Abstract
Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case (X,D(A))_{theta ,infty }. We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on mathbb {R}^d.
Highlights
Before I met Matthias Hieber in person, I already knew him through his work on integrated semigroups
In this paper we shall see that there is a variety of such topologies in real interpolation spaces between a Banach space X and the domain of the generator of a bi-continuous semigroup, called abstract Besov spaces
If X is not reflexive and (T (t))t≥0 = (e− Al ψ (t A))t≥0 is a C0-semigroup in X and D ⊆ X is dense in X the dual group (T (t) )t≥0 is bi-continuous for the topology τ induced by the seminorms px (x ) := | x, x | with x ∈ D, since τ restricted to the closed unit ball BX of X coincides with the restriction of τw∗ to BX . 3
Summary
Before I met Matthias Hieber in person (at Karlsruhe in October 1996), I already knew him through his work on integrated semigroups. In this paper we shall see that there is a variety of such topologies in real interpolation spaces between a Banach space X and the domain of the generator of a bi-continuous semigroup, called abstract Besov spaces. Our results add a third one to the list of properties with respect to which operators behave better in abstract Besov spaces associated with it than in the general case. Since all this may sound very abstract, a word on the motivation for this paper seems to be in order.
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