Local properties of Hilbert spaces of Dirichlet series
LLOCAL PROPERTIES OF HILBERT SPACES OFDIRICHLET SERIES
JAN-FREDRIK OLSEN
Abstract.
We show that the asymptotic behavior of the partial sums of asequence of positive numbers determine the local behavior of the Hilbert spaceof Dirichlet series defined using these as weights. This extends results recentlyobtained describing the local behavior of Dirichlet series with square summablecoefficients in terms of local integrability, boundary behavior, Carleson measuresand interpolating sequences. As these spaces can be identified with functionsspaces on the infinite-dimensional polydisk, this gives new results on the Dirich-let and Bergman spaces on the infinite dimensional polydisk, as well as the scaleof Besov-Sobolev spaces containing the Drury-Arveson space on the infinite di-mensional unit ball. We use both techniques from the theory of sampling inPaley-Wiener spaces, and classical results from analytic number theory. introduction The theory of Dirichlet series, i.e. functions of the form f ( s ) = (cid:80) n ∈ N a n n − s with s = σ + i t as the complex variable, offers a bridge between number theory andanalysis. Perhaps the most appealing example of the power of this connection isgiven by the tauberian approach to the classical prime number theorem. One wayto state the prime number theorem is to say that the Chebyshev-type inequalities A x (log x ) α ≤ (cid:88) n ≤ x w n ≤ B x (log x ) α , (1)with coefficients w n = (cid:26) n is a prime otherwise , and α = 1 , holds for any A, B > as long as x > is taken to be sufficiently large.Originally due to Ikehara, the general idea of the tauberian approach is to connectthe function theoretic properties of the Riemann zeta function ζ ( s ) = (cid:80) n ∈ N n − s to the growth of these partial sums (see e.g. [34, p. 245]). As is well-known, theproperties of the Riemann zeta function is closely related to the behavior of theprime numbers through the Euler product formula ζ ( s ) = (cid:89) p prime − p − s . We study the connection between the asymptotic behavior in terms of the in-equalities (1) for general sequences ( w n ) n ∈ N of non-negative numbers, and local a r X i v : . [ m a t h . C V ] N ov JAN-FREDRIK OLSEN function theoretic properties of the Hilbert spaces H w = (cid:40)(cid:88) n ∈ N a n n − s : (cid:88) | a n | w n < ∞ (cid:41) . (By convention, if w n = 0 , we exclude the basis vector n − s from this definition.)The recent interest in the theory of these types of spaces began with a paperby Hedenmalm, Lindqvist and Seip [11], where in particular the local behaviorof functions in the Dirichlet-Hardy space H , which corresponds to the choice w n ≡ , is studied. By the Cauchy-Schwarz inequality, the space H is seen toconsist of functions analytic on the half-plane C / = { Re s > / } . The resultsof this and later contributions [2, 16, 24, 25] can be summarised as saying thatlocally H looks much like the classical Hardy space H ( C / ) = (cid:40) f analytic on C / : sup σ> / π (cid:90) R | f ( σ + i t ) | < ∞ (cid:41) . One of the starting points of the function theory for the Dirichlet-Hardy spaceis a simple, but striking, local connection indicated by comparing reproducingkernels, i.e. functions k w such that (cid:104) f | k w (cid:105) = f ( w ) for all f in the space, andpoints w in the domain of definition. For the space H , the reproducing kernelat w ∈ C / is the translate k w ( s ) := ζ ( s + ¯ w ) of the Riemann zeta function. TheRiemann zeta function is known to be a meromorphic function with a single poleof residue one at s = 1 . This yields the formula k w ( s ) = 1 s + ¯ w − h ( s + ¯ w ) , where h is an entire function. This reveals that k w is an analytic perturbation ofthe reproducing kernel for H ( C / ) , namely the Szegő-kernel k Sw ( s ) = ( s + ¯ w − .The following results strengthens this local