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# $H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions

Published:2008-10-01
Printed: Oct 2008
• José E. Galé
• Pedro J. Miana
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## 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 $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-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.
 Keywords: functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers
 MSC Classifications: 47A60 - Functional calculus 47D03 - Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 46J15 - Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30] 26A33 - Fractional derivatives and integrals 47L60 - Algebras of unbounded operators; partial algebras of operators 47B48 - Operators on Banach algebras 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc.

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