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THE COMPLEX INVERSION FORMULA REVISITED MARKUS HAASE (June 25, 2007) Abstract We give a simplified proof of the complex inversion formula for semigroups and — more generally — solution families for scalar-type Volterra equations, including the stronger versions on UMD spaces. Our approach is based on (elementary) Fourier analysis. Keywords and phrases: C 0 -semigroup, integrated semigroup, UMD space, complex inversion, Laplace transform,
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  THE COMPLEX INVERSION FORMULA REVISITED MARKUS HAASE (June 25, 2007) Abstract We give a simplified proof of the complex inversion formula for semigroups and — more generally— solution families for scalar-type Volterra equations, including the stronger versions on UMDspaces. Our approach is based on (elementary) Fourier analysis. Keywords and phrases : C  0 -semigroup, integrated semigroup, UMD space, complex inversion,Laplace transform, Volterra equation. 1. Introduction In this paper we are concerned with the following question: Let X,Y  beBanach spaces and let S  : [0 , ∞ ) −→ L ( X,Y  ) be a strongly continuous mappingof finite exponential type ω 0 ( S  ). In what sense and under what conditions doesthe complex inversion inversion formulalim N  →∞ 12 πi   ω + iN ω − iN  e tz ( L S  )( z )d z = S  ( t ) ( t > 0) (1.1)hold true? (Here ω > ω 0 ( S  ) is fixed and L S  denotes the Laplace transform of  S  ).Actually we are interested in the case that S  is a solution family to ascalar-type Volterra equation (see Section 4 below), in particular that S  isa C  0 -semigroup. However, as in [1, Theorem 2.3.4] we do not confine to theseapplications but start very generally.Theorem 2.3.4 from [1] states that (1.1) holds in an “integrated form”. Fromthis one can then derive the standard result on semigroups (strong convergenceon the domain of the generator). In the paper [3] Driouich and El-Mennaoui showed that in case that X  has the UMD property the convergence is strongon all of  X  . This was subsequently generalised from semigroups to solutionfamilies for scalar-type Volterra equations by Cioranescu and Lizama in [2].The aim of the present paper is to present new and much shorter proofs of these results, eventually even generalising them. Our approach uses some ele-mentary Fourier analysis and has the advantage that the recent “UMD-results” c  XXXX Australian Mathematical Society 0263-6115/XX $A2 . 00 + 0 . 00 1  become at least as simple as the classical ones, if not simpler. The resultsobtained are also more specific than the existing ones with respect to whathappens with the approximation for small times (compare for example Theo-rem 3.5 below with [4, Corollary III.5.15]).All our results on the complex inversion formula remain true when we let thelower and the upper bound of the integral in (1.1) tend to infinity independently.One has to replace the Dirichlet kernel in our discussion by a somewhat morecomplicated expression, but the proofs are essentially the same. Preliminary remarks and definitions Here and in the following, X,Y,Z  always denote complex Banach spaces. Thesymbol 1 is used to denote the characteristic function of the positive real axis,that is 1 = χ [0 , ∞ ) . So 1  = δ 0 in the distributional sense, where δ 0 is the Diracmeasure at 0. We write simply t to denote the real coordinate ( t −→ t ). Allfunctions that live on [0 , ∞ ) are tacitly extended to R by 0 on ( −∞ , 0). For amapping S  : [0 , ∞ ) −→ L ( X,Y  ) and ω ∈ R we define its exponential shift  S  ω by S  ω ( t ) := e − ωt S  ( t ) ( t ≥ 0) . The exponential type of  S  is ω 0 ( S  ) := inf  { ω ∈ R | ∃ M  ≥ 0 :  S  ( t )  ≤ M  e ωt ( t ≥ 0) } . If is S  strongly measurable and of finite exponential type, we denote by( L S  )( z ) := strong −   ∞ 0 e − zt S  ( t )d t (Re z > ω 0 ( S  ))its Laplace transform  . If  S  : [0 , ∞ ) −→ L ( X,Y  ) and T  : [0 , ∞ ) −→ L ( Y,Z  ) areboth strongly measurable and of finite exponential type, then the convolution  S  ∗ T  : [0 , ∞ ) −→ L ( X,Y  ) given by( T  ∗ S  )( t ) x :=   ∞ 0 T  ( t − s ) S  ( s ) x d s ( x ∈ X  )is well-defined, strongly continuous and of finite exponential type; furthermore,one has( T  ∗ S  ) ω = T  ω ∗ S  ω and L ( T  ∗ S  ) = ( L T  )( L S  ) . (1.2)Beside this type of convolution we will encounter (in Section 4) µ ∗ S  , where S  : [0 , ∞ ) −→ L ( X,Y  ) is strongly continuous and µ is a locally finite complexBorel measure on [0 , ∞ ). The convolution µ ∗ S  : [0 , ∞ ) −→ L ( X,Y  ) is thengiven as( µ ∗ S  )( t ) x :=   t 0 S  ( t − s ) xµ (d s ) ( x ∈ X  )and is again strongly continuous. With the obvious definition of  µ ω we have( µ ∗ S  ) ω = µ ω ∗ S  ω ; if  µ ω happens to be a bounded measure, L µ is defined inthe obvious way, and one has L ( µ ∗ S  ) = ( L µ )( L S  ).A third situation envolves functions on the whole real line, and is describedin the following. A strongly measurable mapping S  : R −→ L ( X,Y  ) is said2  to be strongly  - L 2 , if  S  ( · ) x ∈ L 2 ( R ; Y  ) for every x ∈ X  . By the closed graphtheorem one then has  S   2 := sup x ∈ X,  x ≤ 1  S  ( · ) x  L 2 ( R ; Y  ) < ∞ . The mapping S  is said to be uniformly- L 2 if there exists a function g ∈ L 2 ( R )such that g ≥ 0 and  S  ( t ) x  Y  ≤ g ( t )  x  X ( t ∈ R , x ∈ X  ) . The function g is said to be a scalar majorant of  S  . We will have occasion touse the following form of Young’s inequality. Lemma 1.1. (Young’s inequality) Let  X,Y  be Banach spaces, let  S  : R −→ L ( X,Y  ) be strongly- L 2 , and let  T  : R −→ L ( Y,Z  ) be uniformly- L 2 with scalar majorant  g . Then the convolution  T  ∗ S  defined by  ( T  ∗ S  )( t ) x :=   R T  ( t − s ) S  ( s ) x d s ( x ∈ X, t ∈ R ) exists and satisfies ( T  ∗ S  ) ∈ C 0 ( R ; L s ( X,Z  )) with  sup t ≥ 0  ( T  ∗ S  )( t )  L ( X,Y  ) ≤  g  2  S   2 . (1.3)One may choose Y  = Z  and T  ( s ) = f  ( s ) I  , g ( t ) = | f  ( t ) | in the lemma, sothe estimate (1.3) becomessup t ≥ 0  ( f  ∗ S  )( t )  L ( X,Y  ) ≤  f   2  S   2 , and this shows that with S  fixed the mapping( f  −→ f  ∗ S  ) : L 2 ( R ) −→ C 0 ( R ; L ( X,Y  ))is continuous. On the other hand, if we choose X  = C , then (1.3) shows thatwith fixed T  the mapping( f  −→ T  ∗ f  ) : L 2 ( R ; Y  ) −→ C 0 ( R ; Z  )is continuous.For N > 0 we denote by D N  the Dirichlet kernel  , that is D N  ( t ) :=sin( Nt ) πt ( t ∈ R ) . Then, as is well known (or by a short computation), D N  ∗ f  =12 π   N  − N  e ist  f  ( s )d s, (1.4)where f  is integrable and  f  = F  f  denotes its Fourier transform. (The function f  may be vector- or operator-valued, of course.) The following is an easyconsequence of Plancherel’s theorem. Lemma 1.2. Let  f  ∈ L 2 ( R ) . Then  D N  ∗ f  → f  in  L 2 ( R ) as N  → ∞ . 3  2. General Laplace transforms Let X,Y  be Banach spaces and let S  : [0 , ∞ ) −→ L ( X,Y  ) be a stronglycontinuous mapping of finite exponential type ω 0 ( S  ). Note that K  N  ( t ) :=12 πi   ω + iN ω − iN  e tz ( L S  )( z )d z = e ωt 12 π   N  − N  e ist ( L S  )( ω + is )d s = e ωt ( D N  ∗ S  ω )( t ) (2.1)whence ( K  N  ) ω = D N  ∗ S  ω . If we replace S  by a ∗ S  with a scalar function a wearrive at our first result. Proposition 2.1. Let  X,Y  be Banach spaces, let  S  : [0 , ∞ ) −→ L ( X,Y  ) be strongly continuous, and let  a ∈ L 1 loc [0 , ∞ ) be a scalar function, both  a and  S  of finite exponential type. Then for every  ω > ω 0 ( S  ) ,ω 0 ( a ) one has lim N  →∞ 12 πi   ω + iN ω − iN  e tz L ( a ∗ S  )( z )d z = a ∗ S  in  L ( X,Y  ) , uniformly in  t from compact subsets of  [0 , ∞ ) . Proof. Replace S  by a ∗ S  in (2.1) to obtaine − ωt K  N  ( t ) = D N  ∗ ( a ∗ S  ) ω = D N  ∗ a ω ∗ S  ω by (1.2). Now D N  ∗ a ω → a ω in L 2 ( R ) and hence, by Young’s inequality, D N  ∗ a ω ∗ S  ω → a ω ∗ S  ω in L ( X  ), uniformly in t ≥ 0. Multiplying everythingby e ωt concludes the proof.Proposition 2.1 does not quite cover [1, Theorem 2.3.4]; however, it willsuffice for the applications we have in mind, and it is certainly more generalthan [2, Lemma 5], where the authors need a ∈ C 1 and assert only strongconvergence and uniformity only in t from compact subsets of (0 , ∞ ).We would like to point out that we do not claim Proposition 2.1 to be new,although it might be (as we do not know of a reference). Our emphasis is on theidea of the proof, which can be put as follows. The complex inversion formulais nothing else but the convergence of the partial inverse Fourier transforms.In a first step one establishes L 2 -convergence; then a convolution with another L 2 -term yields uniform convergence to something which — with some luck —is just a weighted form of what one is interested in.This idea will in the following be applied to the case of the complex inversionformula in its bare (that is, non-integral) form. The following observation willalso be helpful. Lemma 2.2. Let  X,Y  be a Banach space, let  S  : [0 , ∞ ) −→ L ( X,Y  ) bestrongly continuous of finite exponential type, and let  ω > ω 0 ( S  ) . Then  t ( D N  ∗ S  ω ) ∼ ( D N  ∗ [ tS  ( t )] ω )4
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