This paper is devoted to presenting some W1,1\documentclass[12pt]{minimal}
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\begin{document}$$W^{1,1}$$\end{document}-regularity properties of higher order maximal commutator and its fractional variant. More precisely, let k≥1,\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 1,$$\end{document}α∈[0,1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in [0,1)$$\end{document} and b∈Lloc1(R)\documentclass[12pt]{minimal}
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\begin{document}$$b\in L_{\textrm{loc}}^1 ({\mathbb {R}})$$\end{document}. We consider the following k-th order fractional maximal commutator Mb,αkf(x)=supt>0(2t)α-1∫x-tx+t|b(x)-b(z)|k|f(z)|dz,x∈R,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathfrak {M}}_{b,\alpha }^kf(x)=\sup \limits _{t>0}(2t)^{\alpha -1}\int _{x-t}^{x+t}|b(x)-b(z)|^k|f(z)|dz,\quad x\in {\mathbb {R}}, \end{aligned}$$\end{document}which includes the k-th order maximal commutator Mbk,\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {M}}_b^k,$$\end{document} corresponding to the critical case α=0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =0$$\end{document}. We establish the boundedness and continuity of the map f↦(Mb,αkf)′\documentclass[12pt]{minimal}
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\begin{document}$$f\mapsto ({\mathfrak {M}}_{b,\alpha }^kf)'$$\end{document} from W1,1(R)\documentclass[12pt]{minimal}
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\begin{document}$$W^{1,1}({\mathbb {R}})$$\end{document} to Lq(R),\documentclass[12pt]{minimal}
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\begin{document}$$L^{q}({\mathbb {R}}),$$\end{document} provided that α∈[0,1),\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in [0,1),$$\end{document}q∈(1,∞),\documentclass[12pt]{minimal}
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\begin{document}$$q\in (1,\infty ),$$\end{document}b∈Lip(R)\documentclass[12pt]{minimal}
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\begin{document}$$b\in {Lip}({\mathbb {R}})$$\end{document} and b′∈L1(R)\documentclass[12pt]{minimal}
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\begin{document}$$b'\in L^1({\mathbb {R}})$$\end{document}. We emphasize that our work not only improves essentially some known results, but also provides a new and simpler proof of some known ones. It should be also pointed out that the above results are new, even in the special case k=1\documentclass[12pt]{minimal}
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\begin{document}$$k=1$$\end{document}.
