Multiple summing operators on C(K) spaces

In this paper, we characterize, for 1 <= p<infinity, the multiple (p, 1)-summing multi-linear operators on the product of C(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators on C(K) in terms of their representing measures and a new multilinear characterization of L-infinity spaces. We also solve a problem stated by M.S. Ramamijan and E. Schock, improve a result of H. P. Rosenthal and S.J.Szarek, and give new results about polymeasures.