We survey some recent results on functional calculus for generators of holomorphic semigroups, which have been obtained using versions of fractional derivation of Riemann-Liouville or Weyl type. Such a calculus allows us to give tight estimates even in concrete L¹ examples.

Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^p\left(I\right)$, 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$ and their preduals.

Consider I = [0,1] as a compact topological semigroup with max multiplication and usual topology, and let $C\left(I\right),L^p\left(I\right),1\le p\le \infty$, be the associated algebras. The aim of this paper is to study the spaces $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$, r > p, and their preduals.

We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^1\left(AN\right),$ but also for $L^p\left(AN\right)$, where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.

We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in [CGHM] and [A].

Soit $\mathrm{Re}A$, l’espace de Banach des fonctions continues sur $T$ qui sont parties réelles de fonctions de l’algèbre du disque $A\left(D\right)$. On étudie les ensembles de $T$ de synthèse pour $\mathrm{Re}A$ et l’algèbre des multiplicateurs de $\mathrm{Re}A$. On en déduit des théorèmes d’approximation dans $A\left(D\right)$ par des produits de Blaschke.