Banach Journal of Mathematical Analysis

Norm estimates for random polynomials on the scale of Orlicz spaces

Andreas Defant and Mieczysław Mastyło

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Abstract

We prove an upper bound for the supremum norm of homogeneous Bernoulli polynomials on the unit ball of finite-dimensional complex Banach spaces. This result is inspired by the famous Kahane–Salem–Zygmund inequality and its recent extensions; in contrast to the known results, our estimates are on the scale of Orlicz spaces instead of p-spaces. Applications are given to multidimensional Bohr radii for holomorphic functions in several complex variables, and to the study of unconditionality of spaces of homogenous polynomials in Banach spaces.

Article information

Source
Banach J. Math. Anal., Volume 11, Number 2 (2017), 335-347.

Dates
Received: 2 February 2016
Accepted: 30 May 2016
First available in Project Euclid: 28 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1485572421

Digital Object Identifier
doi:10.1215/17358787-0000006X

Mathematical Reviews number (MathSciNet)
MR3603344

Zentralblatt MATH identifier
06694357

Subjects
Primary: 46B70: Interpolation between normed linear spaces [See also 46M35]
Secondary: 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

Keywords
homogeneous polynomials interpolation spaces Orlicz spaces

Citation

Defant, Andreas; Mastyło, Mieczysław. Norm estimates for random polynomials on the scale of Orlicz spaces. Banach J. Math. Anal. 11 (2017), no. 2, 335--347. doi:10.1215/17358787-0000006X. https://projecteuclid.org/euclid.bjma/1485572421


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References

  • [1] F. Bayart, Maximum modulus of random polynomials, Q. J. Math. 63 (2012), 21–39.
  • [2] H. Boas, Majorant series, J. Korean Math. Soc. 37 (2000), 321–337.
  • [3] A. Defant, D. García, and M. Maestre, Bohr’s power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197.
  • [4] A. Defant, D. García, and M. Maestre, Maximum moduli of unimodular polynomials in several variables, J. Korean Math. Soc. 41 (2004), 209–230.
  • [5] L. A. Harris, “Bounds on the derivatives of holomorphic functions of vectors” in Analyse Fonctionnelle et Applications (Rio de Janeiro, 1972), Actualités Aci. Indust. 1367, Hermann, Paris, 1975, 145–163.
  • [6] J. P. Kahane, Some Random Series of Functions, 2nd ed., Cambridge Stud. Adv. Math. 5, Cambridge Univ. Press, Cambridge, 1985.
  • [7] M. Krasnoselskii and Y. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
  • [8] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II, Ergeb. Math. Grenzgeb. 97, Springer, Berlin, 1979.
  • [9] V. I. Ovchinnikov, Interpolation theorems resulting from Grothendieck’s inequality (in Russian), Funktsional Anal. i Prilozhen. 10 (1976), 45–54; English translation in Funkctional Anal. Appl. 10 (1976), 287–294.
  • [10] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Pure Appl. Math. 146, Marcel Dekker, New York, 1991.
  • [11] V. A. Rodin and E. M. Semyonov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), no. 3, 207–222.
  • [12] A. Zygmund, Trigonometric Series, I, II, 2nd ed., Cambridge Univ. Press, New York, 1959.