When are the Hardy-Littlewood inequalities contractive?

Autor: Cavalcante, W. V., Nogueira, T., Pellegrino, D. M., Santos, J., Rueda, P.
Rok vydání: 2017
Předmět:
Druh dokumentu: Working Paper
Popis: 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.
Comment: This preprint was incorporated in the Arxiv preprint number arXiv:1409.6769
Databáze: arXiv