Cotype and nonlinear absolutely summing mappings

In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every $n$% -linear mapping from ${\mathcal{L}}_{\mathrm{\infty }}$-spaces into $\mathbb{K}$ is $% (2;2,…,2,\infty)$-summingandalsoshownthatevery$n$-linearmappingfrom$$\mathcal{L}$_{\infty}$-spacesinto$F$is$(q;2,…,2)$-summingwhenever$F$hascotype$q.$ We also give new examples of analytic summing mappings and polynomial and multilinear versions of a linear Extrapolation Theorem.