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Czechoslovak Mathematical Journal

, Volume 66, Issue 1, pp 251–269 | Cite as

Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

  • Hongbin WangEmail author
Article

Abstract

Let Ω ∈ L s (S n−1) for s ⩾ 1 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
$$\left[ {b,{\mu _\Omega }} \right](f)(x) = {\left( {\int_0^\infty {{{\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}\left[ {b(x) - b(y)} \right]f(y){\text{d}}y} } \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}$$
. In this paper, the author proves the (L p(·)(ℝ n ),L p(·)(ℝ n ))-boundedness of the Marcinkiewicz integral operator \({\mu _\Omega }\) and its commutator [b, \({\mu _\Omega }\)] when p(·) satisfies some conditions. Moreover, the author obtains the corresponding result about \({\mu _\Omega }\) and [b, \({\mu _\Omega }\)] on Herz spaces with variable exponent.

Keywords

Herz space variable exponent commutator Marcinkiewicz integral 

MSC 2010

42B20 42B35 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  1. 1.School of ScienceShandong University of TechnologyZhangdian, Zibo, ShandongP.R.China

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