1. [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. Zbl0103.33402
2. [2] S. Chanillo and R. L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), 277-294. Zbl0598.34019
3. [3] Y. S. Han, On some properties of s-function and Marcinkiewicz integrals, Acta Sci. Natur. Univ. Pekinensis 5 (1987), 21-34. Zbl0648.35012
4. [4] L. Hörmander, Translation invariant operators, Acta Math. 104 (1960), 93-139.
5. [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. Zbl0579.42011
6. [6] D. S. Kurtz, Littlewood-Paley operators on BMO, Proc. Amer. Math. Soc. 99 (1987), 657-666.
7. [7] S. G. Qiu, Boundedness of Littlewood-Paley operators and Marcinkiewicz integral on ${\epsilon }^{\alpha },p$, J. Math. Res. Exposition 12 (1992), 41-50. Zbl0773.42014
8. [8] E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. Zbl0105.05104
9. [9] E. M. Stein, Interpolation of linear operators, ibid. 83 (1956), 482-492. Zbl0072.32402
10. [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
11. [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J. 1971. Zbl0232.42007
12. [12] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, Calif., 1986. Zbl0621.42001
13. [13] A. Torchinsky and Shilin Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235-243.
14. [14] K. Yabuta, Boundedness of Littlewood-Paley operators, Math. Japon. 43 (1996), 134-150. Zbl0843.42008
15. [15] Shilin Wang, Boundedness of the Littlewood-Paley g-function on $Li{p}_{\alpha }\left({ℝ}^{n}\right)$ (0 < α < 1), Illinois J. Math. 33 (1989), 531-541. Zbl0671.42018
16. [16] Silei Wang, Some properties of the Littlewood-Paley g-function, in: Contemp. Math. 42, Amer. Math. Soc., 1985, 191-202.

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