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

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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 \(2\).

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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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Authors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA , Delft, The Netherlands

    Jan van Neerven & Mark Veraar

  2. Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76128,  Karlsruhe, Germany

    Lutz Weis

Authors
  1. Jan van Neerven
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  2. Mark Veraar
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  3. Lutz Weis
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Correspondence to Mark Veraar.

Additional information

J. van Neerven is supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).

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van Neerven, J., Veraar, M. & Weis, L. On the \(R\)-boundedness of stochastic convolution operators. Positivity 19, 355–384 (2015). https://doi.org/10.1007/s11117-014-0302-8

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