# Some applications of the Regularity Principle in sequence spaces

## TL;DR

This paper uses a recent Regularity Principle to provide a simple proof of Hardy--Littlewood inequalities for m-linear forms, establishing monotonicity of constants and their bounds across different parameters.

## Contribution

It offers a straightforward proof of Hardy--Littlewood inequalities leveraging a new Regularity Principle, and derives monotonicity properties of the involved constants.

## Key findings

- Proves the Hardy--Littlewood inequality using the Regularity Principle.
- Shows that the constants are non-decreasing with respect to p.
- Establishes bounds relating constants for different m and p values.

## Abstract

The Hardy--Littlewood inequalities for $m$-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q.J. Math, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For $m<p\leq2m$ it asserts that there is a constant $D_{m,p}^{\mathbb{K}}\geq1$ such that \[ \left( \sum_{j_{1},\cdots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\cdots,e_{j_{m}})\right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq D_{m,p}^{\mathbb{K}}\left\Vert T\right\Vert , \] for all $m$--linear forms $T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and all positive integers $n$. Using a Regularity Principle recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a straightforward proof of the Hardy--Littewood inequality and show that: (1) If $m<p_{1}<p_{2}\leq2m$ then $D_{m,p_{1}}^{\mathbb{K}}\leq D_{m,p_{2}}^{\mathbb{K}}$; (2) $D_{m,p}^{\mathbb{K}}\leq D_{m-1,p}^{\mathbb{K}}$ whenever $m<p\leq 2\left( m-1\right) $ for all $m\geq3$.