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A remark on the abstract Cauchy problem on spaces of Hölder continuous functions. (English) Zbl 0765.34042
Let $$C^ \alpha(\mathbb{R}^ n)$$ $$(0<\alpha<1)$$ be the space of $$\alpha$$- Hölder continuous functions in $$\mathbb{R}^ n$$ endowed with its usual norm. The authors show that no unbounded operator $$A$$ in $$C^ \alpha(\mathbb{R}^ n)$$ can be a semigroup generator. Then they consider the case $$A$$= elliptic differential operator with constant coefficients, and show that if the symbol $$p(\xi)$$ has real part bounded above, $$A$$ generates a $$\beta$$-times integrated semigroup if $$\beta>n/2+1$$. The index $$\beta$$ can be considerable improved in particular cases; for instance, for the Laplacian, we may take $$\beta>0$$ arbitrary.

##### MSC:
 34G10 Linear differential equations in abstract spaces 47D03 Groups and semigroups of linear operators
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