Hieber, Matthias; Räbiger, Frank A remark on the abstract Cauchy problem on spaces of Hölder continuous functions. (English) Zbl 0765.34042 Proc. Am. Math. Soc. 115, No. 2, 431-434 (1992). 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. Reviewer: H.O.Fattorini (Los Angeles) Cited in 2 Documents MSC: 34G10 Linear differential equations in abstract spaces 47D03 Groups and semigroups of linear operators Keywords:unbounded operator; semigroup generator; elliptic differential operator with constant coefficients; symbol PDFBibTeX XMLCite \textit{M. Hieber} and \textit{F. Räbiger}, Proc. Am. Math. Soc. 115, No. 2, 431--434 (1992; Zbl 0765.34042) Full Text: DOI