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Final Activity Report Summary - SEMIGROUPS CALCULUS (Functional Calculus for operators and semigroups under resolvent or boundedness conditions)

We studied the functional calculus for bounded holomorphic semigroups (i.e. for sectorial operators) and their discrete analogue, Tadmor-Ritt operators. This subject is closely linked with the question of maximal regularity of PDE's associated to the semigroup. It is well-known that, in general, no $H^\infty$ calculus exists, $H^\infty$ being the algebra of all bounded holomorphic functions on the right half plane. We proved that a $B^0_{\infty 1}$ calculus exists, $B^0_{\infty 1}$ being the non-homogeneous Besov algebra, that is, the subalgebra of $H^\infty$ consisting of functions $f$ satisfying $\int_0^\infty \max_{y \in \mathbb{R}} |f'(x+iy)| dx < \infty$, endowed with the corresponding norm. The same holds for Tadmor-Ritt operators with the usual Besov algebra $B^0_{\infty 1}$ on the unit disc.

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