Some applications of the regularity principle in sequence spaces

Abstract

The Hardy–Littlewood inequalities for m-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q J Math 5:241–254, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For $m<p\le 2m$ it asserts that there is a constant $D_{m,p}^{\mathbb {K}}\ge 1$ such that

$$\begin{aligned} \left( \sum _{j_{1},\ldots ,j_{m}=1}^{n}\left| T(e_{j_{1}},\ldots ,e_{j_{m} })\right| ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\le D_{m,p} ^{\mathbb {K}}\left\| T\right\| , \end{aligned}$$

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–Littlewood inequality and show that:

• (1) If $m<p_{1}\le p_{2}\le 2m$ then $D_{m,p_{1}}^{\mathbb {K}}\le D_{m,p_{2}}^{\mathbb {K}}$;

• (2) $D_{m,p}^{\mathbb {K}}\le D_{m-1,p}^{\mathbb {K}}$ whenever $m<p\le 2\left( m-1\right)$ for all $m\ge 3$.