In the weighted theory of multilinear operators, the weights class which usually has been considered is the product of Ap\documentclass[12pt]{minimal}
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\begin{document}$$A_p$$\end{document} weights. However, it is known that ∏k=12Apk(Rn)/⊆Ap→(R2n)\documentclass[12pt]{minimal}
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\begin{document}$$\prod _{k=1}^2A_{p_k}({\mathbb {R}}^n)\varsubsetneq A_{\vec {p}}({\mathbb {R}}^{2n})$$\end{document}, and w→=(w1,w2)∈Ap→(R2n)\documentclass[12pt]{minimal}
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\begin{document}$$\vec {w}=(w_1,\,w_2)\in A_{\vec {p}}({\mathbb {R}}^{2n})$$\end{document} does not imply that wk∈Lloc1(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$w_k\in L^1_{\mathrm{loc}}({\mathbb {R}}^n)$$\end{document} for k=1,2\documentclass[12pt]{minimal}
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\begin{document}$$k=1,\,2$$\end{document}. Therefore, it is very interesting to study the weighted theory of multilinear operators with the weights in Ap→(R2n)\documentclass[12pt]{minimal}
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\begin{document}$$A_{\vec {p}}({\mathbb {R}}^{2n})$$\end{document}. In this paper, we consider the weights class Ap→/r→(R2n)\documentclass[12pt]{minimal}
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\begin{document}$$A_{\vec {p}/\vec {r}}({\mathbb {R}}^{2n})$$\end{document}, which is more general than Ap→(R2n)\documentclass[12pt]{minimal}
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\begin{document}$$A_{\vec {p}}({\mathbb {R}}^{2n})$$\end{document}. If w→=(w1,w2)∈Ap→/r→(R2n)\documentclass[12pt]{minimal}
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\begin{document}$$\vec {w}=(w_1,\,w_2)\in A_{\vec {p}/\vec {r}}({\mathbb {R}}^{2n})$$\end{document}, we show that the bilinear Fourier multiplier operator Tσ\documentclass[12pt]{minimal}
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\begin{document}$$T_{\sigma }$$\end{document} is bounded from Lp1(w1)×Lp2(w2)\documentclass[12pt]{minimal}
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\begin{document}$$L^{p_1}(w_1)\times L^{p_2}(w_2)$$\end{document} to Lp(νw→)\documentclass[12pt]{minimal}
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\begin{document}$$L^p(\nu _{\vec {w}})$$\end{document} when the symbol σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} satisfies the Sobolev regularity that supκ∈Z‖σk‖Ws1,s2(R2n)<∞\documentclass[12pt]{minimal}
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\begin{document}$$\sup _{\kappa \in {\mathbb {Z}}}\Vert \sigma _k\Vert _{W^{s_1,s_2}({\mathbb {R}}^{2n})} <\infty $$\end{document} with s1,s2∈(n2,n].\documentclass[12pt]{minimal}
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\begin{document}$$ s_1,s_2\in (\frac{n}{2},\,n].$$\end{document}
