Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Wang, Hongbin

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 251-269.

Résumé

Let $\mathrm{\Omega }\in L^s({\mathrm{S}}^{n-1})$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\mathrm{\Omega }}$ 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 ${\mu }_{\mathrm{\Omega }}$ and its commutator $[b,{\mu }_{\mathrm{\Omega }}]$ when $p(\cdot )$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\mathrm{\Omega }}$ and $[b,{\mu }_{\mathrm{\Omega }}]$ on Herz spaces with variable exponent.

Détail

DOI : 10.1007/s10587-016-0254-1

Classification : 42B20, 42B35
Mots-clés : Herz space; variable exponent; commutator; Marcinkiewicz integral