×

zbMATH — the first resource for mathematics

Some applications of the regularity principle in sequence spaces. (English) Zbl 06861658
Summary: The Hardy-Littlewood inequalities for \(m\)-linear forms have their origin with the seminal paper of G. H. Hardy and J. E. Littlewood [Q. J. Math., Oxf. Ser. 5, 241–254 (1934; Zbl 0010.36101)]. Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For \(m<p\leq 2m\) it asserts that there is a constant \(D_{m,p}^{\mathbb K}\geq 1\) such that \[ \left(\sum_{j_1,\dots,j_m=1}^{n}| T(e_{j_1},\dots,e_{j_m})| ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq D_{m,p} ^{\mathbb K}\| T\| , \] 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: cm
(1)
If \(m<p_1\leq p_2\leq 2m\) 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 \(m<p\leq 2(m-1)\) for all \(m\geq 3\).

MSC:
47H60 Multilinear and polynomial operators
11Y60 Evaluation of number-theoretic constants
47A63 Linear operator inequalities
46G25 (Spaces of) multilinear mappings, polynomials
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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:10.1007/s11117-017-0486-9 · Zbl 06816318
[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) · Zbl 1410.46027
[3] Araújo, G; Pellegrino, D; Silva e Silva, DDP, On the upper bounds for the constants of the Hardy-Littlewood inequality, J. Funct. Anal., 267, 1878-1888, (2014) · Zbl 1298.26066
[4] Araújo, G; Pellegrino, D, Lower bounds for the complex polynomial Hardy-Littlewood inequality, Linear Algebra Appl., 474, 184-191, (2015) · Zbl 1327.46045
[5] Bayart, F.: Multiple Summing Maps: Coordinatewise Summability, Inclusion Theorems and \(p\)-Sidon Sets. arXiv:1704.04437 (2017) · Zbl 1391.46057
[6] Bohnenblust, HF; Hille, E, On the absolute convergence of Dirichlet series, Ann. Math., 32, 600-622, (1931) · Zbl 0001.26901
[7] Caro, N., Núñez-Alarcón, D., Serrano-Rodríguez, D.: On the generalized Bohnenblust-Hille inequality for real scalars, to appear in Positivity. doi:10.1007/s11117-017-0478-9 · Zbl 06816310
[8] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995) · Zbl 0855.47016
[9] Dimant, V; Sevilla-Peris, P, Summation of coefficients of polynomials on \(ℓ _{p}\) spaces, Publ. Mat., 60, 289-310, (2016) · Zbl 1378.46032
[10] Hardy, G; Littlewood, JE, Bilinear forms bounded in space \([p, q]\), Q. J. Math., 5, 241-254, (1934) · Zbl 0010.36101
[11] Littlewood, JE, On bounded bilinear forms in an infinite number of variables, Q. J. Math., 1, 164-174, (1930) · JFM 56.0335.01
[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) · Zbl 1378.32001
[13] Nunes, A, A new estimate for the constants of an inequality due to Hardy and Littlewood, Linear Algebra Appl., 526, 27-34, (2017) · Zbl 1376.46033
[14] Pellegrino, D., Santos, J., Serrano-Rodríguez, D., Teixeira, E.V.: Regularity Principle in Sequence Spaces and Applications. arXiv:1608.03423 [math.CA] (2016) · Zbl 1404.46041
[15] Pérez-García, D.: Operadores Multilineales Absolutamente Sumantes, Ph.D. thesis, Universidad Complutense de Madrid (2004) · Zbl 1298.26066
[16] Praciano-Pereira, T, On bounded multilinear forms on a class of \(ℓ _{p}\) spaces, J. Math. Anal. Appl., 81, 561-568, (1981) · Zbl 0497.46007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.