When are the Hardy-Littlewood inequalities contractive?

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 $\lfloor x\rfloor :=\max \{n\in \mathbb{N}:n\le x\}.$ In particular, if $\epsilon >0$ and $M:=M(m)\le m^{1-\epsilon },$ then the Bohnenblust-Hille inequality restricted to sums over $M$ indexes is contractive.

This preprint was incorporated in the Arxiv preprint number arXiv:1409.6769