On Laplace Carleson embedding theorems

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1 Available online at Journal of Functional Analysis 264 (23) On Laplace Carleson embedding theorems Birgit Jacob a, Jonathan R. Partington b, Sandra Pott c, a Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstraße 2, 4297 Wuppertal, Germany b School of Mathematics, University of Leeds, Leeds LS2 9JT, UK c Centre for Mathematics, Faculty of Science, Lund University, Sölvegatan 8, 22 Lund, Sweden Received 23 December 2; accepted 3 November 22 Available online 4 December 22 Communicated by K. Ball Abstract This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. The little Hankel operators on these Bergman spaces are also considered. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases. 22 Elsevier Inc. All rights reserved. Keywords: Hardy space; Weighted Bergman space; Laplace transform; Carleson measure. Introduction and notation This paper brings together two ideas which are fundamental to the study of Banach spaces of analytic functions. First, there is the classical Paley Wiener theorem (see, for example [9]). This asserts that the Laplace transform (respectively, the Fourier transform) provides an isometric isomorphism between the space L 2 (, ) and the Hardy space H 2 of the right-hand half-plane (respectively, the upper half-plane). There are many generalizations of this result * Corresponding author. addresses: jacob@math.uni-wuppertal.de (B. Jacob), J.R.Partington@leeds.ac.uk (J.R. Partington), Sandra.Pott@math.lu.se (S. Pott) /$ see front matter 22 Elsevier Inc. All rights reserved.

2 784 B. Jacob et al. / Journal of Functional Analysis 264 (23) given, as mentioned below; for example the weighted space L 2 (, ; dt/t) corresponds to the Bergman space A 2 ( ). The strongest known generalization of this result is to the class of socalled Zen spaces (in fact we provide a slightly stronger generalization below), determined by an appropriate measure ν on the half-plane. The other theme that features in this paper is the notion of a Carleson embedding. The classical Carleson theorem is to do with finding a simple condition for the boundedness of the canonical injection H 2 ( ) L 2 (μ), where μ is a Borel measure on, and has many applications in function theory and harmonic analysis (again, [9] is a good reference for this). The embedding theorem has also been generalized to the Bergman space. Bringing these ideas together, we characterize the boundedness of Carleson embeddings for the very general class of Zen spaces (subject to a necessary technical condition on the measure ν), prove new results about the little Hankel operators on these spaces, and discuss in addition those embeddings induced by the Laplace transform. Ultimately, this allows us to give precise results even in the case of L p for p 2, when there is no exact generalization of the Paley Wiener theorem. Our results have applications in terms of interpolation in certain spaces of holomorphic functions and also admissibility and controllability in diagonal semigroups, as outlined in Section 4. (Full details will be presented elsewhere [4].) So, let w denote a weight function on the imaginary axis ir,letμ be a positive regular Borel measure on the right half-plane and let p,q. Embeddings of the form L p v (ir) Lq (,μ), () where a locally integrable function f in the weighted L p space L p v (ir) on the imaginary axis ir is mapped to its Poisson extension on the right half-plane, are known as Carleson embeddings, and have been much studied in the literature. In linear control, another, related class of embeddings plays an important role, namely embeddings of the form H p β,w (, ) Lq (,μ), f Lf = e t f(t)dt, given by the Laplace transform L. Here, H p β,w denotes the Sobolev space of index β and weight w: the case β = corresponds to a weighted L p space. We shall refer to such embeddings as Laplace Carleson embeddings. In the easiest case, β =, w and p = q = 2, the Laplace transform maps H, 2 (R +) = L 2 (R + ) isometrically up to a constant to H 2 ( ), which is a closed subspace of L 2 (ir), and we only have to deal with the unweighted classical Carleson embedding theorem for p = 2. This can be found in many places, for example, [7,9]. Theorem. (Carleson embedding theorem). Let μ be a positive regular Borel measure on the right half-plane. Then the following are equivalent:. The natural embedding H p ( ) L p (,μ) is bounded for some (or equivalently, for all) p<.

3 B. Jacob et al. / Journal of Functional Analysis 264 (23) There exists a constant C>such that k λ (z) 2 dμ(z) C k λ 2 H 2 for all λ, where k λ (z) = 2π z+λ for λ,z. 3. μ(q I ) C I for all intervals I ir, where Q I denotes the Carleson square In this case, μ is called a Carleson measure. Q I = { z = x + iy : iy I, <x< I }. A further relatively easy case is p = q = 2 and w a power weight, w(t) = t α with α<. This case corresponds to the classical embedding theorem for standard weighted Bergman spaces on the half-plane by Duren, see e.g. [4]. For more general weights w, the Laplace Carleson embedding corresponds to a new embedding theorem for weighted Bergman spaces A 2 ν on the half-plane with a translation-invariant measure ν, which is the subject of Section 2. Themain embedding theorem here is Theorem 2., which gives necessary and sufficient conditions for the boundedness of the embedding A p ν Lp (,μ). In the case of general p,q, p>2, the Laplace Carleson embeddings are very subtle even in the case that w =, due to the oscillatory part of the Laplace transform integral kernel. A general characterization seems out of reach at the moment, but with additional conditions on the support of the measure μ, a full characterization can sometimes be given. This is the content of Section3: herethemaintheoremsincludetheorems 3.3 and3.5, whichgivepreciseconditions for the boundedness of a Laplace Carleson embedding L : L p (, ) L q (,μ), f Lf, for sectorial measures μ. Some applications are outlined in Section 4. The reproducing kernel functions for H 2 ( ) are denoted by k λ, λ, where k λ (z) = 2π z+λ for z, and satisfy f(λ)= f,k λ for f H 2 ( ). Note that k λ 2 = 4π Re λ.we will also frequently use the Poisson kernel p λ (t) = π y x 2 + (y t) 2 (z = x + iy,t R), which is related to the reproducing kernel k λ by k λ 2 k λ (it) 2 = p λ (t) for t R.

