When are the Hardy-Littlewood inequalities contractive?

TL;DR

This paper investigates conditions under which the Hardy-Littlewood and Bohnenblust-Hille inequalities have contractive optimal constants, showing that restricting sums to certain index sets can ensure contractivity.

Contribution

It establishes new conditions on the size of index sets that guarantee contractive optimal constants for these inequalities, advancing understanding of their behavior.

Findings

Optimal constants are not contractive in general.

Contractivity can be achieved when sums are over index sets with size M satisfying M log M = o(m).

For M ≤ m^{1-ε}, the inequalities are 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 $\lfloor x\rfloor:=\max\{n\in\mathbb{N}:n\leq x\}.$ 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.