Endpoint regularity of discrete multilinear fractional nontangential maximal functions

Given m >= 1, and a discrete vector-valued function f ->=(f1, with each fj:Zd -> R, we consider the discrete multilinear fractional nontangential maximal operator M alpha,B lambda(f ->)(n ->)=supr>0,x ->is an element of Rd|n ->-x ->|<=lambda r1N(Br(x ->))m-alpha dj=1m n-ary sumation k ->is an element of Br(x ->)boolean AND Zd|fj(k -> Br(x ->)centered at x ->is an element of Rd with radius r, and N(Br(x ->) is the number of lattice points in the set Br(x ->) We show that the operator f -> where B is the collection of all open balls B. Rd, Br ( x) is the open ball in Rd centered at x. Rd with radius r, and N(Br ( x)) is the number of lattice points in the set Br ( x). We show that the operator f . |. M. a, B( f)| is bounded and continuous from 1(Zd) x 1(Zd) x center dot center dot center dot x 1(Zd) to q(Zd) if 0 = a < md and q = 1 such that q > d md- a+ 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.