4 786 B. Jacob et al. / Journal of Functional Analysis 264 (23) Embedding theorems for weighted Bergman spaces In this section, we will be interested in embeddings A p ν () L p (,μ), where A p ν ( ) is a weighted Bergman space defined below and ν is a translation-invariant positive regular Borel measure on, that is, ν = ν λ, where λ denotes Lebesgue measure and ν is a positive regular Borel measure on [, ). This corresponds to the case of radial measures on the unit disc. The investigation of such embeddings has a long history, starting with [3] for the case of the standard weighted Bergman space on with d ν(t) = t β dt for β>, respectively the standard weights ( z ) β on the disc. Oleinik [2] observed already in 974 that for measures ρ(t)dt, where the weight ρ decreases very fast towards t =, such as ρ(t) = e /t+γ, γ>, it is not sufficient to compare the weights of Carleson squares μ(q I ) and ν(q I ) (or equivalently, to compare the measures of Euclidean balls D centered on the imaginary axis). Instead, in the example above one has to consider the measures of Euclidean balls away from the imaginary axis, ) (Re z)+γ D z = D (z, ( + Re z) γ for z. Roughly speaking, the faster the weight ρ(t) decreases for t, the smaller the radius of the ball D z in relation to the distance of z to the imaginary axis, and the more detailed information on the measure μ is required. Recently, necessary and sufficient conditions have been found for the case even faster decreasing weights ρ(t), such as double exponentials [2]. Our aim in this section is somewhat different: we want to find a class of measures ν as large as possible, for which a characterization in terms of Carleson squares, and in terms of testing on powers of reproducing kernels k z, z, is possible. Our approach relies on a dyadic decomposition of the half-plane adapted to the measure which is somewhat technical, but has the advantage that no smoothness or continuity properties of the measure are required. Clearly, we need a growth condition on ν in and the most natural condition to impose is the well-known ( 2 ) (doubling) condition at. One motivation to treat this general setting, apart from the interest of Carleson-type embedding theorems in their own right, is to obtain Laplace Carleson embedding theorems on a large class of weighted spaces L 2 w (, ) (see Theorem 2.4 below), which are important in Control Theory. This is exploited in [4]. On the other hand, the general embedding theorem makes it also possible to study other important operators of analytic function spaces, such as Hankel operators and Volterra-type integration operators, in a more general setting. Such operators on weighted Bergman spaces for various classes of weights have for example been studied in [6,] and recently in [2,6]. As an example of this type of application, we study little Hankel operators on Zen spaces in Section Carleson measure on Zen spaces Let ν be a positive regular Borel measure on [, ) satisfying the following ( 2 )-condition: ν[, 2t) R := sup t> ν[,t) <. ( 2)

5 B. Jacob et al. / Journal of Functional Analysis 264 (23) This is sometimes referred to as a doubling condition, and such measures have been studied in the theory of harmonic analysis and partial differential equations for many years (an early reference is [23]). Let ν be the positive regular Borel measure on =[, ) R given by dν = d ν dλ, where λ denotes Lebesgue measure. In this case, for p<, we call A p ν {f = : C analytic: sup ε> } f(z+ ε) p dν(z) < a Zen space on.if ν({}) >, then by standard Hardy space theory, f has a well-defined boundary function f L p (ir), and we can give meaning to the expression f(z) p dν(z). Therefore, we may write ( f p Aν = f(z) p dν(z)) /p. Note that this expression makes sense in the case that ν({}) = (e.g. the Bergman space), since f is still defined ν-a.e. on. Clearly the space A 2 ν is a Hilbert space. Well-known examples of Zen spaces are Hardy space H p ( ), where ν is the Dirac measure in, or the standard weighted Bergman spaces A p α, where d ν(t) = t α dt, α>. Some further examples constructed from Hardy spaces on shifted half-planes were given by Zen Harper in [,]. Note that by the ( 2 )-condition, there exists N N such that kλ N Ap ν for all λ and all p<. Here is our embedding theorem for Zen spaces. Theorem 2.. Let p<, leta p ν be a Zen space on, with measure ν = ν λ as above, and let μ be a positive regular Borel measure on. Then the following are equivalent:. The embedding A p ν L p (,μ) is well-defined and bounded for one, or equivalently for all, p<. 2. For one, or equivalently for all, p<, and some sufficiently large N N, there exists a constant C p > such that ( k λ (z) ) N p dμ(z) C p ( k λ (z) ) N p dν(z) for each λ. (2) 3. There exists a constant C> such that μ(q I ) Cν(Q I ) for each Carleson square Q I. (3) Proof. The implication () (2) for fixed p is immediate. For the implication (2) (3), we use a standard argument using the decay of reproducing kernels. Given an interval I in ir, and let λ denote the centre of the Carleson square Q I.Note that ( k λ (z) ) N (2π) N (4Reλ) N for z Q I.

6 788 B. Jacob et al. / Journal of Functional Analysis 264 (23) Hence ( k λ (z) ) N p dμ(z) (8π) pn (Re λ) pn μ(q I ). It only remains to estimate (k λ (z)) N p dν(z) in terms of ν(q I ).LetR be the constant from the ( 2 )-condition. For k N,let2 k I denote the interval with the same centre as I and the 2 k fold length. By the ( 2 )-condition, ν(q 2 k I ) Rk 2 k ν(q I ). Note that Hence kλ N p A p ν k= ( k λ (z) ) N (2π) N (2 k Re λ) N for z Q 2 k I \Q 2 k I. ( ) ν(q 2 k I ) p (2π) N (2 k Re λ) N ν(q I ) (2π) pn (Re λ) Np k= 2 k R k 2 (k )Np. Choosing N sufficiently large, depending on R, we find that the sum on the right converges to a constant K N,p depending on N and p. Hence (8π) pn (Re λ) pn μ(q I ) ( k λ (z) ) N p dμ(z) C p ( k λ (z) ) N p dν(z) = C p K N,p (2π) pn (Re λ) Np ν(q I ), (4) and we obtain μ(q I ) C N,p ν(q I ), with a constant C N,p depending only on N and p (and hence on the ( 2 )-condition constant R). (3) () Our strategy is to deduce the boundedness of the embedding from the classical Carleson embedding theorem via a suitable decomposition. Let a>. In the following we denote by H p (C a ) the Hardy space of the shifted halfplane C a. We write H p a for the norm H p (C a ). Suppose for the moment that ν({}) = and that there exists a strictly increasing sequence (a n ) n Z in R + such that:. There exists >c> with a n+ a n a n+ c for all n Z; (5) 2. (2R) 3 ν ( [a n,a n ) ) ν ( [a n,a n+ ) ) 2R ν ( [a n,a n ) ). (6) Write β n = ν([a n,a n+ )). Notice that by (6),

