Multilinear extensions of Grothendieck's theorem

Bombal, F.

doi:10.1093/qjmath/55.4.441

Cited by 32 publications

(58 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the particular case of paraproducts, Marcinkiewicz-Zygmund inequalities were obtained by C. Benea and C. Muscalu in [1] and [2]. The results in [4] extend the previous ones in [10] and [3].…”

Section: A Vector-valued Extension Of Theorem 110supporting

confidence: 64%

Weighted mixed weak-type inequalities for multilinear operators

Ombrosi

Picardi

2019

Studia Math.

View full text Add to dashboard Cite

In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let w = (w1, ..., wm) and ν = w 1 m 1 ...w 1 m m , the main result of the paper sentences that under different conditions on the weights we can obtainwhere T is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the m-fold product of the Hardy-Littlewood maximal operator M , and also for M( f )(x): the multi(sub)linear maximal function introduced in [13].As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calderón-Zygmund operators.where T is Hardy-Littlewood maximal function or any Calderón-Zygmund operator.Quantitative estimates of these mixed weighted results can be found in [16]. Moreover, it was conjectured in [7] that the conclusion of the previous theorem still holds if a weaker and more general hypothesis is satisfied. That is, if we have the following conditions on the weights, u ∈ A 1 and v ∈ A ∞ . Recently, in [14] the first two authors and C. Pérez solved such conjecture. Namely, the following theorem was proved, which constitutes the more difficult case of this class of mixed weighted inequalities.

show abstract

Section: A Vector-valued Extension Of Theorem 110supporting

confidence: 64%

Weighted mixed weak-type inequalities for multilinear operators

Ombrosi

Picardi

2019

Studia Math.

View full text Add to dashboard Cite

show abstract

“…Items (ia) and (ic) are consequences of Theorem 3.1, then we only need to prove (ib). If k q,p (r) < ∞ then k q,p (r) < ∞ (by Proposition 1.4) and by Theorem 1.2(ii) it follows that 2 ≤ r ≤ p. For the converse, take 2 ≤ r ≤ p. If q ≤ r ≤ p then k q,p (r) = 1 by Remark 2.3, while if 2 ≤ r ≤ q then k q,p (r) < ∞ by Corollary 2.6, since k q,p (2) < ∞ (this is the particular case of the Marcinkiewicz-Zygmund inequalities proved in [5,18]) and k q,p (q) = 1 < ∞. Now we prove (ii).…”

Section: 2mentioning

confidence: 93%

Multilinear Marcinkiewicz-Zygmund Inequalities

Carando

Mazzitelli

Ombrosi

2017

J Fourier Anal Appl

View full text Add to dashboard Cite

Abstract. We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on ℓ r -valued extensions of linear operators. We show that for certain 1 ≤ p, q 1 , . . . , q m , r ≤ ∞, there is a constant C ≥ 0 such that for every bounded multilinear operator T :, the following inequality holds.In some cases we also calculate the best constant C ≥ 0 satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.

show abstract

“…Observe that there is no restriction to assume r k < q. Indeed, any linear map with value in a cotype q space is always (q, 1)-summing and we may apply Theorem 2.3 to deduce that any multilinear map with value in a cotype q space is always multiple (q, 1)-summing, a result already observed in [8,Theorem 3.2] Our second more appealing result happens when we start from a (r k , p k )-separately summing map (namely |C k | = 1 for all k) with 1…”

Section: 2mentioning

confidence: 56%

“…, X m , Y Banach spaces and T : X 1 × · · · × X m → Y mlinear. Following [8] and [17], for r ≥ 1 and p = (p 1 , . .…”

Section: Introductionmentioning

confidence: 99%