connection. The first is [11, Theorem4.11], which was found independently by Montgomery [22, p. 140] in the contextof analytic number theory. Theorem (Local embedding theorem [11, 22]) . Given a bounded interval I , thereexists C > , depending only on the length of I , such that for all F ∈ H we have sup σ> / (cid:82) I | F ( σ + i t ) | d t ≤ C (cid:107) F (cid:107) H . It is an immediate consequence of this theorem that if F ∈ H then F ( s ) /s ∈ H ( C / ) . In particular, this implies that functions in H have non-tangentialboundary values almost everywhere on the abscissa σ = 1 / . The second theoremwe mention is in some sense dual to the previous one, and describes the spacespanned by the boundary functions. Theorem (Local boundary function property [24]) . Given a bounded interval I and a function f ∈ H ( C / ) , there exists F ∈ H such that F − f has an analyticcontinuation across the segment / I . OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 3
With these results in hand, it is not difficult to show that a compactly supportedpositive measure µ on C / is a Carleson measure for H if and only if it is aCarleson measure for H ( C / ) (see also the proof of Theorem 3). Recall that if H is a Hilbert space of functions on C / , we say that a positive Borel measure isCarleson for H if there exists C > such that for all f ∈ H we have (cid:90) C | f ( s ) | d µ ( s ) ≤ C (cid:107) f (cid:107) H . Finally, we mention the following result on interpolating sequences. Recall thata sequence ( s j ) j ∈ N is called interpolating for a Hilbert space H of functions analyticon some domain Ω , which admits a reproducing kernel k w at all w ∈ Ω , if for allsequences ( w j ) j ∈ N satisfying (cid:80) | w j | / (cid:107) k w j (cid:107) < ∞ there exists a solution f ∈ H tothe problem f ( s j ) = w j . Theorem (Local interpolation theorem [25]) . Let S = ( s j ) be a bounded sequenceof distinct points in C / . Then S is interpolating for H if and only if it isinterpolating for H ( C / ) . (See [8, 21, 23, 29] for further results on functions spaces of Dirichlet series.)Such precise results are perhaps surprising in view of a deep feature of thetheory, which dates back to H. Bohr [4]. He observed that Dirichlet series can beidentified in a natural way with power series of countably infinitely many variablesby identifying the i ’th complex variable z i with the Dirichlet monomial p − si , where p i is the i ’th prime number. Therefore the study of the spaces H w can be seenas the study of Hilbert spaces of functions in countably infinitely many variables.Namely, the space H w , introduced above, is identified with H w = (cid:40)(cid:88) ν a ν z ν : (cid:88) ν | a ν | /w ν < ∞ (cid:41) . Here ν = ( ν , ν , . . . ) is a multi-index, z ν = z ν z ν · · · , and we only sum over ν with finite non-zero entries in N . In particular, the Dirichlet-Hardy space H is identified with the Hardy space on the infinite dimensional polydisk, H ( D ∞ ) ,which corresponds to the choice w ν ≡ .For completeness, we briefly discuss the space H ( D ∞ ) , or rather, its morenatural counterpart H ( T ∞ ) , where T ∞ = (cid:110) ( z , z , . . . ) : z i ∈ T (cid:111) is the countably infinite dimensional torus. T ∞ is more natural to work on than D ∞ , since it is a compact abelian group under coordinate-wise multiplication,and therefore admits a unique normalized Haar measure µ . It follows that we maydefine the space L ( T ∞ ) in the usual way. To define the analytic subspace H ( T ∞ ) ,we identify each χ ∈ T ∞ with a multiplicative function determined uniquely by χ ( p j ) = z j , where p j is the j ’th prime number. The function χ is extended toall the positive rational numbers Q + by setting χ (1 /n ) = χ ( n ) . The Fourier JAN-FREDRIK OLSEN spectrum of f ∈ L ( T ∞ ) is in this way identified with Q + . In light of this, wedefine H ( T ∞ ) to be the