This chapter provides a detailed study of the notion of (Rademacher) R-boundedness, its Gaussian analogue of γ-boundedness, and some relatives – essential tools in deeper manipulations of random sums arising in their applications to various domains of analysis. We discuss both the general operator-theoretic mechanisms of creating R-bounded families, and concrete sources and applications of R-boundedness in classical analysis. One section is dedicated to the central role of R-boundedness in the theory of Fourier multipliers, and another one to the R-boundedness of integral means and the range of sufficiently smooth operator-valued functions. In the final section, we characterise the situations in which R-boundedness coincides with other types of boundedness.
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