Illinois Journal of Mathematics

The boundedness of Marcinkiewicz integral with variable kernel

Chin-Cheng Lin, Ying-Chieh Lin, Xiangxing Tao, and Xiao Yu

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Abstract

In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by $$\mu_{\Omega,\alpha}(f)(x)=\bigg(\int_{0}^{\infty}\bigg| {\int_{|x-y|\leq t}}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}f(y)\,dy\bigg|^{2} \frac{dt}{t^{3-\alpha}}\bigg)^{1/2}, %\\ %\quad0<\alpha\leq2. $$where $0<\alpha\leq2$. We first prove that $\mu_{\Omega,\alpha}$ is bounded from $L^{{2n}/{n+\alpha}}(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$ without any smoothness assumption on the kernel $\Omega$. Then we show that, if the kernel $\Omega$ satisfies a class of Dini condition, $\mu_{\Omega,\alpha}$ is bounded from $H^p(\mathbb{R}^n)$ ($p \le1$) to $H^q(\mathbb{R}^n)$, where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{2n}$. As corollary of the above results, we obtain the $L^p-L^q$ ($1< p<2$) boundedness of this fractional Marcinkiewicz integral.

Article information

Source
Illinois J. Math., Volume 53, Number 1 (2009), 197-217.

Dates
First available in Project Euclid: 22 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1264170846

Digital Object Identifier
doi:10.1215/ijm/1264170846

Mathematical Reviews number (MathSciNet)
MR2584942

Zentralblatt MATH identifier
1185.42011

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces

Citation

Lin, Chin-Cheng; Lin, Ying-Chieh; Tao, Xiangxing; Yu, Xiao. The boundedness of Marcinkiewicz integral with variable kernel. Illinois J. Math. 53 (2009), no. 1, 197--217. doi:10.1215/ijm/1264170846. https://projecteuclid.org/euclid.ijm/1264170846


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