L2L^{2} boundedness for commutators of fractional differential type Marcinkiewicz integral with rough variable kernel and BMO Sobolev spaces

In this paper, for {0<\gamma<1} and {b\in I_{\gamma}(\mathrm{BMO})}, the authors give the {L^{2}({\mathbb{R}}^{n})} boundedness of {\mu_{\gamma;b}}, the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.