Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities
Studia Mathematica, Tome 125 (1997) no. 3, pp. 271-287
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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 $T: L_q(μ) → L_p(ν)$, each n ∈ ℕ and functions $f_1,...,f_n ∈ L_q(μ)$, $( ʃ(∑^{n}_{k=1} |Tf_{k}|^r)^{p/r} dν)^{1/p} ≤ c∥T∥(ʃ(∑^{n}_{k=1} |f_k|^{r})^{q/r} dμ)^{1/q}$. 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.
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author = {Andreas Defant and },
title = {Best constants and asymptotics of {Marcinkiewicz-Zygmund} inequalities},
journal = {Studia Mathematica},
pages = {271--287},
publisher = {mathdoc},
volume = {125},
number = {3},
year = {1997},
doi = {10.4064/sm-125-3-271-287},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-125-3-271-287/}
}
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Andreas Defant; . Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities. Studia Mathematica, Tome 125 (1997) no. 3, pp. 271-287. doi: 10.4064/sm-125-3-271-287
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