In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz–Sobolev space condition. The work is motivated by the recent result of the first author and Slavíková [12] where an analogous version of classical Hörmander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if mn/2<s<mn\documentclass[12pt]{minimal}
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\begin{document}$$mn/2<s<mn$$\end{document}, then ‖Tσ(f1,⋯,fm)‖Lp(Rn)≲supk∈Z‖σ(2k·)Ψ(m)^‖Lsmn/s,1(Rmn)‖f1‖Lp1(Rn)⋯‖fm‖Lpm(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \big \Vert T_{\sigma }(f_1,\dots ,f_m)\big \Vert _{L^p(\mathbb {R}^n)}\lesssim \sup _{k\in \mathbb {Z}}\big \Vert \sigma (2^k\;\mathbf {\cdot }\;)\widehat{\Psi ^{(m)}}\big \Vert _{L_{s}^{mn/s,1}(\mathbb {R}^{mn})}\Vert f_1\Vert _{L^{p_1}(\mathbb {R}^n)}\cdots \Vert f_m\Vert _{L^{p_m}(\mathbb {R}^n)} \end{aligned}$$\end{document}for certain p,p1,⋯,pm\documentclass[12pt]{minimal}
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\begin{document}$$p,p_1,\dots ,p_m$$\end{document} with 1/p=1/p1+⋯+1/pm\documentclass[12pt]{minimal}
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\begin{document}$$1/p=1/p_1+\dots +1/p_m$$\end{document}. We also show that the above estimate is sharp, in the sense that the Lorentz–Sobolev space Lsmn/s,1\documentclass[12pt]{minimal}
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\begin{document}$$L_s^{mn/s,1}$$\end{document} cannot be replaced by Lsr,q\documentclass[12pt]{minimal}
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\begin{document}$$L_{s}^{r,q}$$\end{document} for r<mn/s\documentclass[12pt]{minimal}
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\begin{document}$$r<mn/s$$\end{document}, 0<q≤∞\documentclass[12pt]{minimal}
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\begin{document}$$0<q\le \infty$$\end{document}, or by Lsmn/s,q\documentclass[12pt]{minimal}
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\begin{document}$$L_s^{mn/s,q}$$\end{document} for q>1\documentclass[12pt]{minimal}
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\begin{document}$$q>1$$\end{document}.
