Positivity

, Volume 22, Issue 1, pp 191–198

# Some applications of the regularity principle in sequence spaces

Article

## 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. (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. (2)

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

### Keywords

Multilinear forms Hardy–Littlewood inequalities Regularity principle

### Mathematics Subject Classification

47H60 11Y60 47A63 46G25

### References

1. 1.
Albuquerque, N., Nogueira, T., Núñez-Alarcón, D., Pellegrino, D., Rueda, P.: Some applications of the Hölder inequality for mixed sums, to appear in Positivity. doi:
2. 2.
Albuquerque, N., Araújo, G., Maia, M., Nogueira, T., Pellegrino, D., Santos, J.: Optimal Hardy–Littlewood Inequalities Uniformly Bounded by a Universal Constant. arXiv:1609.03081 (2016)
3. 3.
Araújo, G., Pellegrino, D., Silva e Silva, D.D.P.: On the upper bounds for the constants of the Hardy–Littlewood inequality. J. Funct. Anal. 267, 1878–1888 (2014)
4. 4.
Araújo, G., Pellegrino, D.: Lower bounds for the complex polynomial Hardy–Littlewood inequality. Linear Algebra Appl. 474, 184–191 (2015)
5. 5.
Bayart, F.: Multiple Summing Maps: Coordinatewise Summability, Inclusion Theorems and $$p$$-Sidon Sets. arXiv:1704.04437 (2017)
6. 6.
Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32, 600–622 (1931)
7. 7.
Caro, N., Núñez-Alarcón, D., Serrano-Rodríguez, D.: On the generalized Bohnenblust–Hille inequality for real scalars, to appear in Positivity. doi:
8. 8.
Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)
9. 9.
Dimant, V., Sevilla-Peris, P.: Summation of coefficients of polynomials on $$\ell _{p}$$ spaces. Publ. Mat. 60, 289–310 (2016)
10. 10.
Hardy, G., Littlewood, J.E.: Bilinear forms bounded in space $$[p, q]$$. Q. J. Math. 5, 241–254 (1934)
11. 11.
Littlewood, J.E.: On bounded bilinear forms in an infinite number of variables. Q. J. Math. 1, 164–174 (1930)
12. 12.
Maia, M., Nogueira, T., Pellegrino, D.: Bohnenblust–Hille inequality for polynomials whose monomials have uniformly bounded number of variables. Integral Equ. Oper. Theory 88, 143–149 (2017)
13. 13.
Nunes, A.: A new estimate for the constants of an inequality due to Hardy and Littlewood. Linear Algebra Appl. 526, 27–34 (2017)
14. 14.
Pellegrino, D., Santos, J., Serrano-Rodríguez, D., Teixeira, E.V.: Regularity Principle in Sequence Spaces and Applications. arXiv:1608.03423 [math.CA] (2016)
15. 15.
16. 16.
Praciano-Pereira, T.: On bounded multilinear forms on a class of $$\ell _{p}$$ spaces. J. Math. Anal. Appl. 81, 561–568 (1981)