7 n Z B. Jacob et al. / Journal of Functional Analysis 264 (23) β n f p Ha p n+ f(z) p dν = n Z [a n,a n+ ) β n f p H p (2R) 3 β n f p an H p. a n+ n Z n Z f(t+ iω) p d ν(t)dω Thus both ( n Z β n f p H p a n+ ) /p and ( n Z β n f p H p an ) /p give us equivalent expressions for the A p ν -norm. In addition, notice that if μ, ν satisfy the Carleson-type condition (3), then also μ, n Z β nδ an λ, satisfy the condition with the same constant C. Therefore, we assume without loss of generality that ν = n Z β n δ an, where δ an denotes the Dirac measure at a n. For simplicity of notation we will also assume that the constant C in the condition (3) equals. Our next step is a decomposition of μ into n Z μ n, where each μ n satisfies the Carlesontype condition (3) with respect to ν n = β n δ an λ. Lemma 2.2. Let N Z and suppose that μ is supported on the closed half-plane C an. Then there exist positive regular Borel measures μ n, n N, such that. μ = μ n ; (7) n=n 2. μ n is supported on the closed half-plane C an ; (8) 3. There exists a constant C > such that for all intervals I ir, μ n (Q I ) C ν n (Q I ) (n>n), μ N (Q I ) C N k= ν k (Q I ). (9) Moreover, μ n is a Carleson measure for the shifted half-plane C an, with Carleson constant C an (μ n ) C c β n (n > N), C an (μ N ) C c N k= where c is the constant appearing in the definition of the sequence (a n ) above. Proof. First, we prove that μ n, n N, exist, satisfying the conditions (7), (8) and (9). By replacing ν N with ( N n= β n )δ an λ, we can assume without loss of generality that ν is supported on C an. We begin by constructing a family of Carleson rectangles, on which (9) can be checked. Fix a dyadic grid D N of half-open intervals in ir with minimal intervals of length a N, and denote β k,

8 79 B. Jacob et al. / Journal of Functional Analysis 264 (23) these intervals on length a N as intervals of generation. The remaining intervals in the family D N will be parents, grandparents, etc. of the minimal intervals. We can assume without loss that c 2. () By (5), for any k. If we choose an integer l 2 such that and add l intermediate points c γ = a N+k+ a N+k c a N+k+ ( an+k+ a N+k a N+k ( c) 2, ) /l < ( c) 2 γa N+k,...,γ l a N+k between a N+k and a N+k+. In this way, we create a strictly increasing sequence (b j ) j such that b = a N, all terms of the sequence (a n ) n N appear as terms of the sequence (b j ) j, and c b j b j+ < ( c) 2 b j for all j. () The family F of Carleson squares we want to consider is formed by rectangles of the form (,b j+ ) I, where I D N, j, for which the eccentricity is bounded above and below by < b j+ 2. (2) 2 I Such a rectangle is denoted by Q I,j. Note that by () and (), for each I D N there exits j with (2), and for each j there exists exactly one size of intervals I in D N such that (2) holds. Note that different Carleson squares in this family F can have the same base I D N.It is easy to see that any Carleson square Q over an interval in ir can be covered by a bounded number of elements in F, with comparable base length. Therefore, up to a possible change of constant, it is sufficient to check the Carleson-type conditions (3) and (9) on the elements of F. The family F gives rise to a family T of right halves of the Carleson rectangles in F, which we will call tiles, and which will form a decomposition of the closed half-plane C an into disjoint sets. These are rectangles of the form T I,j =[b j,b j+ ) I, where Q I,j F, see Fig.. We say that an interval I D N belongs to generation j, if it is the base for a Carleson square Q I,j F. Thus each I D N belongs to at least one, and possibly more than one, generation.

9 B. Jacob et al. / Journal of Functional Analysis 264 (23) Fig.. The tiles T I,j. The idea of the construction below is to define μ N as the largest possible part of μ which can be dominated by ν N, in terms of condition (9). Recall that by the Carleson-type condition (3) we are given, we have in particular μ(t I, ) = μ(q I, ) ν(q I, ) = ν N (Q I, ) for all intervals I of generation in D N. For such intervals, we define the remaining part of ν N by νn {an } I = ν N(Q I, ) μ(t I, ) ν N {an } I. ν N (Q I, ) This defines a measure ν N on {a N} ir. In the second step, we define a measure ν N on {a N } ir by letting νn {an } I = νn (Q I, ) μ(t I, ) νn (Q I, ) ν N {an } I if ν N (Q I,)>μ(T I,), ifνn (Q I,) μ(t I,),

10 792 B. Jacob et al. / Journal of Functional Analysis 264 (23) for intervals I D N of generation. In the next and all following steps, having already defined the measure ν j N on {a N } ir for some j, we let ν j+ N {an } J ν j N (Q J,j+) μ(t J,j+ ) ν ν j N N (Q {an } J if ν j N (Q J,j+)>μ(T J,j+ ), J,j+) = ifν j N (Q J,j+) μ(t J,j+ ), ν j N (Q J,j+)>, ifν j N {a N } J =, (3) for intervals of generation j + ind N, thereby defining the measure ν j+ N on {a N } ir. Depending on which case in (3) appears, we say that (J, j + ) is of type, 2 or 3, respectively. We are finally ready to define the measure μ N. We start by letting μ N TI, = μ TI, for intervals I D N of generation. For T I,j+ T, j, let μ TI,j+ if ν j N (Q I,j+)>μ(T I,j+ ), ν μ N TI,j+ = j N (Q I,j+) μ(t I,j+ ) μ T I,j+ if ν j N (Q I,j+) μ(t I,j+ ), ν j N (Q I,j+)>, ifν j N T I,j+ =. Since the tiles in T form a decomposition of C an, this defines a measure μ N on C an. By construction, we have for each Q I,j F, (I, j) of type 2 or type 3: If (I, j) is of type, then μ N (Q I,j ) = ν N ( {an } I ). μ N (Q I,j )<ν N ( {an } I ). Hence μ N satisfies Condition 3 in Theorem 2. with respect to ν N. Let us now look at the Carleson condition for μ μ N. If (I, j) is of type, then μ N (T I,j ) = μ(t I,j ) by construction, and (μ μ N )(T I,j ) =. Therefore, by decomposing the Carleson square Q I,j into ( Q I,j = T I,j I I,Q I,j F Q I,j and iterating if necessary, we see that we have to check Condition 3 only for Carleson squares Q I,j with (I, j) of type 2 or type 3. But in this case, )

