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Square Function Estimates and Functional Calculi
In this paper the notion of an abstract square function (estimate) is
introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is
a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators.
By the seminal work of Kalton and Weis, this definition is a coherent
generalisation of the classical notion of square function appearing in the
theory of singular integrals. Given an abstract functional calculus (E, F, Phi)
on a Banach space X, where F (O) is an algebra of scalar-valued functions on a
set O, we define a square function Phi_gamma(f) for certain H-valued functions
f on O. The assignment f to Phi_gamma(f) then becomes a vectorial functional
calculus, and a "square function estimate" for f simply means the boundedness
of Phi_gamma(f). In this view, all results linking square function estimates
with the boundedness of a certain (usually the H-infinity) functional calculus
simply assert that certain square function estimates imply other square
function estimates. In the present paper several results of this type are
proved in an abstract setting, based on the principles of subordination,
integral representation, and a new boundedness concept for subsets of Hilbert
spaces, the so-called ell-1 -frame-boundedness. These abstract results are then
applied to the H-infinity calculus for sectorial and strip type operators. For
example, it is proved that any strip type operator with bounded scalar
H-infinity calculus on a strip over a Banach space with finite cotype has a
bounded vectorial H-infinity calculus on every larger strip.Comment: 49
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