Multilinear Riesz potential operators on Herz-type spaces and generalized Morrey spaces

Let in, it be integers with n >= 2, m >= 1, the multilinear Riesz potential operators be defined by I-alpha((m))(f)(x)=integral((Rn)m) f(1)(y(1))...f(m)(y(m))/vertical bar(x-y(1),...,x-y(m))vertical bar(mn-alpha) dy, where y = (y(1),...y(m)) and f = (f(1),...f(m)). In the first part of this paper, the boundedness for the operator I-alpha((m)) on the homogeneous Herz-Morrey product spaces, M(K) over dot(p1,q1)(n(1-1/q1),lambda 1)(R-n) x ... x M(K) over dot(pm,qm)(n(1-1/qm),) (lambda m)(R-n), and on the Herz-type Herdy product spaces, H(K) over dot(q1)(sigma 1,p1)(R-n) x...x H(K) over dot(qm)(sigma m,pm)(R-n) for sigma(i) > n(1-1/q(i)), are established respectively. The second goal of the paper is to extend the known L-p-boundedness of I-alpha((m)) to generalized Morrey spaces, L-p,L-phi(R-n), where p is an element of [1, + infinity) and phi is the suitable doubling and integral functions.