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 .
|Number of pages||30|
|Journal||Positivity: an international journal devoted to the theory and applications of positivity in analysis|
|Publication status||Published - 2015|
- Stochastic convolutions
- Maximal regularity
- Hardy–Littlewood maximal function
- UMD Banach function spaces