On the UMD constant of the space $ℓ₁^{N}$

Adam Osękowski

On the UMD constant of the space

ℓ ₁^{N}

Adam Osękowski

Colloquium Mathematicae, Tome 144 (2016), p. 135-147 / Harvested from The Polish Digital Mathematics Library

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Résumé

Let N ≥ 2 be a given integer. Suppose that $d f = {(d f ₙ)}_{n \geq 0}$ is a martingale difference sequence with values in $ℓ ₁^{N}$ and let ${(ε ₙ)}_{n \geq 0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $ℙ (s u p_{n \geq 0} | | \sum_{k = 0}^{n} ε_{k} d f_{k} {| |}_{ℓ ₁^{N}} \geq 1) \leq (l n N + l n (3 l n N)) / (1 - {(2 l n N)}^{- 1}) s u p_{n \geq 0} | | \sum_{k = 0}^{n} d f_{k} {| |}_{ℓ ₁^{N}}$ . It is shown that this result is asymptotically sharp in the sense that the least constant $C_{N}$ in the above estimate satisfies $l i m_{N \to \infty} C_{N} / l n N = 1$ . The novelty in the proof is the explicit verification of the ζ-convexity of the space $ℓ ₁^{N}$ .

Publié le : 2016-01-01

Zbl 06497303

EUDML-ID : urn:eudml:doc:283650

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     author = {Adam Os\k ekowski},
     title = {On the UMD constant of the space $l1^{N}$
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     journal = {Colloquium Mathematicae},
     volume = {144},
     year = {2016},
     pages = {135-147},
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Adam Osękowski. On the UMD constant of the space $ℓ₁^{N}$
            . Colloquium Mathematicae, Tome 144 (2016) pp. 135-147. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-1-7/