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

On the $$R$$ R -boundedness of stochastic convolution operators The $$R$$ 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 $$L^p$$ L p -regularity, $$2<p<\infty$$ 2 < p < ∞ , for certain classes of sectorial operators acting on spaces $$X=L^q(\mu )$$ X = L q ( μ ) , $$2\le q<\infty$$ 2 ≤ q < ∞ . This paper presents a systematic study of $$R$$ R -boundedness of such families. Our main result generalises the afore-mentioned $$R$$ R -boundedness result to a larger class of Banach lattices $$X$$ X and relates it to the $$\ell ^{1}$$ ℓ 1 -boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the $$\ell ^{1}$$ ℓ 1 -boundedness of these operators and the boundedness of the $$X$$ X -valued maximal function. This analysis leads, quite surprisingly, to an example showing that $$R$$ R -boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type $$2$$ 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 19 (2) – Jul 25, 2014
30 pages

/lp/springer_journal/on-the-r-r-boundedness-of-stochastic-convolution-operators-yqksjCRwAr
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0302-8
Publisher site
See Article on Publisher Site

Abstract

The $$R$$ 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 $$L^p$$ L p -regularity, $$2<p<\infty$$ 2 < p < ∞ , for certain classes of sectorial operators acting on spaces $$X=L^q(\mu )$$ X = L q ( μ ) , $$2\le q<\infty$$ 2 ≤ q < ∞ . This paper presents a systematic study of $$R$$ R -boundedness of such families. Our main result generalises the afore-mentioned $$R$$ R -boundedness result to a larger class of Banach lattices $$X$$ X and relates it to the $$\ell ^{1}$$ ℓ 1 -boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the $$\ell ^{1}$$ ℓ 1 -boundedness of these operators and the boundedness of the $$X$$ X -valued maximal function. This analysis leads, quite surprisingly, to an example showing that $$R$$ R -boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type $$2$$ 2 .

Journal

PositivitySpringer Journals

Published: Jul 25, 2014

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