Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Czechoslovak Mathematical Journal

Hongbin ¹ 

[1] School of Science, Shandong University of Technology, No. 266 Xincunxi Road, Zhangdian, Zibo, Shandong, 255000, P. R. China

关键词: Herz space;    variable exponent;    commutator;    Marcinkiewicz integral;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

Let $\Omega\in L^s({\mathrm S}^{n-1})$ for $s\geq1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral $\mu_\Omega$ and $b$ is defined by \begin{equation*} \displaystyle[b,\mu_\Omega] (f)(x)=\biggl(\int^\infty_0\biggl|\int_{|x-y|\leq t} \frac{\Omega(x-y)}{|x-y|^{n-1}}[b(x)-b(y)]f(y)  d y\bigg|^2\frac{ d t}{t^3}\bigg)^{\!1/2}. \end{equation*} In this paper, the author proves the $(L^{p(\cdot)}(\mathbb{R}^n),L^{p(\cdot)}(\mathbb{R}^n))$-boundedness of the Marcinkiewicz integral operator $\mu_\Omega$ and its commutator $[b,\mu_\Omega]$ when $p(\cdot)$ satisfies some conditions. Moreover, the author obtains the corresponding result about $\mu_\Omega$ and $[b,\mu_\Omega]$ on Herz spaces with variable exponent.

【 授权许可】

Unknown