AbstractGiven$m\geq 1$m≥1,$0\leq \lambda \leq 1$0≤λ≤1, and a discrete vector-valued function$\vec{f}=(f_{1},\ldots,f_{m})$f→=(f1,…,fm)with each$f_{j}:\mathbb{Z} ^{d}\rightarrow \mathbb{R}$fj:Zd→R, we consider the discrete multilinear fractional nontangential maximal operator$$ \mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }(\vec{f}) (\vec{n})=\mathop{\sup_{r>0, \vec{x}\in \mathbb{R}^{d}}}_{ \vert \vec{n}-\vec{x} \vert \leq \lambda r}\frac{1}{N(B _{r}(\vec{x}))^{m-\frac{\alpha }{d}}} \prod _{j=1}^{m}\sum_{\vec{k}\in B_{r}(\vec{x})\cap \mathbb{Z}^{d}} \bigl\vert f_{j}(\vec{k}) \bigr\vert , $$Mα,Bλ(f→)(n→)=supr>0,x→∈Rd|n→−x→|≤λr1N(Br(x→))m−αd∏j=1m∑k→∈Br(x→)∩Zd|fj(k→)|,where$\mathcal{B}$Bis the collection of all open balls$B\subset \mathbb{R}^{d}$B⊂Rd,$B_{r}(\vec{x})$Br(x→)is the open ball in$\mathbb{R}^{d}$Rdcentered at$\vec{x}\in \mathbb{R}^{d}$x→∈Rdwith radiusr, and$N(B_{r}(\vec{x}))$N(Br(x→))is the number of lattice points in the set$B_{r}(\vec{x})$Br(x→). We show that the operator$\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }(\vec{f})|$f→↦|∇Mα,Bλ(f→)|is bounded and continuous from$\ell ^{1}(\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z} ^{d})\times \cdots \times \ell ^{1}(\mathbb{Z}^{d})$ℓ1(Zd)×ℓ1(Zd)×⋯×ℓ1(Zd)to$\ell ^{q}(\mathbb{Z} ^{d})$ℓq(Zd)if$0\leq \alpha < md$0≤α<mdand$q\geq 1$q≥1such that$q>\frac{d}{md- \alpha +1}$q>dmd−α+1. We also prove that the same result also holds for the discrete multilinear fractional nontangential maximal operators associated with cubes. These results we obtained represent significant and natural extensions of what was known previously.