Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Abstract

Let Ω ∈ L ^s(Sⁿ⁻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(·)(ℝⁿ),L ^p(·)(ℝⁿ))-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.