Sobre algumas desigualdades clássicas e espaços de sequências

Abstract

This work is divided into three parts. In the first, we study the behavior of constants that satisfy Hardy–Littlewood inequalities to multilinear forms defined in sequence spaces. Initially, we present the optimal constants for a particular type, called mixe (l p/ p−1 , l2) Littlewood inequality. Then, for other inequalities, we see what happens to the constants when we disturb the optimal exponents. In the second part, we solve definitively a problem raised by Carando, Defant and Sevilla–Peris: given the Bohnenblust–Hille inequality for complex m-homogeneous polynomials whose monomials have a number of variables uniformly bounded by a positive integer M, we show that the optimal constants are uniformly bounded, regardless of the value of m. In the third part, we study lineability in sequence spaces. We show that certain subsets of some spaces of invariant sequences contain, except for the null sequence, a closed subspace of infinite dimension.