On the R -boundedness of stochastic convolution operators

Journal article (2015)

Authors

J.M.A.M. van Neerven Analysis - EEMCS
M.C. Veraar Analysis - EEMCS
L Weis Karlsruhe Institut für Technologie
Research Group
Analysis (EEMCS) (TU Delft)
R-boundedness Maximal regularity Hardy–Littlewood maximal function Stochastic convolutions UMD Banach function spaces
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Published Date

2015

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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
, 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 

.