closed subspace whose Fourier spectrum is supportedonly on N . Similarly, for any p > we obtain the spaces H p ( T ∞ ) . The Bohridentification now yields a family of spaces Dirichlet-Hardy spaces that we denoteby H p . We refer the reader to [2, 5, 11] for further details. In particular, in [5], itis explained how functions in H p ( T ∞ ) can be identified with analytic functions on D ∞ ∩ (cid:96) , thereby justifying the use the notation H p ( D ∞ ) . As a consequence, onedirection of the Bohr correspondence can be understood as evaluating a functionin H p ( T ∞ ) at the points (2 − s , − s , − s , . . . ) for Re s > / .Analogues of the three theorems mentioned above have also been obtained in forthe choice of weights w n = (log n ) α [24, 25]. However, observe that these spaces,which were introduced by McCarthy in [21], correspond to spaces of functions ininfinite variables where monomials of the same degree may differ in norm. Ourapproach in this paper allows us to consider more general choices of weights w n ,which correspond to more well-known spaces of infinite variables. In fact, we areable determine the local behavior of spaces of Dirichlet series corresponding toimportant classical spaces. These include the Dirichlet and Bergman spaces onthe infinite dimensional polydisk, and the Drury-Arveson space, as well as thegeneral scale of analytic Besov-Sobolev spaces, on the infinite dimensional unitball. (See examples 1 through 7 below.)The structure of the paper is as follows. Our results are presented as theorems1 to 4 in the following section, where we also include a detailed treatment of theexamples mentioned above, as well as a few additional ones. In Section 3 we recallsome background results on sampling theory needed in the proofs, and establish asimple lemma. The proofs of our theorems are given in sections 4 to 7. In Section8, we make some concluding remarks.2. Results
We begin with some notation. Recall that we denote the complex variable by s = σ + i t , and C σ = { σ > σ } . In addition, for a bounded interval I ⊂ R , weset C I = { s ∈ C : i( s − / / ∈ R \ I } . That is, C I is the complex plane with tworays on the abscissa σ = 1 / removed. Also, we take f ( x ) ∼ g ( x ) to mean that g ( x ) /f ( x ) → for x approaching some given limit, and by f ( x ) (cid:39) g ( x ) we meanthat there exists constants A, B > such that Af ( x ) ≤ g ( x ) ≤ Bf ( x ) for all x .If only one of the inequalities hold, we use the symbols (cid:46) and (cid:38) . We denote theLebesgue measure in the plane by d m .Next, we review the definition of the classical scale of spaces D α ( C / ) , whichcontains the Bergman ( α = − ), Hardy ( α = 0 ) and Dirichlet ( α = 1 ) spaceson the half-plane C / . Accordingly, we set D ( C / ) := H ( C / ) . For α < ,the space D α ( C / ) consists of the functions f analytic on the half-plane C / and OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 5 finite in the norm (cid:107) f (cid:107) D α ( C / ) = (cid:90) C / | f ( s ) | (cid:18) σ − (cid:19) − α − d m ( s ) . For < α ≤ , the space consists of functions analytic on C / for which f ( σ ) → as σ → ∞ , and which are finite in the norm (cid:107) f (cid:107) D α ( C / ) = (cid:90) C / | f (cid:48) ( s ) | (cid:18) σ − (cid:19) − α +1 d m ( s ) . The spaces D α ( C / ) are reproducing kernel spaces. I.e., given α ≤ and ξ ∈ C / ,there exists a function k ξ ( s ) , such that (cid:104) f | k ξ (cid:105) = f ( ξ ) . For α < these reproducingkernels are now given by k αξ ( s ) = c α ( s + ¯ ξ − α − , for the constants c α = ( − α )2 − α − when α < and c α = 2 α − (1 − α ) − for < α < . In the limiting case α = 1 , we have k αξ ( s ) = 1 π log 1 s + ¯ ξ − . To simplify the statements of our theorems, we define the following notions oflocal embeddings. Here we use the notation Ω I = (1 / , × I , where I ⊂ R is abounded interval. Definition 1.