11 B. Jacob et al. / Journal of Functional Analysis 264 (23) μ N (Q I,j ) = ν N ( {an } I ) and thus by the original Condition 3 in Theorem 2. for μ and ν, μ μ N satisfies Condition 3 with respect to ν ν N, again with constant on Carleson squares in F. Finally, again by condition (3) in Theorem 2. for μ and ν, we see that for all tiles T I,j contained in the strip {z C: a N Re z<a N+ }, μ N TI,j = μ TI,j. Therefore, μ μ N is supported on the closed half-plane C an+. We can now make an induction step by applying the same procedure to the measures μ μ N, ν ν N with respect to the half-plane C an+ to construct μ N+, etc. We thus obtain a decomposition μ = n=n satisfying the conditions (7), (8) and (9) in the lemma. It remains to be shown that for each n>n, μ n is a Carleson measure with respect to the shifted half-plane C an, with the appropriate estimate of the Carleson constant. Let Q be a Carleson square in C an over the interval I. Recall that by (9), there exists C > with μ n ( Q) C ν n ( Q) for each Carleson square Q in and that (a n a n ) ca n. If the sidelength I of Q is less than a n a n, then Q has empty intersection with the support of μ n, and μ n (Q) =. If I a n a n, then Q can be covered by a Carleson square Q in with sidelength at most c I. Thus μ n μ n (Q) μ n ( Q) C ν n ( Q) C c β n I, and we obtain the desired result. The result for μ N is shown in the same way. Now let f A p ν. Note that by the ( 2 )-condition, each of the norms f H p an, denoting the norms on the Hardy spaces H p (C an ) of the shifted half-planes C an, is finite. Restricting μ to some closed half-plane C an, and using the decomposition in Lemma 2.2, we obtain C an f(z) p dμ(z) = n=n C p C c f(z) p dμ n (z) β n f p Ha p n n=n (2R) 3 C C p c = (2R) 3 C C p c β n f p Ha p n n=n ν ( [a n,a n ) ) f p Ha p n n=n (2R) 3 C C p f(z) p c A p ν.

12 794 B. Jacob et al. / Journal of Functional Analysis 264 (23) Here, we use that for any f A p ν,themap is non-increasing. Using that r f(z) p dμ(z) = sup f(r+ it) p dt N Z C an f(z) p dμ(z), we obtain the desired estimate. Now we can finish the proof of the theorem by showing that a sequence of positive numbers (a n ) n Z with the required properties (5), (6) exists, and that we can also treat the case ν({})>. Let R be the ( 2 ) constant of the measure ν, and let F be the function given by F :[, ) R, F(r)= ν ( [,r) ). F is left continuous, and also right continuous up to countably many jumps. By the ( 2 )- condition, F(2t) RF (t), so in particular, a jump of F at t maybenomorethan(r )F (t). If ν({}) =, then let for n Z a n = sup { r : F(r) (2R) 2n}, provided that the supremum is finite. If the supremum is infinite, we stop the sequence at the corresponding n. By the right continuity of F, a n > for all n Z, and by the condition on jumps of F, and therefore Hence a n+ 2a n and (2R) 2n F(a n ) R (2R)2n ν([,a n+ )) ν([,a n )) (2R)2n+2 4R. R(2R) 2n Furthermore, a n+ a n a n+ 2.

13 B. Jacob et al. / Journal of Functional Analysis 264 (23) (2R) 2n (4R ) ν ( [a n,a n+ ) ) = F(a n+ ) F(a n ) (2R) 2n+2, hence 2R ν([a n,a n+ )) ν([a n,a n )) (2R)3. If ν({}), then let a =, β = ν({}), and let a n = sup { r : F(r) (2R) 2(n+) ν ( {} )} for n N. We see in the same way as before that properties (5), (6) hold, if we replace ν([a,a )) by ν((a,a )). We write β = ν((a,a )) and β n = ν([a n,a n )) for n 2. Then β f p H p a + n= β n f p Ha p n+ f(z) p dν(z) β f p H p a + β n f p Han p, and the same construction as before applies. If the sequence (a n ) is finite to the right, an analogous argument can be made. The following proposition is elementary and appears for special cases in [,]. Partial results are also given in [2,5]. Proposition 2.3. Let A 2 ν be a Zen space, and let w : (, ) R + be given by w(t) = 2π e 2rt d ν(r) (t > ). Then the Laplace transform defines an isometric map L : L 2 w (, ) A2 ν. Note that the existence of the integral is guaranteed by the ( 2 )-condition. Proof. Let f L 2 w (, ). Then sup Lf(z+ ε) 2 dν(z) = sup ε> ε> = sup ε> = sup ε> n= (Lf )(ε + r + ) 2 L 2 (ir) d ν(r) ( F e (r+ε) f ) 2 L 2 (R) d ν(r) 2π e (r+ε) f 2 L 2 (, ) d ν(r)

14 796 B. Jacob et al. / Journal of Functional Analysis 264 (23) = sup = ε> f(t) 2 2π e 2(r+ε)t d ν(r)dt f(t) 2 w(t)dt by isometry of the Fourier transform and the dominated convergence theorem. Here is a Laplace Carleson embedding theorem, which is an immediate consequence. Theorem 2.4. Let A 2 ν be a Zen space, ν = ν dλ, and let w : (, ) R + be given by Then the following are equivalent: w(t) = 2π e 2rt d ν(r) (t > ). (4). The Laplace transform L given by Lf(z)= e tz f(t)dt defines a bounded linear map L : L 2 w (, ) L2 (,μ). 2. For a sufficiently large N N, there exists a constant C> such that ( Lt N e λt) (z) 2 dμ(z) C t N e λt 2 w(t)dt for each λ. 3. There exists a constant C> such that μ(q I ) Cν(Q I ) for each Carleson square Q I. Proof. Noticing that L(t N e λt ) is a scalar multiple of (k λ ) N, this follows immediately from Theorem Hankel operators on Zen spaces The boundedness of little Hankel operators on weighted Bergman spaces has been studied in various settings, see e.g. [2,3,5,29], though mostly for standard weights. Generally, boundedness results for Hankel operators are often connected to Carleson embedding results, at least concerning sufficient conditions for boundedness. We will show in this subsection that the boundedness of Hankel operators on Zen spaces can be deduced from a Carleson measure condition on the symbol. In the case of the Hardy space or the standard weighted Bergman spaces, this condition is also necessary.

15 B. Jacob et al. / Journal of Functional Analysis 264 (23) Theorem 2.5. Let A 2 ν be a Zen space, dν = d ν dλ. Let b : C be analytic, b H 2 ( ). If the measure b (z) 2 Re z ν ( [, Re z) ) da(z) (5) is a ν-carleson measure on, then the little Hankel operator A 2 ν A2 ν, f Q ν bf is bounded. Here, Q ν denotes the orthogonal projection Q ν : L 2 (,dν) A 2 ν. Proof. For the proof, we follow the lines of the proof of the Fefferman Stein duality theorem in [7]. Suppose that b (z) 2 Re z ν([, Re z)) da(z) is a ν-carleson measure and let f,g A 2 ν. Then Q ν bf,ḡ A 2 ν = = where we use the Littlewood Paley identity ir b(r + t)f(r + t)g(r + t)dt d ν(r) b ( ) (z) (z + r) f(r+ )g(r + ) Re zda(z)d ν(r), f(r+ t) 2 dt d ν(r) = f (z + r) 2 Re zda(z)d ν(r) ir and its polarization. Now = b (z + r)f(r + z)g (r + z) Re zda(z)d ν(r) ( ( ( b (z + r) 2 f(r+ z) ) /2 2 Re zda(z)d ν(r) g (z + r) ) /2 2 Re zda(z)d ν(r) b (z) 2 f(z) [ Re z ] 2 Re(z r)d ν(r) da(z) ) /2

