# Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Studia Mathematica (1997)

• Volume: 125, Issue: 3, page 271-287
• ISSN: 0039-3223

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## Abstract

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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 $T:{L}_{q}\left(\mu \right)\to {L}_{p}\left(\nu \right)$, each n ∈ ℕ and functions ${f}_{1},...,{f}_{n}\in {L}_{q}\left(\mu \right)$, $\left(ʃ\left({\sum }_{k=1}^{n}|T{f}_{k}{|}^{r}{\right)}^{p/r}{d\nu \right)}^{1/p}\le c\parallel T\parallel \left(ʃ\left({\sum }_{k=1}^{n}|{f}_{k}{|}^{r}{{\right)}^{q/r}d\mu \right)}^{1/q}$. 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  are solved; as a by-product we obtain best constants of several important inequalities from the theory of summing operators.

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