Fix α ≤ . We say the space H w is locally embedded in the space D α ( C / ) if for every finite interval I there exists a constant C > such that, if α < then (cid:90) Ω I | F ( s ) | (cid:18) σ − (cid:19) − α − d m ( s ) ≤ C (cid:107) F (cid:107) H w , if α = 0 , then sup σ> / (cid:90) I | F ( σ + i t ) | d t ≤ C (cid:107) F (cid:107) H w , and if < α ≤ , then (cid:90) Ω I | F (cid:48) ( s ) | (cid:18) σ − (cid:19) − α +1 d m ( s ) ≤ C (cid:107) F (cid:107) H w . We can now formulate our first theorem. It generalizes the local embeddingtheorem mentioned in the introduction.
Theorem 1.
Let ( w n ) n ∈ N be a sequence of non-negative numbers, and α ∈ ( −∞ , .The following statements are equivalent.(a) There exists a constant C > such that for all x ≥ , (cid:88) n ≤ x w n ≤ C x (log x ) α . (b) H w is embedded locally into the space D α ( C / ) . JAN-FREDRIK OLSEN
By analogy to the prime number theorem, the inequality in ( a ) can be consideredas an upper Chebyshev-type inequality.Although we defer most proofs to later sections, we now give the simplest possi-ble illustration of how Chebyshev-type inequalities are connected to the functiontheoretic properties of the spaces H w . The argument is very similar to the one in[11]. Proof of ( a ) ⇒ ( b ) when α = 0 . For F ∈ H w and σ > / , we calculate by duality (cid:18)(cid:90) I | F ( σ + i t ) | d t (cid:19) / = sup g ∈ L (cid:107) g (cid:107) =1 (cid:90) I F ( σ + i t ) g (i t )d t = sup g ∈ L (cid:107) g (cid:107) =1 N (cid:88) n =1 a n n − σ (cid:90) I g (i t ) n − i t d t = √ π sup g ∈ L (cid:107) g (cid:107) =1 N (cid:88) n =1 a n ˆ g (log n ) n σ . If we multiply and divide by √ w n , apply the Cauchy-Schwarz inequality, and takethe appropriate limits, this yields (cid:18)(cid:90) I | F (1 / t ) | d t (cid:19) / (cid:46) (cid:107) F (cid:107) H w sup g ∈ L ( I ) (cid:107) g (cid:107) =1 (cid:88) n ≥ | ˆ g (log n ) | n w n (cid:124) (cid:123)(cid:122) (cid:125) ( ∗ ) . (2)The functions ˆ g are Fourier transforms of functions with compact support in afixed interval in R , which implies that they are very regular in the sense that for ξ ∈ ( k, k + 1) we get the easy estimate | ˆ g ( ξ ) | ≤ | ˆ g ( k ) | + (cid:107) ˆ g (cid:48) (cid:107) L ( k,k +1) . This issufficient to conclude, since by this estimate, the upper Chebyshev inequality for ( w n ) , and basic properties of the Fourier transform, we obtain ( ∗ ) = ∞ (cid:88) k =1 (cid:88) n ∈ (e k , e k +1 ) | ˆ g (log n ) | n w n ≤ ∞ (cid:88) k =1 | ˆ g ( k ) | + (cid:107) ˆ g (cid:48) (cid:107) L ( k,k +1) e k (cid:88) n ≤ e k +1 w n (cid:46) (cid:107) g (cid:107) L ( I ) . (cid:3) The following result generalizes the theorem on boundary functions mentionin the introduction (see also Theorem 4 below), and demonstrates the functiontheoretic significance of lower Chebyshev-type inequalities.
Theorem 2.
