Dominated polynomials on $\mathcal{L}_P$ -spaces

Abstract.

It is well known that continuous bilinear forms on C(K) ×  C(K) are 2-dominated. This paper shows that generalizations of this result are not to be expected. The main result asserts that for every $\mathcal{L}_P$ -space E(1≦ p ≦∞), every n ≧ 2, every r > 0 and every Banach space F , there exists an n-homogeneous polynomial P : E  →  F such that P is not of type [Π _r ], hence P is neither r-dominated nor r-semi-integral (if n  = 2 and p  =  ∞, F is supposed to contain an isomorphic copy of some $\ell _q$ , 1≦ _q   <  ∞).