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

top
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

top

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 -

References

top
  1. [1] G. Baumbach and W. Linde, Asymptotic behaviour of p-summing norms of identity operators, Math. Nachr. 78 (1977), 193-196. Zbl0384.46005
  2. [2] 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. 
  3. [3] 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. Zbl0785.47021
  4. [4] 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. Zbl0793.47035
  5. [5] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, 1993. Zbl0774.46018
  6. [6] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995. Zbl0855.47016
  7. [7] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 104, North-Holland, 1985. 
  8. [8] D. J. H. Garling, Absolutely p-summing operators in Hilbert space, Studia Math. 38 (1970), 319-331. Zbl0203.45502
  9. [9] J. Gasch and L. Maligranda, On vector-valued inequalities of Marcinkiewicz-Zygmund, Herz and Krivine type, Math. Nachr. 167 (1994), 95-129. Zbl0842.47019
  10. [10] E. Gené, M. B. Marcus and J. Zinn, A version of Chevet's theorem for stable processes, J. Funct. Anal. 63 (1985), 47-73. Zbl0654.60009
  11. [11] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1956), 1-79. Zbl0074.32303
  12. [12] C. Herz, The theory of p-spaces with application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69-82. Zbl0216.15606
  13. [13] J. Hoffmann-Jøorgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159-186. Zbl0265.60005
  14. [14] M. Junge, Geometric applications of the Gordon-Lewis property, Forum Math. 6 (1994), 617-635. Zbl0809.52009
  15. [15] 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. 
  16. [16] J. L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les sphères, Adv. Math. 31 (1979), 16-30. Zbl0413.46054
  17. [17] S. Kwapień, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193-201. Zbl0211.43505
  18. [18] S. Kwapień, On operators factoring through L p -space, Bull. Soc. Math. France Mém. 31-32 (1972), 215-225. 
  19. [19] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Ergeb. Math. Grenzgeb. 23, Springer, 1991. Zbl0748.60004
  20. [20] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in p -spaces and applications, Studia Math. 29 (1968), 275-326. Zbl0183.40501
  21. [21] J. Marcinkiewicz et A. Zygmund, Quelques inégalités pour les opérations linéaires, Fund. Math. 32 (1939), 113-121. Zbl65.0506.02
  22. [22] B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces L p , Astérisque 11 (1974). Zbl0278.46028
  23. [23] B. Maurey et G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), 45-90. Zbl0344.47014
  24. [24] R. E. A. C. Paley, On a remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-264. Zbl0005.24806
  25. [25] A. Pietsch, Absolutely p-summing operators in L r -spaces, Bull. Soc. Math. France Mém. 31-32 (1972), 285-315. 
  26. [26] A. Pietsch, Operator Ideals, North-Holland, 1980. 
  27. [27] P. Saphar, Applications p-décomposantes et p-absolument sommantes, Israel J. Math. 11 (1972), 164-179. Zbl0241.47016
  28. [28] H. Vogt, Komplexifizierung von Operatoren zwischen L p -Räumen, Diplomarbeit, Oldenburg, 1995. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.