Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Defant, Andreas; Junge, Marius

Studia Mathematica, Tome 122 (1997), p. 271-287 / Harvested from The Polish Digital Mathematics Library

Résumé

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T : L_{q} (μ) \to L_{p} (ν)$ , each n ∈ ℕ and functions $f_{1}, . . ., f_{n} \in L_{q} (μ)$ , $(ʃ (\sum_{k = 1}^{n} | T f_{k} |^{r})^{p / r} {d ν)}^{1 / p} \leq c ∥ T ∥ (ʃ (\sum_{k = 1}^{n} | f_{k} |^{r} {)^{q / r} d μ)}^{1 / q}$ . This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception: the important case 1 ≤ p < r = 2 < q ≤ ∞); if such an inequality does not hold, then we give asymptotically optimal estimates for the graduation of these constants in n. Two problems of Gasch and Maligranda from [9] are solved; as a by-product we obtain best constants of several important inequalities from the theory of summing operators.

Publié le : 1997-01-01

Zbl 0916.47018

EUDML-ID : urn:eudml:doc:216438

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     author = {Andreas Defant and Marius Junge},
     title = {Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {271-287},
     zbl = {0916.47018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv125i3p271bwm}
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Defant, Andreas; Junge, Marius. Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities. Studia Mathematica, Tome 122 (1997) pp. 271-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv125i3p271bwm/

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