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Positivity

, Volume 19, Issue 2, pp 355–384 | Cite as

On the \(R\)-boundedness of stochastic convolution operators

  • Jan van Neerven
  • Mark VeraarEmail author
  • Lutz Weis
Article

Abstract

The \(R\)-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal \(L^p\)-regularity, \(2<p<\infty \), for certain classes of sectorial operators acting on spaces \(X=L^q(\mu )\), \(2\le q<\infty \). This paper presents a systematic study of \(R\)-boundedness of such families. Our main result generalises the afore-mentioned \(R\)-boundedness result to a larger class of Banach lattices \(X\) and relates it to the \(\ell ^{1}\)-boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the \(\ell ^{1}\)-boundedness of these operators and the boundedness of the \(X\)-valued maximal function. This analysis leads, quite surprisingly, to an example showing that \(R\)-boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type \(2\).

Keywords

Stochastic convolutions Maximal regularity \(R\)-boundedness Hardy–Littlewood maximal function UMD Banach function spaces 

Mathematics Subject Classification (2000)

Primary 60H15 Secondary 42B25 46B09 46E30 60H05 

Notes

Acknowledgments

We thank Tuomas Hytönen for his kind permission to present his short proof of Proposition 4.5 here. We thank the anonymous referee for carefully reading and providing helpful comments.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of MathematicsKarlsruhe Institute of Technology (KIT) KarlsruheGermany

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