Abstract view
$H^\infty$ Functional Calculus and MikhlinType Multiplier Conditions


Published:20081001
Printed: Oct 2008
José E. Galé
Pedro J. Miana
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 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
functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers

MSC Classifications: 
47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22 show english descriptions
Functional calculus Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} Banach algebras of differentiable or analytic functions, $H^p$spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30] Fractional derivatives and integrals Algebras of unbounded operators; partial algebras of operators Operators on Banach algebras Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
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.
