Some applications of the Regularity Principle in sequence spaces

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 $m\geq3$.