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Classes of distribution semigroups. (English) Zbl 1156.47039
The authors discuss regularity properties of distribution semigroups, the main interest being in their behavior at the origin. Consider $$E$$ a Banach space and $$L(E)$$ the space of bounded linear operators from $$E$$ to $$E$$. They prove that if $${\mathcal G} \in {\mathcal D}_+^{\prime}(L(E))$$, then $${\mathcal G}$$ is a distribution semigroup if and only if it is a quasi-distribution semigroup. In particular, a distribution semigroup is a representable distribution semigroup. If $$A$$ is a closed linear operator in $$E$$, then the “density index” is $$n(A):=\inf\{k\in {\mathbb N}_0\mid \forall m \geq k: {\overline {D(A^m)}}={\overline {D(A^k)}}\}$$.
If $${\mathcal G}$$ is a strong distribution semigroup, then it is a distribution with stationary dense infinitesimal generator $$A$$ satisfying $$n(A)\leq 1$$. The authors characterize strong distribution semigroups via the value $$0$$ at the origin in the sense of Łojasiewicz for their primitive. Some generalizations of the classes of smooth distribution semigroups are presented.
Another result characterizes distribution semigroups on $${\mathcal F}_0$$ in terms of integrated semigroups. In this context, $${\mathcal F}_0$$ is the completion of $${\mathcal D}((0,\infty))$$ under a certain sequence of seminorms. The authors give applications to Schrödinger equations in $$C_b({\mathbb R}^n)$$, $$L^{\infty}({\mathbb R}^n)$$ and $$\text{BMO}({\mathbb R}^n$$) (i.e., the space of functions of bounded mean oscillation modulo constants).

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47D62 Integrated semigroups
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