Estruturas robustas em classes de operadores e a desigualdade multipolinomial de Kahane-Salem-Zygmund

Abstract

This work presents results in two distinct lines of research: lineability/spaceability and multipolynomial versions of Kahane-Salem-Zygmund inequalities. First, we present the basic notions and properties of the theories and central topics of the thesis, which form Chapter 1. In Chapter 2, we provide results on the contemporary notions of pointwise lineability/ spaceability and (α, β)-lineability/spaceability, which are more restrictive than the classical notion in the context of multilinear summing operators taking values on (quasi-)Banach sequence spaces. In particular, we prove that the class of multilinear operators taking values on ℓq (0 < q ≤ ∞) that are absolutely but not multiple summing is pointwise c-spaceable when non-empty. In Chapter 3, we expose a criterion of spaceability within the general setting of multi-ideal operators that generalizes the one presented by F. L. Hern´andez et al. and some applications. Finally, we reserve for Chapter 4 the presentation of the multipolynomial versions of the Kahane-Salem-Zygmund inequalities. An initial variant of a multilinear version with optimal exponents is obtained, and given certain restrictions, we invoke other multilinear versions, and their multipolynomial extensions are provided.