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1997 | 125 | 3 | 271-287
Tytuł artykułu

Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

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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 $f_1,...,f_n ∈ L_q(μ)$, $( ʃ(∑^{n}_{k=1} |Tf_{k}|^r)^{p/r} dν)^{1/p} ≤ c∥T∥(ʃ(∑^{n}_{k=1} |f_k|^{r})^{q/r} dμ)^{1/q}$. 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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Bibliografia
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  • [9] J. Gasch and L. Maligranda, On vector-valued inequalities of Marcinkiewicz-Zygmund, Herz and Krivine type, Math. Nachr. 167 (1994), 95-129.
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  • [17] S. Kwapień, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193-201.
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  • [21] J. Marcinkiewicz et A. Zygmund, Quelques inégalités pour les opérations linéaires, Fund. Math. 32 (1939), 113-121.
  • [22] B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^p$, Astérisque 11 (1974).
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  • [24] R. E. A. C. Paley, On a remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-264.
  • [25] A. Pietsch, Absolutely p-summing operators in $L_r$-spaces, Bull. Soc. Math. France Mém. 31-32 (1972), 285-315.
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  • [27] P. Saphar, Applications p-décomposantes et p-absolument sommantes, Israel J. Math. 11 (1972), 164-179.
  • [28] H. Vogt, Komplexifizierung von Operatoren zwischen $L_p$-Räumen, Diplomarbeit, Oldenburg, 1995.
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