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<∞  , for certain classes of sectorial operators acting on spaces X=L q (μ)  , 2≤q<∞  . 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 ℓ 1   -boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the ℓ 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  .
Original languageEnglish
Pages (from-to)355-384
Number of pages30
JournalPositivity: an international journal devoted to the theory and applications of positivity in analysis
Volume19
Issue number2
DOIs
Publication statusPublished - 2015

Keywords

  • Stochastic convolutions
  • Maximal regularity
  • R-boundedness
  • Hardy–Littlewood maximal function
  • UMD Banach function spaces

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