Multilinear Hardy-CesA ro operator and commutator on the product of Morrey-Herz spaces

We obtain necessary and sufficient conditions on functions s(1)(t), ..., s(m)(t) and psi(t) such that the weighted multilinear Hardy-Cesaro operator (f(1), ..., f(m)) bar right arrow integral([0,1]n) (Pi(m)(k=1) f(k)(s(k)(t)x))psi(t) dt is bounded from (K) over dot(q1)(alpha 1, p1) (omega(1)) x ... x (K) over dot(qm)(alpha m, pm) (omega(m)) to (K) over dot(q)(alpha, p) (omega) and from M(K) over dot(p1,q1)(alpha 1,lambda 1) (omega(1)) x ... x M(K) over dot(pm,qm)(alpha m,lambda m) (omega(m)) to M (K) over dot(p,q)(alpha,omega) (omega). Sharp bounds are also obtained for both cases 0 < p < 1 and 1 <= p < infinity. Provided b(1), ..., b(m) are Lipschitz functions we give a sufficient condition on functions s(1)(t), ..., s(m)(t), psi(t) such that the commutator of the weighted Hardy-Cesaro operator (f(1), ..., f(m)) bar right arrow integral([0,1]n) (Pi(m)(k=1) f(k)(s(k)(t)x)) (Pi(m)(k=1) (b(k)(x) - b(k)(s(k)(t)x))) psi(t) dt is bounded from M(K) over dot(p1,q1)(alpha 1,lambda 1) (omega(1)) x ... x M(K) over dot(pm,qm)(alpha m,lambda m) (omega(m)) to M(K) over dot(p,q)(alpha',lambda)(omega) for both cases 0 < p < 1 and 1 <= p < infinity. As a consequence, when m = n = 1 and (s)1(t) = t, we obtain an improvement of a recent result by Tang, Xue and Zhou.