On the 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<∞ 

, 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 

.