16 798 B. Jacob et al. / Journal of Functional Analysis 264 (23) since ( f A 2 ν g A 2 ν, g (z + r) ) /2 2 Re zda(z)d ν(r) b (z) Re z 2 Re(z r)d ν(r)da(z) b (z) 2 Re z ν ( [, Re z) ) da(z) and the latter is a ν-carleson measure. The second term b (z + r)f (r + z) g(r + z) Re zda(z)d ν(r) is estimated accordingly. Hence the Hankel operator is bounded. In the case of the Hardy space H 2 ( ),thatis,if ν is the Dirac measure in, the Carleson condition on the measure (5) is a well-known characterization of the boundedness of the Hankel operator, see e.g. [7]. For further applications of the theorem above, recall the Bloch space B( ) on, { B( ) = f Hol( ): f B = sup f (z) } Re z<. z Let us consider the following inverse ( 2 )-condition on ν: sup inf M> r> ν([,mr)) ν([,r)) >. (6) Corollary 2.6. Let A 2 ν () be a Zen space with ν satisfying the inverse ( 2 )-condition (6) and let b B( ). Then the little Hankel operator A 2 ν A2 ν, f Q ν bf is bounded. Proof. Let b B( ) and let Q I be a Carleson square. We write F(r)= ν([,r)) for r. Then Q I b (z) 2 Re z ν ( [, Re z) ) da(z) b 2 B I I Q I Re z Re z s F(s)ds. d ν(r)da(z)

17 B. Jacob et al. / Journal of Functional Analysis 264 (23) To finish the proof, it remains only to be shown that there exists C> with x F(s)ds CF(x) s for all x [, ). Choosing M,γ > with ν([,mr)) inf γ, r> ν([,r)) we find that x s F(s)ds M n x n= M (n+) x n= (M ) s F(s)ds ( M n M (n+)) xm n+ F ( M n x ) x γ n F(x)= n= γ(m ) γ F(x). Hence the ν-carleson condition for the measure (5) in Theorem 2.5 applies, and the little Hankel operator is bounded. Remark. In the case of the standard weighted Bergman spaces A 2 α () with α>, we have d ν(r) = r α dr, and the inverse ( 2 )-condition (6) holds. It is well-known that the little Hankel operator with symbol b is bounded on A 2 α, if and only if b B() (see e.g. [3]). Therefore, the ν-carleson condition for the measure (5) in Theorem 2.5, respectively the Bloch condition on the symbol in Corollary 2.6, are also necessary in this case. 3. L p L q embeddings As mentioned above, there is no known full characterization of boundedness of Laplace Carleson embeddings L p (, ) L q (,μ), f Lf = e t f(t)dt. However, characterizations are possible in case p 2, if the conjugate index p satisfies p <q, and in some cases with additional information on the support on the measure. We list some results for natural spectral inclusion conditions which appear naturally in the theory of operator semigroups. In the cases we consider here, the oscillatory part of the Laplace transform can be discounted, and a full characterization of boundedness can be achieved. First, let us make the following simple observation.

18 8 B. Jacob et al. / Journal of Functional Analysis 264 (23) Proposition 3.. Let μ be a positive regular Borel measure on,let p,q < and suppose that the Laplace Carleson embedding L : L p (, ) L q (,μ), f Lf, is well-defined and bounded. Then there exists a constant C p,q > such that for all intervals I ir, μ(q I ) C p,q I q/p if p>, μ(q I ) C p,q if p =. (7) Proof. Let Q I be a Carleson square with centre λ I. Note that ( Le λ I ) q L q (,μ) Q I ( Le λ I ) q dμ (4Reλ I ) q μ(q I ) = 2 q I q μ(q I ), and e λ I p p = p Re λ I = 2 p I for p,q <. Hence there exists a constant C p,q > such that μ(q I ) C p,q I q/p if p> and μ(q I ) C p,q if p =. This concludes the proof. The proposition immediately yields the following theorem: Theorem 3.2. Let μ be a positive regular Borel measure supported in the right half-plane, and let <p q<, p 2. Then the following are equivalent:. The Laplace Carleson embedding L : L p (, ) L q (,μ), f Lf, is well-defined and bounded. 2. There exists a constant C> such that μ(q I ) C I q/p for all intervals I ir. (8) 3. There exists a constant C> such that Le z L q μ C e z L p for all z. Proof. Obviously () (3), and (3) (2) by the proposition. This leaves (2) (), which is also easy: By the Hausdorff Young inequality, the map L : L p (, ) L p (ir) is bounded. By Duren s theorem, the Poisson extension L p (ir) L q μ( ) is bounded, given the Carleson condition (7). The composition of both gives the boundedness of the Laplace transform L : L p (, ) L q (,μ).this concludes the proof.

19 B. Jacob et al. / Journal of Functional Analysis 264 (23) The case p>2 is much more subtle: in particular the Laplace transform no longer maps L p (, ) into H p ( ). We give two special cases here, that of the measure μ being supported in a strip and that of μ being supported in a sector. 3.. Sectorial measures If the measure μ is supported on a sector S(θ) ={z : arg z <θ} for some <θ< π 2, then the oscillatory part of the Laplace transform can be discounted, and a full characterization of boundedness can be achieved (see also [9, Theorem 3.2] for an alternative characterization by means of a different measure). Theorem 3.3. Let μ be a positive regular Borel measure supported in a sector S(θ), <θ< π 2, and let q p>. Then the following are equivalent:. The Laplace Carleson embedding L : L p (, ) L q (,μ), f Lf, is well-defined and bounded. 2. There exists a constant C> such that μ(q I ) C I q/p for all intervals in I ir which are symmetric about. 3. There exists a constant C> such that Le z L q μ C e z L p for all z R There exists a constant C> such that Le 2n L q μ C e 2n L p for all n N. Proof. (2) () For n Z,let T n = { x + iy :2 n <x 2 n, 2 n <y 2 n }. That is, T n is the right half of the Carleson square Q In over the interval I n ={y R, y 2 n }. The T n are obviously pairwise disjoint. Without loss of generality we assume <θ<arctan( 2 ), in which case S(θ) n= T n. Now let z T n for some n Z. Then we obtain, for f L p (, ), Lf(z) e zt f(t) dt e 2 n t f(t) dt CΘ 2 n+ Mf ( 2 n+), where C Θ > is a constant dependent only on the integration kernel Θ(t) = χ [, ) (t + )e t and Mf is the Hardy Littlewood maximal function. We refer to e.g. [22, p. 57, Eq. (6)] for a pointwise estimate between the maximal function induced by the kernel Θ and M. We can easily dominate Θ by a positive, radial, decreasing L function here. Consequently, S(θ) Lf(z) q dμ(z) n= 2 q( n+)( Mf ( 2 n+)) q μ(tn )

