Multiple weakly summing multilinear mappings and polynomials

In this paper, we introduce and study a new class containing absolutely summing multilinear mappings and polynomials, which we call multiple weakly summing multilinear mappings and polynomials. We investigate some interesting properties about multiple weakly (p; q₁,...,q_k)-summing multilinear mappings and polynomials defined on Banach spaces: In particular, we prove a kind of Dvoretzky-Rogers' Theorem and an ideal property for multiple weakly (p; q₁..., q_k)-summing multilinear mappings and polynomials. We also prove that the Aron-Berner extensions of multiple weakly (p;q₁....q_k)-summing murtilinear mappings and polynomials are also multiple weakly (p;q₁,...,q_k)-summing.