arXiv subjects

T. Nogueira

Publications and source records attributed to T. Nogueira.

3 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↗

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↗

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↗