Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Andreas Defant; Marius Junge

ArticleOriginal scientific text

Title

Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Authors ¹, ²

Affiliations

Fachbereich Mathematik, Universität Oldenburg, D-26111 Oldenburg, Germany
Mathematisches Seminar, Universität Kiel, D-24098 Kiel, Germany

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) \to L_{p} (ν)

, each n ∈ ℕ and functions

f_{1}, ..., f_{n} \in L_{q} (μ)

,

{(ʃ {(\sum^{n}_{k = 1} {| T f_{k} |}^{r})}^{\frac{p}{r}} d ν)}^{\frac{1}{p}} \leq c ∥ T ∥ {(ʃ {(\sum^{n}_{k = 1} {| f_{k} |}^{r})}^{\frac{q}{r}} d μ)}^{\frac{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.

G. Baumbach and W. Linde, Asymptotic behaviour of p-summing norms of identity operators, Math. Nachr. 78 (1977), 193-196.
B. Carl and A. Defant, An inequality between the p- and (p,1)-summing norm of finite rank operators from C(K)-spaces, Israel J. Math. 74 (1991), 323-335.
B. Carl and A. Defant, Tensor products and Grothendieck type inequalities of operators in $L_{p}$ -spaces, Trans. Amer. Math. Soc. 331 (1992), 55-76.
A. Defant, Best constants for the norm of the complexification of operators between $L_{p}$ -spaces, in: K. D. Bierstedt, A. Pietsch, W. M. Ruess and D. Vogt (eds.), Functional Analysis, Proc. Essen Conf., 1991, Lecture Notes in Pure and Appl. Math. 150, Dekker, 1993, 173-180.
A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, 1993.
J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995.
J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 104, North-Holland, 1985.
D. J. H. Garling, Absolutely p-summing operators in Hilbert space, Studia Math. 38 (1970), 319-331.
J. Gasch and L. Maligranda, On vector-valued inequalities of Marcinkiewicz-Zygmund, Herz and Krivine type, Math. Nachr. 167 (1994), 95-129.
E. Gené, M. B. Marcus and J. Zinn, A version of Chevet's theorem for stable processes, J. Funct. Anal. 63 (1985), 47-73.
A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1956), 1-79.
C. Herz, The theory of p-spaces with application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69-82.
J. Hoffmann-Jøorgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159-186.
M. Junge, Geometric applications of the Gordon-Lewis property, Forum Math. 6 (1994), 617-635.
H. König, On the complex Grothendieck constant in the n-dimensional case, in: P. F. X. Müller and W. Schachermeyer (eds.), Proc. Strobl Conference on "Geometry of Banach spaces", London Math. Soc. Lecture Note Ser. 158, Cambridge Univ. Press, 1990, 181-199.
J. L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les sphères, Adv. Math. 31 (1979), 16-30.
S. Kwapień, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193-201.
S. Kwapień, On operators factoring through $L_{p}$ -space, Bull. Soc. Math. France Mém. 31-32 (1972), 215-225.
M. Ledoux and M. Talagrand, Probability in Banach Spaces, Ergeb. Math. Grenzgeb. 23, Springer, 1991.
J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $ℒ_{p}$ -spaces and applications, Studia Math. 29 (1968), 275-326.
J. Marcinkiewicz et A. Zygmund, Quelques inégalités pour les opérations linéaires, Fund. Math. 32 (1939), 113-121.
B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^{p}$ , Astérisque 11 (1974).
B. Maurey et G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), 45-90.
R. E. A. C. Paley, On a remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-264.
A. Pietsch, Absolutely p-summing operators in $L_{r}$ -spaces, Bull. Soc. Math. France Mém. 31-32 (1972), 285-315.
A. Pietsch, Operator Ideals, North-Holland, 1980.
P. Saphar, Applications p-décomposantes et p-absolument sommantes, Israel J. Math. 11 (1972), 164-179.
H. Vogt, Komplexifizierung von Operatoren zwischen $L_{p}$ -Räumen, Diplomarbeit, Oldenburg, 1995.

Title

Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Affiliations

Abstract

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