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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).

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