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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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