20 82 B. Jacob et al. / Journal of Functional Analysis 264 (23) C q Θ = C q Θ n= n= C q Θ 2q/p ( f q L p. 2 q( n+) 2 nq/p ( Mf ( 2 n+ )) q 2 q/p ( 2 ( n+) ( Mf ( 2 n+) p)) q/p n= 2 ( n+)( Mf ( ) q/p 2 n+)) p Note that in the case <p 2 (but not for p>2) this result can also easily be deduced from the Hausdorff Young inequality and Duren s theorem [4]. (4) (2) Let I ir be an interval which is symmetric about. We can assume without loss that I =2 n. It is easy to see that ( Le 2n )(z) = 2 n + z 2 n+ for z Q I. Thus μ(q I ) ( 2 n+) q (Le 2 n )(z) q dμ(z) C ( 2 n+) q e 2 n q p 2nq 2 nq/p = 2 nq/p. () (3) and (3) (4) are obvious. Remark 3.4. Let μ, θ, p and q be as in Theorem 3.3. In[9, Theorem 3.2], essentially the equivalence of the following statements is shown for discrete measures:. The Laplace Carleson embedding L : L p (R + ) L q (,μ), f Lf, is well-defined and bounded. 2. There exists a constant C> such that μ(q I ) C I q/p for all intervals in I ir which are symmetric about, where d μ(z) = z q dμ( z ). The proof also uses the method of the maximal function. Now let us consider the case p>qfor sectorial measures μ. We will, among others, obtain a condition in terms of the balayage S μ of μ (compare this with the characterization of bounded H p ( ) L q (,μ)embeddings for p>qin [7]). Recall that the balayage S μ of a positive

21 B. Jacob et al. / Journal of Functional Analysis 264 (23) Borel measure μ on is given by S μ (t) = p z (t) dμ(z), where p z is the Poisson kernel, defined in the introduction as p z (t) = π y x 2 + (y t) 2 (z = x + iy,t R). To look at a dyadic version, let T n = { x + iy :2 n <x 2 n, 2 n <y 2 n } as in the previous proof, and for k Z, lett n,k = T n + ik2 n,sothet n,k are the translates of T n parallel to the imaginary axis. Similarly, let I n,k = I n + k2 n be the translates of I n := {y R: y 2 n }.The{T n,k : n, k Z} then form a dyadic tiling of the right half-plane. Let S d μ (t) = n,k Z χ In,k (t) μ(t n,k) 2 n. S d μ is called the dyadic balayage of μ. Note that Sd μ 2πS μ pointwise, since S μ (t) = p z (t) dμ(z) n= k= n= k= χ In,k (t) p z (t) dμ(z) T n,k χ In,k (t) inf z T n,k { pz (t) } μ(t n,k ) 2π n,k Z χ In,k (t) μ(t n,k) 2 n. (9) In the special case that μ is sectorial with opening angle θ<π/2, the measure μ is supported on n Z T n, and we get a particularly simple form of the dyadic balayage S d μ, namely where S d μ (t) = n= χ In (t) μ(t n) 2 n = S d μ,k (t) = n= k= n= χ In \I n (t) μ(t n+k) 2 n+k = χ In \I n (t) μ(t n+k) 2 n+k = Sμ, d ( 2 k t ). Sμ,k d (t), Let us now look at an estimate from above for S μ in terms of the S d μ,k.lett I n\i n. Then k= max p z (t) z T k This easily implies that for t I n \I n, { 2 k π2 2n if n>k, π2 k if n k.

22 84 B. Jacob et al. / Journal of Functional Analysis 264 (23) Hence, for t R S μ (t) π = π = π S μ (t) π ( j= ( j= ( = π j= ( j= ( j= 2 n+j π2 2n μ(t n+j ) + 2 2j μ(t n+j ) 2 n+j + 2 2j S d μ,( 2 j t ) + j= j= ) 2 n+j μ(t n+j ) μ(t n+j ) 2 n+j ) Sμ,( d 2 j t )). j= 2 2j S d μ,( 2 j t ) + j= Sμ, d ( 2 j t )) 2 2j Sμ, d ( 2 j t ) ) + Sμ d (t). (2) Theorem 3.5. Let μ be a positive regular Borel measure supported in a sector S(θ), <θ< π 2 and let q<p<. Then the following are equivalent:. The embedding L : L p (R + ) L q (,μ), f Lf, is well-defined and bounded. 2. The sequence (2 nq/p μ(s n )) is in l p/(p q) (Z). HereS n = k Z T n,k ={z C: 2 n < Re z 2 n }. 3. The sequence (2 n/p Lk 2 n L q μ ) is in l qp/(p q) (Z). If p <q, then the above is also equivalent to 4. t q(2 p)/p S μ L p/(p q) (R). Remark. If p q, then the sweep S μ may be infinite everywhere, so we cannot expect a characterization in terms of S μ. Proof of Theorem 3.5. We will start by showing (2) (). Recall that S(θ) is contained in n,k Z, k N T n,k for some N N. Suppose that (2) holds. Now as in the proof of Theorem 3.3, we obtain for z S n Lf(z) e zt f(t) dt e 2n t f(t) dt C Θ 2 n+ Mf ( 2 n+). Note that χ I n Mf (2 n+ ) χ I n Mf. Consequently

23 Lf(z) q dμ(z) B. Jacob et al. / Journal of Functional Analysis 264 (23) n= ( n= 2 q( n+) Mf ( 2 n+) q μ(sn ) n= 2 ( n+) Mf ( ) q/p ( 2 n+) p f q L p ( 2 nq/p μ(s n ) ) (p/q) n= 2 q( n+) Mf ( 2 n+) q μ(sn ) 2 (q/p )( n+)(p/q) μ(s n ) (p/q) ) /(p/q) by the boundedness of the Hardy Littlewood maximal function on L p (, ). () (2) For λ> we write k λ for the L p (, ) function given by k λ (t) = λ /p e λt (t ), noting that k λ L p. From a result of Gurariĭ and Macaev in [8], it can be deduced that ( ) /p α n k 2 n α n p (2) p n Z for any l p sequence (α n ). More precisely, we claim that there are constants A,B > such that for all scalars (α k ) we have A α n p p α n k 2 n B α n p. n n p n By the change of variable x = e t we have α n k 2 n n p p = n Z α n 2 n/p x 2n /p n Recall that a sequence (m j ) is a lacunary sequence if inf m j+ /m j = r>. Now, the result of Gurariĭ and Macaev in [8] asserts the following: If (n j + /p) is lacunary, then the sequence of functions (t (n j + /p) /p t n j ) in L p (, ) is equivalent to the standard basis of l p. Writing n j = 2 j /p for j Z, we have the conditions of the Gurariĭ Macaev theorem, and the claim follows. Now, denoting by (Ω, dω) the probability space of sequences (ɛ n ) taking values in, with equal probability, equipped with the standard product σ -algebra and probability measure, we obtain p dx.

