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N. Albuquerque

Publications and source records attributed to N. Albuquerque.

8 recordsLinked to original sources
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↗
arXiv2016

Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant

The Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces and $m<p\leq 2m$ asserts that \begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*} for all continuous $m$-linear forms $T:\ell _{p}\times \cdots \times \ell _{p}\rightarrow \mathbb{R}$ or $\mathbb{C}.$ The case $m=2$ recovers a classical inequality proved by Hardy and Littlewood in 1934. As a consequence of the results of the present paper we show that the same inequality is valid with $2^{\frac{m-1}{2}}$ replaced by $2^{\frac{\left( m-1\right) \left( p-m\right) }{p}}$. In particular, for $m<p\leq m+1$ the optimal constants of the above inequality are uniformly bounded by $2.$

math.FA↗
arXiv2015

Absolutely summing multilinear operators via interpolation

We use an interpolative technique from \cite{abps} to introduce the notion of multiple $N$-separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust-Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sepúlveda. More precisely, as a consequence of our main result, for $1\leq t<2$ and $m\in \mathbb{N}$ we prove that $$ \left(\sum_{i_{1},\dots,i_{m}=1}^{\infty}\left\vert U\left(e_{i_{1}},\dots,e_{i_{m}}\right) \right\vert^{\frac{2tm}{2+(m-1)t}}\right)^{\frac{2+(m-1)t}{2tm}} \leq \left[\prod_{j=2}^{m}Γ\left(2-\frac{2-t}{jt-2t+2}\right) ^{\frac{t(j-2)+2}{2t-2jt}}\right] \left\Vert U\right\Vert $$ for all complex $m$-linear forms $U:c_{0}\times \cdot \cdot \cdot \times c_{0}\rightarrow \mathbb{C}$.

math.FA↗
arXiv2015

A subexponential vector-valued Bohnenblust-Hille type inequality

Bayart, Pellegrino and Seoane recently proved that the polynomial Bohnenblust--Hille inequality for complex scalars is subexponential. We show that a vector valued polynomial Bohnenblust-Hille inequality on complex Banach lattices is also subexponential for some special cases. Our main result result recovers the best known constants of the classical polynomial inequality provided in \cite{bps}.

math.FA↗
arXiv2015

Hölder's inequality: some recent and unexpected applications

Hölder's inequality, since its appearance in 1888, has played a fundamental role in Mathematical Analysis and it is, without any doubt, one of the milestones in Mathematics. It may seem strange that, nowadays, it keeps resurfacing and bringing new insights to the mathematical community. In this expository article we show how a variant of Hölder's inequality (although well-known in PDEs) was essentially overlooked in Functional Analysis and has had a crucial (and in some sense unexpected) influence in very recent and major breakthroughs in Mathematics. Some of these recent advances appeared in 2012-2014 and include the theory of Dirichlet series, the famous Bohr radius problem, certain classical inequalities (such as Bohnenblust--Hille or Hardy--Littlewood), or even Mathematical Physics.

math.FA↗
arXiv2014

Peano curves on topological vector spaces

The starting point of this paper is the existence of Peano curves, that is, continuous surjections mapping the unit interval onto the unit square. From this fact one can easily construct of a continuous surjection from the real line $\mathbb{R}$ to any Euclidean space $\mathbb{R}^n$. The algebraic structure of the set of these functions (as well as extensions to spaces with higher dimensions) is analyzed from the modern point of view of lineability, and large algebras are found within the families studied. We also investigate topological vector spaces that are continuous image of the real line, providing an optimal lineability result.

math.GN↗