an:00040167
Zbl 0765.34042
Hieber, Matthias; R??biger, Frank
A remark on the abstract Cauchy problem on spaces of H??lder continuous functions
EN
Proc. Am. Math. Soc. 115, No. 2, 431-434 (1992).
00007264
1992
j
34G10 47D03
unbounded operator; semigroup generator; elliptic differential operator with constant coefficients; symbol
Let \(C^ \alpha(\mathbb{R}^ n)\) \((0<\alpha<1)\) be the space of \(\alpha\)- H??lder 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.
H.O.Fattorini (Los Angeles)