24 86 B. Jacob et al. / Journal of Functional Analysis 264 (23) ( ) q/p ( α n p ) q L ε n α n k n dω(ε) n n L q μ Ω = ε n α n 2 n/p q 2 n + z dμ(z)dω(ε) Ω n ( = k n = n S k n ) α n 2 2 2n/p q/2 2 n + z 2 dμ(z) ( n ) α n 2 2 2n/p q/2 2 n + z q dμ(z) α n q 2 nq/p dμ(z) 2qn S n α n q 2 nq/p μ(s n ). Here, we have used the fact that μ is supported in a sector S(θ) in the last inequality. Thus (2 nq/p μ(s n )) is an l (p/q) sequence, and we have proved the desired implication. A simple argument, again using sectoriality, shows that Hence (3) (2). For () (3), note that for any n, n Lk 2 n q q 2 nq μ(s n). Re Lk 2 n Lk2 n(z) for z S(θ). Hence for any sequence (α n ) l p/q (Z), α n for all n, ( α n 2 n/p Lk 2 n q L q (μ) αn /q 2 n/p Lk 2 n(z) q ) dμ(z) n ( Re n n α /q n 2 n/p Lk 2 n(z)) q dμ(z) ( ) L αn /q 2 n/p q k 2 n (z) dμ(z) n αn /q 2 n/p q k 2 n α /q n p q p = α n p/q by (2). This proves the equivalence of the first three statements.

25 B. Jacob et al. / Journal of Functional Analysis 264 (23) (2) (4) Again, we can assume without loss that <θ<π/2, in which case S(θ) n= T n.ift q(2 p)/p S μ L p/p q (R), then by (9) t q(2 p)/p S d μ, Lp/(p q) (R), and n= 2 nqp/(p (p q)) μ(s n ) p/(p q) = n= n= I n+ \I n 2 n 2 n(p 2)q/(p q)μ(s n) p/(p q) 2 np/(p q) t (p 2)q/p Sμ, d p/(p q) dt = t (p 2)q/p Sμ, d p/(p q) p/(p q) <. Thus (2) holds. Conversely, if (2 nq/p μ(s n )) l p/(p q), then t (p 2)q/p S d μ, Lp/p q (R) by the above calculation. By (2), t (p 2)q/p S μ p/(p q) π One sees easily that ( t (p 2)q/p S d μ (t) p/(p q) + k= 2 2k t (p 2)q/p Sμ, d ( 2 k t ) ) p/(p q). 2 2k t (p 2)q/p Sμ,( d 2 k t ) p/(p q) = 2 2k 2 k(p q)/p 2 k(p 2)q/p t (p 2)q/p Sμ, d p/(p q) = 2 k(q+ q/p) t (p 2)q/p Sμ, d p/(p q), thus the second term in the sum converges and is controlled by the expression For the first term, write t (p 2)q/(p q) Sμ, d p/(p q). S d μ (t) = n χ In (t) μ(t n) 2 n = k= n χ In \I n (t) μ(t n+k) 2 n+k as before. For each k, it follows that ( t q(2 p)/p n χ In \I n (t) μ(t ) p/(p q) n+k) 2 n+k dt n = n 2 n 2 nq(2 p)/(p q) 2 (n+k)p/(p q) μ(t n+k ) p/(p q) 2 n(q/p q+)p/(p q) 2 (n+k)p/(p q) μ(t n+k ) p/(p q)

26 88 B. Jacob et al. / Journal of Functional Analysis 264 (23) = n 2 n( q/p +)p/(p q) 2 (n+k)p/(p q) μ(t n+k ) p/(p q) = 2 k( q/p )p/(p q) n 2 (n+k)( q/p )p/(p q) μ(t n+k ) p/(p q) 2 k( q/p )p/(p q). Hence t (p 2)q/p S μ L p/(p q) (R). This concludes the proof A counterexample Let μ denote the measure on the interval [, ) defined by dμ(x) = dx/ x. Clearly, μ is sectorial and contained in a shifted half-plane. Moreover μ satisfies the estimate that for a Carleson square Q of size h one has μ(q) 2h /2. Nonetheless, the Laplace Carleson embedding L : L 2 (, ) L (μ) is unbounded (or equivalently, in this case, the Carleson embedding H 2 ( ) L (μ) is unbounded). This can be seen by noting that μ does not satisfy the condition of [8, Theorem C], since the function t Γ(t) x dμ(x) behaves as x /2 and thus does not lie in L 2.(HereΓ(t)maybetakento be the interval [t, ).) It is constructive to give an explicit counterexample, following the reasoning of the proof of [8, Theorem C]. (Note that counterexamples in the case of the disc are simpler, and can be found in [24].) Define φ : R R by { if t, φ(t)= t /2 ( + log t ) if t. Thus φ L 2 (R) and there is a function F H 2 with Re F(it)= φ(t).now,iff L (μ) we would have Re Fdμ F dμ <, from which, as in [8] we could conclude (by writing Re F(x,) in terms of the Poisson kernel) that and hence which is a contradiction. A := φ(t) t /2 dt = x dx φ(t) x 2 + t 2 dt <, x x dx φ(t) 2x 2 dt A<, x t

