Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Andreas Defant; Marius Junge

Studia Mathematica (1997)

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

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 ( μ ) → L p ( ν ) , each n ∈ ℕ and functions f 1 , . . . , f n ∈ L q ( μ ) , ( ʃ ( ∑ k = 1 n | T f k | r ) p / r d ν ) 1 / p ≤ c ∥ T ∥ ( ʃ ( ∑ 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.

How to cite

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Defant, Andreas, and Junge, Marius. "Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities." Studia Mathematica 125.3 (1997): 271-287. <http://eudml.org/doc/216438>.

@article{Defant1997,
abstract = {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.},
author = {Defant, Andreas, Junge, Marius},
journal = {Studia Mathematica},
keywords = {Marcinkiewicz-Zygmund inequality; Bochner spaces; -summing; -integral; -mixing operators},
language = {eng},
number = {3},
pages = {271-287},
title = {Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities},
url = {http://eudml.org/doc/216438},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Defant, Andreas
AU - Junge, Marius
TI - Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 3
SP - 271
EP - 287
AB - 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.
LA - eng
KW - Marcinkiewicz-Zygmund inequality; Bochner spaces; -summing; -integral; -mixing operators
UR - http://eudml.org/doc/216438
ER -

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