On the $$R$$ R -boundedness of stochastic convolution operators

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 \(2\).