R-boundedness

  • Tuomas Hytönen
  • Jan van Neerven
  • Mark Veraar
  • Lutz Weis
Chapter
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics book series (MATHE3, volume 67)

Abstract

This chapter provides a detailed study of the notion of (Rademacher) R-boundedness, its Gaussian analogue of γ-boundedness, and some relatives – essential tools in deeper manipulations of random sums arising in their applications to various domains of analysis. We discuss both the general operator-theoretic mechanisms of creating R-bounded families, and concrete sources and applications of R-boundedness in classical analysis. One section is dedicated to the central role of R-boundedness in the theory of Fourier multipliers, and another one to the R-boundedness of integral means and the range of sufficiently smooth operator-valued functions. In the final section, we characterise the situations in which R-boundedness coincides with other types of boundedness.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Tuomas Hytönen
    • 1
  • Jan van Neerven
    • 2
  • Mark Veraar
    • 3
  • Lutz Weis
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  3. 3.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  4. 4.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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