Let ( w n ) be a sequence of non-negative numbers and α ∈ ( −∞ , .If the upper Chebyshev-type inequality of (1) holds for this choice of α and ( w n ) ,then the following statements are equivalent: OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 7 (a) There exists a constant such that for all x ≥ , (cid:88) n ≤ x w n (cid:38) x (log x ) α . (b) For intervals I sufficiently small and every f ∈ D α ( C / ) there exists F ∈ H w such that f − F has an analytic continuation across the segment / I .Moreover, for every domain Γ at a positive distance from C \ C I , there existsa constant C such that (cid:107) f − F (cid:107) L ∞ (Γ) ≤ C (cid:107) f (cid:107) D α . The proof relies in a crucial way on the theory of sampling sequences, and isgiven in Section 5. See also remarks 4 and 5 below on the optimality of this result.The following result should be considered an application of the previous twotheorems, and the proof is given in Section 6.
Theorem 3.
Let ( w n ) be a sequence of non-negative numbers and α ∈ ( −∞ , .If both the Chebyshev-type inequalities (1) hold for this choice of α and ( w n ) , thenthe following statements are true.(a) The Carleson measures with compact support for H w and D α ( C / ) coincide.(b) The bounded interpolating sequences of H w and D α ( C / ) coincide. We state and prove a simple lemma which is used in the proof of this theoremas it offers a simple application of the Chebyshev-type inequalities. The proof ofTheorem 3 is given in Section 6.
Lemma 1.
Let ( w n ) be a sequence of non-negative numbers and α ∈ R . If boththe Chebyshev-type inequalities (1) hold for this α and ( w n ) , then for s = σ + i t in C / there are constants such that (cid:107) k D α s (cid:107) D α (cid:46) (cid:88) n ∈ N w n n − σ (cid:46) (cid:107) k D α s (cid:107) D α , as σ → / . Proof.
Denote the k -th partial sum of w n by W k . We sum the left-hand side byparts, and then apply the mean value theorem for σ ∈ (1 / , , to get (cid:88) n ∈ N n − σ w n = (cid:88) n ≥ W n ( n − σ − ( n + 1) − σ ) (cid:39) (cid:88) n ∈ N W n n − σ − . By an application of the Chebyshev-type inequalities, this is seen to be comparableto (cid:88) n ∈ N n − σ (log n + 1) α . The desired conclusion now follows exactly from [25, Lemma 3.1], which gives thebehavior of these weighted zeta-type functions as σ → . (cid:3) Next, we record a stronger version of Theorem 2, as it is more suited for theexamples we consider below. The proof is given in Section 7.
JAN-FREDRIK OLSEN
Theorem 4.
Suppose that for some constant
C > we have (cid:88) n ≤ x w n ∼ C x (log x ) α , as x → ∞ , (3) then part ( b ) of Theorem 2 holds for every finite interval I . Our first example asserts that the above results generalize those mentioned inthe introduction.
