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ESTIMATES FOR MARCINKIEWICZ INTEGRALS IN BMO AND CAMPANATO SPACES

Published online by Cambridge University Press:  09 August 2007

GUOEN HU,
YAN MENG and
DACHUN YANG
GUOEN HU
Affiliation:
Department of Applied Mathematics, University of Information Engineering, P. O. Box 1001-747, Zhengzhou 450002, People's Republic of China e-mail: huguoen@eyou.com
YAN MENG
Affiliation:
School of Information, Renmin University of China, Beijing 100872, People's Republic of China e-mail: mengyan@ruc.edu.cn
DACHUN YANG
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China e-mail: dcyang@bnu.edu.cn
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Abstract

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In this paper, the authors consider the behavior on BMO($\mathbb R^n$) and Campanato spaces for the higher-dimensional Marcinkiewicz integral operator which is defined by\begin{equation*} \mu_{\Omega}(f)(x)=\left (\int^{\infty}_0\Big |\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-1}}f(y)\,dy\Big |^2\,\frac{dt}{t^3}\right)^{1/2}, \end{equation*} where Ω is homogeneous of degree zero, has mean value zero and is integrable on the unit sphere. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO($\mathbb R^n$) or to a certain Campanato space, then [μΩ(f)]2 is either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness is also obtained. The corresponding Lusin area integral is also considered.

Keywords

42B25

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007, pp. 167 - 187
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

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