Desigualdade de Bohnenblust-Hille: estimativas e comportamento assintótico

Abstract

The Bohnenblust{Hille inequality guarantees the existence of a function C : N ! [1;+1), corresponding to each positive integer m, a constant C(m) with the following property: regardless of the choice of the natural N and the bounded m-linear form U : `N 1 `N 1 ! K, the sequence (U(ei1 ; : : : ; eim))N i1;:::;im=1 belongs to ` 2m m+1 and its 2m m+1-norm is bounded by C(m)kUk, where k k denotes the supremum norm. Apart from the intrinsic mathematical interest, for C(m) does not depend on each natural N, the diversity and relevance of the applications enrich the result further. On the actual scenario, recent explicit estimates for the constants C(m) present optimal asymptotic behaviour and subexponencial growth, in contrast to the exponential growth of the known estimates from the last decades. Valuable informations concerning the optimal constants (the lowest ones with the previous property stated) emerge, once proved that these also enjoy of an optimal asymptotic growth, if it exists.