Square functions and radonifying operators
This chapter presents the theory of radonifying operators and explains their use as generalised square functions, which allows the extension of key ideas from classical Littlewood--Paley theory in L p spaces to more general Banach spaces. The space of radonifying operators is shown to display several function-space-like properties, including Hölder-type duality, convergence and Fubini theorems. Pointwise multipliers of this space are characterised in terms of γ-boundedness. We also show that the generalised square function space admits canonical extensions of linear operators bounded on the scalar-valued L 2 space, avoiding the intrinsic difficulties of the L p extension problem discussed in Volume I. Assuming type, cotype or related properties, we even obtain R-bounded extensions of bounded families of Hilbert space operators. In the final section we prove several embedding theorems of classical function spaces into the space of radonifying operators.
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