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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
##### Keywords:
differential operator; symbol; integrated semigroup
Full Text:
##### References:
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