A mapping f , defined on an open subset A of a Banach space E, with values in another Banach space F , such that (f (a + xj) − f (a)) ∞ j=1 is absolutely summable in F , whenever (xj) ∞ j=1 is unconditionally summable (respectively, absolutely summable) in E, is called absolutely summing (respectively, regularly summing) at the point a ∈ A. It is proved that f is regularly summing at a if, and only if, there are M > 0 and δ > 0, such that f (a+x)−f (a) ≤ M x , for all x ≤ δ. This result has as a consequence a characterization of absolutely summing mappings by means of inequalities. This result is analogous to the well know characterization of the linear absolutely summing mappings. Several results and examples show that the existence of non-linear absolutely summing mappings is not a rare phenomena. A Dvoretzky-Rogers Theorem for n-homogeneous polynomials is proved.

ÁBSTRACT. ¡be space of rnulrilinear mappings of nuclear typc (s;r,,...,r,) between Banacb spaces is considered, sorne of lÉs properties are described (including the relaúonship with tensor products) and its topological dual is characterized as a Banach space of absolutely summing mappings.

Abstract. For linear operators, if 1 ≤ p ≤ q < ∞, then every absolutely p-summing operator is also absolutely q-summing. On the other hand, it is well known that for n ≥ 2, there are no general "inclusion theorems" for absolutely summing n-linear mappings or n-homogeneous polynomials. In this paper we deal with situations in which the spaces of absolutely p-summing and absolutely q-summing linear operators coincide, and prove that for 1 ≤ p ≤ q ≤ 2 and n ≥ 2, we have inclusion theorems for absolutely summing n-linear mappings/n-homogeneous polynomials/holomorphic mappings. It is worth mentioning that our results hold precisely in the opposite direction from what is expected in the linear case, i.e., we show that, in some situations, as p increases, the classes of absolutely p-summing mappings becomes smaller.

After proving the Khintchine inequality for the n-Rademacher functions of ARON and GLOBVENIK with constants independent from n E N, applications are given to the theory of polynomials and holomorphic functions between Banach spaces. In particular, the following result is proved: Every entire mapping from a Banach space into another one of cotype q, vanishing at the origin and of r-dominated type at zero for some r > 0 maps unconditionally 2-summable sequences into absolutely q-summable sequences.Partially supported by the Brazilian-German CNPq/GMD-agreement and FAEP-UNICAMP. Universitiit Oldenburg Fachbereich Mathematik 0-261 I Oldenburg Gennnnr IMECC-UnicampCaixa Postal 6065 13.081-970 Campinas, S.P. Bruzil