Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

摘要

summary:Let $\Omega \in L^s({\mathrm S}^{n-1})$ for $s\geq 1$ 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 \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) {\rm d} y\bigg |^2\frac {{\rm 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.