arXiv subjects

W. V. Cavalcante

Publications and source records attributed to W. V. Cavalcante.

4 recordsLinked to original sources

arXiv2019

Extension of Lipschitz-type operators on Banach function spaces

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving integrals. These Lipschitz type inequalities provide the suitable frame to work with operators that take values on Banach function spaces. In the last part of the paper we use some interpolation procedures to extend our study to interpolated Banach function spaces.

math.FA↗

arXiv2018

On summability of multilinear operators and applications

This paper has two clear motivations: a technical and a practical. The technical motivation unifies in a single and crystal clear formulation a huge family of inequalities that have been produced separately in the last 90 years in different contexts. But we do not just join inequalities; our method also create a family of inequalities invisible by previous approaches. The practical motivation is to show that our deeper approach has strength to attack various problems. We provide new applications of our family of inequalities, continuing the recent work by Maia et al., that, by using our main theorem, substantially improved an inequality of Carando et al. which seemed impossible to be achieved by their original method.

math.FA↗

arXiv2018

When are the Hardy-Littlewood inequalities contractive?

The optimal constants of the $m$-linear Bohnenblust-Hille and Hardy-Littlewood inequalities are still not known despite its importance in several fields of Mathematics. For the Bohnenblust-Hille inequality and real scalars it is well-known that the optimal constants are not contractive. In this note, among other results, we show that if we consider sums over $M:=M(m)$ indexes with $M\log M=o(m)$, the optimal constants are contractive. For instance, we can consider% \[ M=\left\lfloor \frac{m}{\left( \log m\right) ^{1+\frac{1}{\log\log\log m}}% }\right\rfloor \] where $\lfloor x\rfloor:=\max\{n\in\mathbb{N}:n\leq x\}.$ In particular, if $\varepsilon>0$ and $M:=M(m)\leq m^{1-\varepsilon},$ then the Bohnenblust-Hille inequality restricted to sums over $M$ indexes is contractive.

math.FA↗

arXiv2017

On the geometry of multilinear forms

We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary steps, and thus it can be extensively applied in both computational and theoretical problems.

math.FA↗