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Integrated semigroups and differential operators on \(L^ p\) spaces. (English) Zbl 0724.34067
In this note we prove that every operator A on \(L^ p({\mathbb{R}}^ n)\) whose symbol P is a purely imaginary elliptic polynomial on \({\mathbb{R}}^ n\) or is of the form \(P(\xi)=i| \xi |^ m\) generates an \(\alpha\)-times integrated semigroup on \(L^ p({\mathbb{R}}^ n)\) \((1<p<\infty)\) whenever \(\alpha\geq n| 1/2-1/p|\) and an \(\alpha\)- times integrated semigroup on \(L^ 1({\mathbb{R}}^ n)\) or \(L^{\infty}({\mathbb{R}}^ n)\) whenever \(\alpha >n/2\). These constants are shown to be optimal for all powers of the Laplacian besides the square root. In that case A generates an \(\alpha\)-times integrated semigroup \((\alpha >0)\) on \(L^ p({\mathbb{R}}^ n)\) \((1<p<\infty)\) if and only if \(\alpha >(n-1)| 1/2-1/p|\) and if and only if \(\alpha >(n-1)/2\) in the cases \(p=1\) or \(p=\infty\).
Reviewer: M.Hieber

MSC:
34G10 Linear differential equations in abstract spaces
42B15 Multipliers for harmonic analysis in several variables
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