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Let 0<β<1 andΩbe a proper open and non-empty subset of Rn.In this paper,the object of our investigation is the multilinear local maximal operator Mβ,defined by Mβ((f))(x)=supQ(∈xQ∈Fβ)Πi=1^m1/|Q|∫Q|fi(yi)|dyi,where Fβ={Q(x,l):x∈Ω,l<βd(x,Ωc)},Q=Q(x