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arXiv · 1705.06307

When are the Hardy-Littlewood inequalities contractive?

W. V. Cavalcante·T. Nogueira·D. M. Pellegrino·J. Santos·P. Rueda
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Abstract

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.

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W. V. Cavalcante, T. Nogueira, D. M. Pellegrino, J. Santos, P. Rueda. 2017-05-17. When are the Hardy-Littlewood inequalities contractive?. https://arxiv.org/abs/1705.06307

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