The R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}-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 Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}-regularity, 2<p<∞\documentclass[12pt]{minimal}
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\begin{document}$$2<p<\infty $$\end{document}, for certain classes of sectorial operators acting on spaces X=Lq(μ)\documentclass[12pt]{minimal}
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\begin{document}$$X=L^q(\mu )$$\end{document}, 2≤q<∞\documentclass[12pt]{minimal}
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\begin{document}$$2\le q<\infty $$\end{document}. This paper presents a systematic study of R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}-boundedness of such families. Our main result generalises the afore-mentioned R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}-boundedness result to a larger class of Banach lattices X\documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document} and relates it to the ℓ1\documentclass[12pt]{minimal}
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\begin{document}$$\ell ^{1}$$\end{document}-boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the ℓ1\documentclass[12pt]{minimal}
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\begin{document}$$\ell ^{1}$$\end{document}-boundedness of these operators and the boundedness of the X\documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document}-valued maximal function. This analysis leads, quite surprisingly, to an example showing that R\documentclass[12pt]{minimal}
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\begin{document}$$R$$\end{document}-boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type 2\documentclass[12pt]{minimal}
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\begin{document}$$2$$\end{document}.
