This is an announcement for the paper "On the R-boundedness of stochastic convolution operators" by Jan van Neerven, Mark Veraar, and Lutz Weis.
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 $2$.
Archive classification: math.FA math.PR
Mathematics Subject Classification: Primary: 60H15, Secondary: 42B25, 46B09, 46E30, 60H05
Submitted from: m.c.veraar@tudelft.nl
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1404.3353
or