Gustavo Araújo

On the upper bounds for the constants of the Hardy–Littlewood inequality☆

Abstract The best known upper estimates for the constants of the Hardy–Littlewood inequality for m-linear forms on l p spaces are of the form ( 2 ) m − 1 . We present better estimates which depend on p and m. An interesting consequence is that if p ≥ m 2 then the constants have a subpolynomial growth as m tends to infinity.

Optimal estimates for summing multilinear operators

We show that given a positive integer m, a real number and the set of non-multiple -summing m-linear forms on contains, except for the null vector, a closed subspace of maximal dimension whenever . This result is optimal since for all m-linear forms on are multiple -summing. In particular, among other results, we generalize a result related to cotype (from 2010) due to Botelho, Michels and the second named author.

Lower bounds for the complex polynomial Hardy–Littlewood inequality☆

Abstract The Hardy–Littlewood inequality for complex homogeneous polynomials asserts that given positive integers m ≥ 2 and n ≥ 1 , if P is a complex homogeneous polynomial of degree m on l p n with m p ≤ ∞ given by P ( x 1 , … , x n ) = ∑ | α | = m a α x α , then there exists a constant C C , m , p pol ≥ 1 (which does not depend on n ) such that ( ∑ | α | = m | a α | ρ ) 1 ρ ≤ C C , m , p pol ⋅ sup z ∈ B l p n ⁡ | P ( z ) | , with ρ = p p − m if m p 2 m and ρ = 2 m p m p + p − 2 m if 2 m ≤ p ≤ ∞ . In this short note we provide nontrivial lower bounds for the constants C C , m , p pol . For instance we prove that, for m ≥ 2 and m p ∞ , C C , m , p pol ≥ 2 m p for m even, and C C , m , p pol ≥ 2 m − 1 p for m odd. Estimates for the case p = ∞ (this is the particular case of the complex polynomial Bohnenblust–Hille inequality) were recently obtained by D. Nunez-Alarcon in 2013.

Lineability in sequence and function spaces

We prove the existence of large algebraic structures - including large vector subspaces or infinitely generated free algebras - inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of discontinuous separately continuous real functions. Lineability in special spaces of sequences is also investigated.Programa de Doutorado Sanduiche no Exterior (Coordenacao de aperfeicoamento de pessoal de nivel superior)

A note on multiple summing operators and applications

ABSTRACT We prove a new result on multiple summing operators and, among other results and applications, we provide a new extension of Littlewood’s 4 / 3 inequality to m-linear forms.