For b∈Lloc(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$$\end{document} and 0<α<1\documentclass[12pt]{minimal}
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\begin{document}$$0<\alpha <1$$\end{document}, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely gΩ,α;b(f)(x)=(∫0∞|1t∫|x-y|≤tΩ(x-y)|x-y|n+α-1(b(x)-b(y))f(y)dy|2dtt)1/2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$\end{document}Here, we obtain the necessary and sufficient conditions for the function b to guarantee that gΩ,α;b\documentclass[12pt]{minimal}
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\begin{document}$$g_{\Omega ,\alpha ;b}$$\end{document} is a bounded operator on L2(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$L^2({\mathbb {R}}^n)$$\end{document}. More precisely, if Ω∈L(log+L)1/2(Sn-1)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}$$\end{document} and b∈Iα(BMO)\documentclass[12pt]{minimal}
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\begin{document}$$b\in I_{\alpha }(BMO)$$\end{document}, then gΩ,α;b\documentclass[12pt]{minimal}
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\begin{document}$$g_{\Omega ,\alpha ;b}$$\end{document} is bounded on L2(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$L^2({\mathbb {R}}^n)$$\end{document}. Conversely, if gΩ,α;b\documentclass[12pt]{minimal}
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\begin{document}$$g_{\Omega ,\alpha ;b}$$\end{document} is bounded on L2(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$L^2({\mathbb {R}}^n)$$\end{document}, then b∈Lipα(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$b \in Lip_\alpha ({\mathbb {R}}^n)$$\end{document} for 0<α<1\documentclass[12pt]{minimal}
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\begin{document}$$0<\alpha < 1$$\end{document}.
