# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1705.06307