1. CJM 2008 (vol 60 pp. 1010)
 Galé, José E.; Miana, Pedro J.

$H^\infty$ Functional Calculus and MikhlinType Multiplier Conditions
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 halfline, 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 Mikhlintype 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 Mikhlintype. As a result, we obtain a new version of the quoted
equivalence.
Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers Categories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22 
