Multiple summing maps: Coordinatewise summability, inclusion theorems and p-Sidon sets

Bayart, Frédéric

doi:10.1016/j.jfa.2017.08.013

Cited by 55 publications

(25 citation statements)

References 23 publications

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“…Then, by the Bohnenblust-Hille inequality [7] (for k = 1) and by the induction hypothesis (otherwise), we know that each T k (x k ) is multiple (r k , 1)−summing. It follows from the results of [26] (see also [14] and [4]) that T is multiple (r, 1)−summing with…”

Section: 2mentioning

confidence: 99%

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

2019

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Grothendieck's theorem asserts that every continuous linear operator from ℓ1 to ℓ2 is absolutely (1, 1)-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by ℓ1 spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of m injective tensor products is solved.

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Section: 2mentioning

confidence: 99%

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

2019

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“…Motivated by the applications of the class of multiple summing operators (see [4, 9, 20]), we will explain in this section that our arguments allow us to prove that some abstract classes of semi‐integral operators are multiple summing in a more general setting. Before stating the results, we recall some geometrical notions from the theory of Banach lattices.…”

Section: The Haar Measure For φ Semi‐integral Multilinear Operatorsmentioning

confidence: 99%

“…We show that these operators are multiple summing operators in a more general sense, underlying the fact that the results that are obtained for φ semi‐integral mappings are actually true in a broader setting. Finally, let us point out that multiple summing operators generated by scales of

ℓ_{p}

‐sequence spaces is an active current area of research that has found several interesting applications (see, e.g., [4, 9, 20]). Our study generalizes some domination properties of multilinear operators associated to

ℓ_{p}

‐spaces to the case of Orlicz sequence spaces.…”

Section: Introductionmentioning

confidence: 99%

Ideals of multilinear mappings via Orlicz spaces and translation invariant operators

Mastyło

Pérez

2021

Mathematische Nachrichten

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We study some new summability properties of multilinear operators. We introduce the concepts of φ‐summing, φ semi‐integral and φ‐dominated multilinear maps generated by Orlicz functions. We prove a variant of Pietsch's domination theorem for φ‐summing operators, providing also a characterization of φ‐dominated operators in terms of factorizations. We analyze vector‐valued inequalities associated to these maps, which are applied to obtain general variants of multiple summing operators. We also study translation invariant multilinear operators acting in products of spaces of continuous functions, proving that a factorization theorem can be obtained for them as a consequence of a suitable representation of the corresponding normalized Haar measure.

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“…The Inclusion Theorem [17, Proposition 3.4] was also proved recently and independently, with a different technique, by F. Bayart [7].…”

Section: A Straightforward Proof Of the Hardy-littlewood Inequality Amentioning

confidence: 99%

Some applications of the regularity principle in sequence spaces

Cavalcante

2017

Positivity

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Abstract. 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 ≤ 2m it asserts that there is a constantand 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:( IntroductionLittlewood's 4/3 inequality [14] is probably the forerunner of the by now socalled Hardy-Littlewood inequalities for multilinear forms. Published in 1930, it asserts that n i,j=1|T (e i , e j )|

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Multiple summing maps: Coordinatewise summability, inclusion theorems and p-Sidon sets

Cited by 55 publications

References 23 publications

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

Ideals of multilinear mappings via Orlicz spaces and translation invariant operators

Some applications of the regularity principle in sequence spaces

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