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H∞ Functional Calculus and Mikhlin-Type Multiplier Conditions
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled)
${{H}^{\infty }}$ calculus for
$T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra
$\Lambda _{\infty ,1}^{\alpha }({{\mathbb{R}}^{+}})$. Such an algebra includes functions defined byMikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for
$T$. In this paper, we use fractional derivation to analyse in detail the relationship between
$\Lambda _{\infty ,1}^{\alpha }$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.
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- Copyright © Canadian Mathematical Society 2008
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