27 B. Jacob et al. / Journal of Functional Analysis 264 (23) Measures supported in a strip Theorem 3.6. Let μ be a positive regular Borel measure supported in a strip C α,α 2 = {z C: α 2 Re z α } for some α 2 α >, and let <p q<, q 2. Then the following are equivalent:. The embedding L : L p (, ) L q (,μ), f Lf, is well-defined and bounded, with a bound only depending on the Carleson Duren constant C and the ratio α 2 α. 2. There exists a constant C>such that μ(q I ) C I q/p for all intervals I ir. (22) 3. There exists a constant C> such that Le z L q μ C e z L p for all z C α,α 2. Proof. Again, obviously () (3). To show (3) (2), we have to remember that the argument in Proposition 3. only works for Carleson squares Q I with centre λ I C α,α 2. Any Carleson square with centre λ I C α /3,α 2 can be covered by a Carleson square of at most triple sidelength with centre in C α,α 2, any Carleson square with centre in λ I C,α /3 has nonempty intersection with the support of μ.ifq I is a Carleson square with centre λ I C α2, then its intersection with the support of μ can be covered by at most [ 2Reλ I α 2 ]+ Carleson squares with centre in C α,α 2. Hence μ(q I ) I α 2 α q/p 2 I p/q. This leaves (2) (). Consider the line parallel to the imaginary axis ir + α /2. Note that is bounded, since ( L : L p (R + ) L 2 ir + α ) 2 Lf L 2 (ir+ α 2 ) = L ( e α /2t f ) L 2 (ir) = e α/2t f 2 e α t /2 2 p p/(p 2) f 2p p α f p. Since the measure μ is supported in C α,α 2, by the Carleson condition (7) we have for each Carleson square in Q I ={z C: i Im z I, α /2 < Re z<α /2 + I } in,α /2: Thus the Poisson embedding μ(q I ) C I q/p C I q(/p /2) I q/2 Cα q(/p /2) 2 I q/2.

28 8 B. Jacob et al. / Journal of Functional Analysis 264 (23) ( L 2 ir + α ) L q (,μ) 2 is bounded by Duren s theorem, with constant Cα /p /2 2. Again, composing both maps yields the Laplace transform L : L p (R + ) L q (,μ α ) 2 p 2p with norm bound Cα α /2 /p 2 = C( α 2 α ) /2 /p Sobolev spaces In this subsection, we will be interested in embeddings L : H p β (, ) Lp (,μ), where for β>the space H p β (, ) is given by H p β {f ( ) (, ) = L p (R + ): d β f(t) dx ( ) f p H p = f p p + d β p f β dx. p dt < Here ( d dx )β f is defined as a fractional derivative via the Fourier transform. It is now easy to find versions of Theorems 3.3 and 3.5 for Sobolev spaces. Corollary 3.7. Let μ be a positive Borel measure supported in a sector S(θ), <θ< π 2, and let q p>. Then the following are equivalent:. The embedding p }, L : H p β (, ) Lq (,μ), f Lf, is well-defined and bounded. 2. There exists a constant C> such that μ q,β (Q I ) C I q/p for all intervals in I ir which are symmetric about.here,dμ q,β (z) = ( + z qβ )dμ(z). 3. There exists a constant C> such that Le z L q μ C e z H p β for all z R +. Proof. Follows immediately from Theorem 3.3 and basic properties of the Laplace transform. Corollary 3.8. Let μ be a positive regular Borel measure supported in a sector S(θ), <θ< π 2 and let q<p, β. Suppose that S μ β,q L p/(p q). Then the embedding

29 B. Jacob et al. / Journal of Functional Analysis 264 (23) L : H p β (, ) Lq (,μ), f Lf, is well-defined and bounded. Proof. Follows immediately from Theorem 3.5. Laplace Carleson embeddings of Sobolev spaces Hβ 2 are easily understood by means of Theorem.: Theorem 3.9. Let μ be a positive Borel measure on the right half-plane and let β>. Then the following are equivalent:. The Laplace Carleson embedding H 2 β (, ) L2 (,μ) is bounded. 2. The measure + z 2β dμ(z) is a Carleson measure on. Proof. The proof is a simple reduction to the Carleson embedding theorem. Note that the map H 2 β (R +) H 2 ( ), f ( + z) β Lf, is an isomorphism. The remainder follows from the holomorphy of ( + z) β and Lf on, and a density argument. 4. An application Suppose that we are given a C -semigroup (T (t)) t defined on a Hilbert space H, with infinitesimal generator A, and consider the system dx(t) dt = Ax(t) + Bu(t), x() = x, t. (23) Here u(t) C is the input at time t, and B : C D(A ),thecontrol operator. We write D(A ) for the completion of H with respect to the norm x D(A ) = (β A) x H, for any β ρ(a). To guarantee that the state x(t) lies in H one asks that B L(C,D(A ) ) and T(t)Bu(t)dt m u L 2 (, ) H for some m > (the admissibility condition for B). The C -semigroup (T (t)) t has an extension to D(A ).

30 82 B. Jacob et al. / Journal of Functional Analysis 264 (23) We refer to the survey [3] and the book [25] for the basic background to admissibility in the context of well-posed systems. For diagonal semigroups, it is linked with the theory of Carleson measures as in [2,26]:ifA has a Riesz basis of eigenvectors, with eigenvalues (λ k ), then a scalar control operator corresponding to a sequence (b k ) is admissible if and only if the measure μ := k b k 2 δ λk is a Carleson measure for H 2 ( ). An extension to normal semigroups has also been made [27]. Generalizations to α-admissibility, in which u lies in L 2 (, ; t α dt) for <α<, were studied by Wynn [28]. He used the fact that the Laplace transform maps L 2 (, ; t α dt) to a weighted Bergman space, for which a Carleson measure theorem is known. The results above enable us to take this generalization further, considering admissibility in the sense of the input lying in much more general spaces L 2 (, ; w(t)dt). Assume that q< and the semigroup (T (t)) t acts on a Banach space X with a q-riesz basis of eigenvectors (φ k ); that is, T(t)φ k = e λkt φ k for each k, and (φ k ) is a Schauder basis of X such that for some C,C 2 > wehave C ak q q a k φ k C2 ak q for all sequences (a k ) in l q. Without loss of generality X = l q and the eigenvectors of the generator of (T (t)) t, denoted by A, are the canonical basis of l q. Suppose also that we have a Banach space Z of functions on (, ), either an L p space or a weighted space L 2 (, ; w(t)dt), whose dual space Z can be regarded, respectively, as either L p (, ) or L 2 (, ; w(t) dt) in a natural way. The following general result appears in [4]. Theorem 4.. Let B be a linear bounded map from C to D(A ) corresponding to the sequence (b k ). The control operator B is Z-admissible for (T (t)) t, that is, there is a constant m > such that T(t)Bu(t)dt m u Z, u Z, X if and only if the Laplace transform induces a continuous mapping from Z into L q (,dμ), where μ is the measure b k q δ λk. This gives a direct application of our results on Laplace Carleson embeddings: full details are given in [4]. Acknowledgments This work was supported by EPSRC grant EP/I62X/. The third author also acknowledges partial support of this work by a Heisenberg Fellowship of the German Research Foundation (DFG).

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