Fractional powers of momentum of a spectral distribution

Jazar, Mustapha

doi:10.1090/s0002-9939-1995-1242090-0

Cited by 5 publications

(1 citation statement)

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“…In the case where U is a bounded group, Theorem 5.1 is due to Jazar [Ja1], [Ja2], who uses spectral calculus for the proof. A slightly more general version of Theorem 5.1 is given in [El2] with a very different and more complicated proof.…”

Section: It Follows From Proposition 52 Thatmentioning

confidence: 99%

Boundary values of holomorphic semigroups

Arendt

Mennaoui

Hieber

1997

Proc. Amer. Math. Soc.

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Abstract. The concept of boundary values of holomorphic semigroups is used to give a new proof of a result due to Hörmander, saying that the operator i∆ generates a C 0 -semigroup on L p (R N ) if and only if p = 2. Using a recent result on Laplace transforms by Prüss one obtains by this theory also a new proof of the classical characterization theorem of holomorphic semigroups. IntroductionThe starting point of the present paper is a classical result of Hörmander [Hö] saying that the operator i∆ generates a C 0 -semigroup on L p (R N ) if and only if p = 2. Hörmander's proof is based on euclidean harmonic analysis. In fact, he shows that for each t > 0, the function ξ → eThere is a completely different way to look at his result. Indeed, observe that the Laplacian ∆ generates a holomorphic C 0 -semigroup of angle π/2 on L p (R N ), 1 ≤ p < ∞. We use the following simple result on the boundary of a holomorphic semigroup (see Section 1 for the proof) to prove Hörmander's result in an elementary way. Proposition 0.1. Let A be the generator of a holomorphic C 0 -semigroup T on some Banach space E of angle π/2. Then iA generates a C 0 -semigroup on E if and only if T is bounded onIn fact, the explicit representation of the semigroup (e z∆ ) Re z>0 as a convolution operator with the Gaussian kernel allows us to show in Section 2 that e zδ is unbounded on Ω if p = 2. In view of Proposition 0.1 we obtain in this way a proof of Hörmander's result.Another interesting example of a holomorphic semigroup of angle π/2 is the Riemann-Liouville semigroup J (describing fractional integration) on L p (0, 1), 1 ≤ p < ∞. It does have a boundary value for 1 < p < ∞. We show this by identifying J(z) with B −z (Re z > 0), where −B generates the translation semigroup on L p (0, 1), and by applying the transference principle due to Coifman and Weiss [C-W].In Sections 4 and 5 we consider the inverse problem: Which semigroups are obtained as boundary values of holomorphic semigroups? Given an operator A

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Section: It Follows From Proposition 52 Thatmentioning

confidence: 99%