Hardy-Littlewood / Bohnenblust-Hille multilinear inequalities and Peano curves on topological vector spaces.

Abstract

This work is divided in two subjects. The first concerns about the Bohnenblust–Hille and Hardy– Littlewood multilinear inequalities. We obtain optimal and definitive generalizations for both inequalities. Moreover, the approach presented provides much simpler and straightforward proofs than the previous one known, and we are able to show that in most cases the exponents involved are optimal. The technique used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this thesis to improve the constants for vector-valued Bohnenblust–Hille type inequalities. The second subject has as starting point the existence of Peano spaces, that is, Haurdorff spaces that are continuous image of the unit interval. From the point of view of lineability we analyze the set of continuous surjections from an arbitrary euclidean spaces on topological spaces that are, in some natural sense, covered by Peano spaces, and we conclude that large algebras are found within the families studied. We provide several optimal and definitive result on euclidean spaces, and, moreover, an optimal lineability result on those special topological vector spaces.