Boundary values of holomorphic semigroups

Abstract

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

Generalized Gaussian estimates and riesz means of Schrödinger groups

“…The GGE (3) for p o = 1 is equivalent to the GE (1) [5, Proposition 2.9]. The central part of this paper is to deduce from the GGE (3) the following generalization and slight improvement of the L p ->• L p -norm estimate (2) (Theorem 1.1 below), which improves a result of Davies [11]: [12], and the ||e~M|| p^p -estimate (4) is optimal also [3].…”

Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

“…Lastly, (f) follows from 19], Theorem 3.1, because g 1; 2 L 1 (I) is nonnegative and nonincreasing for 2 (0 1]. 2 Let us note that if p = 2 and X = H is a Hilbert space with inner product (: :), then L (I H) is again a Hilbert space with inner product Next we study the properties of the domains of the operators of fractional di erentiation D(L ), endowed with the norm kfk R p 0 (I X) := kL fk L p (I X) :…”

“…Then the following lemma ( 37], Lemma 5.2) shows that A is bounded. 2 Lemma 2.7 If (A) is a bounded subset of C and kR( A)k = O(1=j j) as j j ! 1 then A 2 B (X).…”

“…Since F(e i t ) is a continuous function on ( t) 2 0 2 ] I then there exists a constant M > 0 such that jF(e i t )j M, ( t) 2 i t )ju( t)j 2 d dt kuk2 L 2 (I H) : (5.67) So, the operator C(t) is strictly accretive in L 2 (I H), i.e. there exists > 0 s u c h that Re (C(:)u u) L 2 (I H) kuk 2 L 2 (I H) 8u 2 L 2 (I H): (5.68) On the Hilbert space H 1=2 consider the family of operators fA(t)g t2I , de ned by D(A(t)) := H 3=2 A(t) : = C(t)A: Again by the continuity of F, it follows A(:) 2 C(I B(H 3=2 H 1=2 )).…”

“…It remains to apply the following lemma and the proof of Theorem 2.21 is completed 2. Lemma 2.22 Let f( ) : 0 1) !…”