Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T05:07:57.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces

Published online by Cambridge University Press:  11 January 2016

Yong Ding ,
Qingying Xue  and
Kôzô Yabuta
Yong Ding
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing, 100875, P. R. of [email protected]
Qingying Xue*
Affiliation:
Department of Mathematics, Beijing Normal University, Beijing, 100875, P. R. of [email protected]
Kôzô Yabuta
Affiliation:
School of Science and Technology, Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, [email protected]
*
School of Science and Technology Kwansei Gakuin University, Gakuen 2-1 Sanda, Hyogo 669-1337, Japan[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let g(f), S(f), g*λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g*λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g*λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρs(f) and μλ*,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp-Dini type condition.

Keywords

42B25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 181 , March 2006 , pp. 103 - 148
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

Footnotes

The first named author was supported by NSF of China (Grant No. 10571015) and DPFIHE of China (Grant No. 20050027025).

References

[1] Calderóon, A. P., Weiss, M. and Zygmund, A., On the existence of singular integrals, Proc. Sympos. Pure Math., 10 (1967), 56–73.Google Scholar
[2] Colzani, L., Hardy spaces on Spheres, Ph.D thesis, Washington University, St. Louis, MO, 1982.Google Scholar
[3] Colzani, L., Taibleson, M. and Weiss, G., Maximal estimates for Cesàro and Riesz means on spheres, Indiana Univ. Math. J., 33 (1984), 873–889.Google Scholar
[4] Ding, Y., Fan, D. and Pan, Y., Lp-boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta. Math. Sinica (English series), 16 (2000), 593–600.Google Scholar
[5] Ding, Y., Lu, S. and Xue, Q., On Marcinkiewicz integral with homogeneous kernels, J. Math. Anal. Appl., 245 (2000), 471–488.Google Scholar
[6] Ding, Y., Lu, S. and Yabuta, K., A problem on rough parametric Marcinkiewicz functions, J. Austral. Math. Soc, 72 (2002), 13–21.Google Scholar
[7] Ding, Y. and Xue, Q., Weak type (1, 1) bounds for a class of the Littlewood-Paley operators, J. Math. Soc. Japan, 57 (2005), 184–194.Google Scholar
[8] Ding, Y., Xue, Q. and Yabuta, K., Weighted estimate for a class of Littlewood-Paley operators, to appear in Taiwanese J. Math.Google Scholar
[9] Fabes, E. B., Johnson, R. L. and Neri, U., Spaces of harmonic functions representable by Poisson integrals of functions in BMO and Lp, λ , Indiana Univ. Math. J., 25 (1976), 159–170.Google Scholar
[10] Han, Y., On some properties of s-function and Marcinkiewicz integrals, Acta Sci. Natur. Univ. Pekinensis, 5 (1987), 21–34.Google Scholar
[11] Hörmander, L., Translation invariant operators, Acta Math., 104 (1960), 93–139.Google Scholar
[12] Kurtz, D. S., Littlewood-Paley operators on BMO, Proc. Amer. Math. Soc, 99 (1987), 657–666.Google Scholar
[13] Kurtz, D. S. and Wheeden, R. L., Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc, 255 (1979), 343–362.CrossRefGoogle Scholar
[14] Peetre, J., On the theory of Lp, λ spaces, J. Funct. Anal., 4 (1969), 71–87.Google Scholar
[15] Qiu, S., Boundedness of Littlewood-Paley operators and Marcinkiewicz integral on Sα,p , J. Math. Res. Exposition, 12 (1992), 41–50.Google Scholar
[16] Sakamoto, M. and Yabuta, K., Boundedness of Marcinkiewicz functions, Studia Math., 135 (1999), 103–142.Google Scholar
[17] Stein, E. M., On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc, 88 (1958), 430–466.Google Scholar
[18] Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970.Google Scholar
[19] Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.Google Scholar
[20] Sun, Y., On the existence and boundedness of square function operators on Cam-panato spaces, Nagoya Math. J., 173 (2004), 139–151.CrossRefGoogle Scholar
[21] Torchinsky, A., Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, Calif., 1986.Google Scholar
[22] Torchinsky, A. and Wang, S., A note on the Marcinkiewicz integral, Colloq. Math., 60/61 (1990), 235–243.Google Scholar
[23] Wang, S., Boundedness of the Littlewood-Paley g-function on Lipα(ℝn) (0 < α < 1), Illinois J. Math., 33 (1989), 531–541.CrossRefGoogle Scholar
[24] Wang, S., Some properties of the Littlewood-Paley g-function, Contemp. Math., 42 (1985), 191–202.Google Scholar
[25] Wang, S. and Chen, J., Some notes on square function operator, Annals of Mathematics (Chinese), Series A, 11 (1990), 630–638.Google Scholar
[26] Xue, Q., Parametrized Littlewood-Paley operators, Thesis, Beijing Normal University, 2004.Google Scholar
[27] Yabuta, K., Boundedness of Littlewood-Paley operators, Math. Japonica, 43 (1996), 143–150.Google Scholar
[28] Yabuta, K., Some remarks to Marcinkiewicz functions, Kwansei Gakuin Univ. Nat. Sci. Rev., 6 (2002), 9–15.Google Scholar
[29] Yabuta, K., Existence and boundedness of g*λ-function and Marcinkiewicz functions on Campanato spaces, Sci. Math. Jpn., 59 (2004), 93–112.Google Scholar