Some applications of the regularity principle in sequence spaces

Cavalcante, W.

doi:10.1007/s11117-017-0506-9

Cited by 7 publications

(8 citation statements)

References 18 publications

(13 reference statements)

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“…Recently, W.V. Cavalcante has shown that (4.1) is a consequence of the inclusion result for multiple summing operators due to Pellegrino et al combined with Theorem 5 (see [10]). The standard isometries between L(X p , X) and ℓ w p * (X), for 1 < p ≤ ∞, allow us to read the previous Theorems 5, 6, 7 as coincidence results (see [12]).…”

Section: Applications: Hardy-littlewood's Inequalitiesmentioning

confidence: 94%

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Anisotropic regularity principle in sequence spaces and applications

Albuquerque

Rezende

2018

Commun. Contemp. Math.

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We refine a recent technique introduced by Pellegrino, Santos, Serrano and Teixeira and prove a quite general anisotropic regularity principle in sequence spaces. As applications we generalize previous results of several authors regarding Hardy-Littlewood inequalities for multilinear forms. Littlewood inequalities. In this paper we follow this vein, exploring the ideas from the Regularity Principle proven by Pellegrino et al. [21] and providing a couple of applications.The paper is organized as follows. In Section 2, borrowing ideas from [21] we prove an anisotropic inclusion theorem for summing operators that will be useful along the paper. The techniques and arguments explored in Section 2 paves the way to the statement of a kind of anisotropic regularity principle for sequence spaces/series, in Section 3, completing results from [21]. In the final section the bulk of results are combined to prove new Hardy-Littlewood inequalities for multilinear operators.1.1. Summability of multilinear operators. The theory of multiple summing multilinear mappings was introduced in [18, 23]; this class is certainly one of the most useful and fruitful multilinear generalizations of the concept of absolutely summing linear operators, with important connections with the Bohnenblust-Hille and Hardy-Littlewood inequalities and its applications in applied sciences. For recent results on absolutely summing operators and these classical inequalities we refer to [7,11,17] and the references therein. The Hardy-Littlewood inequalities have its starting point in 1930 with Littlewood's 4/3 inequality [16]. In 1934 Hardy and Littlewood extended Littlewood's 4/3 inequality [15] to more general sequence spaces. Both results are for bilinear forms. In 1981 Praciano-Pereira [24] extended these results to the m-linear setting and recently various authors have revisited this subject. In 2016 Dimant and Sevilla-Peris [13] proved the following inequality: for all positive integers m and all m < p ≤ 2m we have (1.1)   ∞ j1,··· ,jm=1|T (e j1 , · · · , e jm )| p p−m   p−m p ≤ √ 2 m−1T for all continuous m-linear forms T : ℓ p × · · · × ℓ p → K. It is also proved that the exponent p p−m cannot be improved. However, this optimality seems to be just apparent, as remarked in some previous works (see [4,11]). Following these lines, the exponent can be potentially improved in the anisotropic viewpoint. In order to do that, the theory of summing operators shall play a fundamental role.Along the years, somewhat puzzling inclusion results for multilinear summing operators were obtained [8,9,23]. In this note we prove an inclusion result for multiple summing operators generalizing recent approaches of 2010 Mathematics Subject Classification. 46G25, 47H60.

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Section: Applications: Hardy-littlewood's Inequalitiesmentioning

confidence: 94%

“…The Hardy-Littlewood inequalities have been investigated in depth in the recent years (see, for instance, [1,2,3,4,10,11,13,17,19,21,22]). Here X p stands for ℓ p if p < ∞ and X ∞ := c 0 .…”

Section: Applications: Hardy-littlewood's Inequalitiesmentioning

confidence: 99%

Anisotropic regularity principle in sequence spaces and applications

Albuquerque

Rezende

2018

Commun. Contemp. Math.

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“…The Hardy-Littlewood inequalities consist in optimal extensions of Littlewood's 4/3 inequality [17] (originally stated for c 0 spaces). In the last years the interest in this subject (which can be considered part of the theory of multiple summing and absolutely summing operators) was renewed and several authors became interested in this field ( [1,2,3,6,7,20,22,23]).…”

Section: Introductionmentioning

confidence: 99%

“…Khintchine inequality is originally a result from probability, but it is also frequently used in Analysis and Topology (see [5,7,10,11,12,13,21]). The importance of the Rademacher functions and the Khintchine inequality in Functional Analysis lies mainly on the fact of its utility in the study of the geometry of Banach spaces (see [5,10,12]).…”

Section: Introductionmentioning

confidence: 99%

The best constants in the multiple Khintchine inequality

Núñez-Alarcón

2018

Linear and Multilinear Algebra

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“…The original purpose of this inequality is to provide a unimodular form on (ℓ n p ) d with relatively small supremum norm but relatively large majorant function [12]. Nowadays this result is a formidable tool in modern analysis with a broad range of applications (see, e.g., [1,2,4,7,13,14,15,16,18,19]).…”

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confidence: 99%