On the -boundedness of stochastic convolution operators

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 -regularity, , for certain classes of sectorial operators acting on spaces , . This paper presents a systematic study of -boundedness of such families. Our main result generalises the afore-mentioned -boundedness result to a larger class of Banach lattices and relates it to the -boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the -boundedness of these operators and the boundedness of the -valued maximal function. This analysis leads, quite surprisingly, to an example showing that -boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type 2.