Example 1 (The Dirichlet-Hardy space and McCarthy’s spaces) . Let w n = (1 +log n ) α . For α = 0 , we have H w = H , and it is trivial to estimate the partialsums. So, theorems 1 and 3 reduce to the local embedding and interpolationtheorem, respectively, of the introduction, as well as the statement on the localequivalence of Carleson measures. Note that Theorem 2 reduces to a weaker resultthan the one on local boundary functions in the introduction, while Theorem 4,which holds in this and all of the following examples, reduces to exactly thistheorem. For general α ≤ , we get the same results, except in this case wehave to compare the space H w to D α ( C / ) . In this case, the spaces H w wereintroduced in [21], and the corresponding results are contained in [24, 25]. Recallthat the Dirichlet-Hardy space is identified with H ( D ∞ ) by Bohr’s observation,but for α (cid:54) = 0 there is no such natural identification as the monomials z n and z m may have different norms even if | m | = | n | for multi-indices m, n .Examples 2 through 5 explore natural analogues on D ∞ for the scale of spaceson D which include the Bergman, Hardy and Dirichlet spaces. To fix notation, welet f ( z ) = (cid:80) n ∈ N a n z n , and define norms by (cid:107) f (cid:107) A β ( D ) = (cid:90) D | f ( z ) | d m β ( z ) = (cid:88) n ∈ N | a n | n !( β + 1)( β + 2) · · · ( β + n ) , (4)where d m β ( z ) = (( β +1) /π )(1 − r ) β r d r d θ is a probability measure on D for β > ,and (cid:107) f (cid:107) D α ( D ) = (cid:88) n ∈ N | a n | ( n + 1) α . (5)(Note that the integral norm in (4) breaks down for β ≤ unless suitably modified.However, for β ∈ ( − , we only consider the coefficient norm.) Here we followthe notation of [10] and [33], respectively. To define the spaces A β ( D d ) and D α ( D d ) for d ∈ N ∪ {∞} while avoiding tedious notation, we content ourselves in sayingthat for the space D α ( D d ) , the monomials z ν . . . z ν d d form an orthogonal basis withnorm the square root of ( ν + 1) α · · · ( ν d + 1) α . For d = ∞ , the union of thesesystems of monomials form the orthogonal basis. For the spaces A β ( D d ) , with β > − , the corresponding statements holds in terms of the coefficient norms,while for β > one retains the identity (cid:107) f (cid:107) A β ( D d ) = (cid:90) D d | f ( z , . . . , z d ) | d m β ( z ) . . . d m β ( z d ) . (6) OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 9
Since (1 + β )(2 + β ) · · · ( n + β ) n ! (cid:39) (1 + n ) β , it follows that on the unit disk, or in fact on any finite polydisk, these spaces haveequivalent norms for α = − β and β > − . This no longer holds on D ∞ . Example 2 (The spaces A β ) . For γ > we define the numbers d γ ( n ) by therelation ζ ( s ) γ = (cid:80) n ∈ N d γ ( n ) n − s . By considering the Euler product, it is not hardto see that d γ ( p ν ) = γ ( γ + 1) . . . ( γ + ν − /ν ! . An explicit formula now extendseasily to n ∈ N since d γ ( kl ) = d γ ( k ) d γ ( l ) whenever k and l are relatively prime.We now define A β := H w for the weights w n = d β +1 ( n ) . By the above discussionand the Bohr correspondence, the spaces A β are isometrically identified with thespace A β ( D ∞ ) . Moreover, the space A β has translates of ζ ( s ) β as its reproducingkernel, and it follows by [14, Theorem 14.9] that for some constant C > , (cid:88) n ≤ x w n ∼ Cx (log x ) β . Hence, in the sense of theorems 1, 2, 3 and 4, the space A β behaves locally like D − β ( C / ) , as could be expected. Example 3.
In the limit as β → − + , the previous example leads us to alsoconsider the case when H w is the space of Dirichlet series with the reproducingkernel given by translates of log ζ ( s ) = (cid:88) p (cid:88) j p − js j := (cid:88) n ∈ N Λ( n )log n n − s . Here Λ( n ) is the von Mangoldt function. By a calculation, which gave the firstproof of the prime number theorem, von Mangoldt [35] found that (cid:88) n ≤ x Λ( n ) ∼ x. (The partial sum on the left-hand side of this asymptotic formula is usually calledthe Chebyshev function and is denoted by ψ ( x ) .) It now follows that the weights w n = Λ( n ) / log n satisfy (cid:88) n ≤ x w n ∼ x log x , whence, by theorems 1, 2, 3 and 4, the space H w behaves locally like D ( C / ) .Observe that by the same arguments, the space H w (cid:48) with weights w (cid:48) n = Λ( n ) ,which has translates of the derivative of log ζ ( s ) as its reproducing kernel, behaveslocally like the Hardy space H ( C / ) . Example 4 (The spaces D α ) . Let d ( n ) denote the number of divisors of the n ’thinteger. Explicitly, if n = p ν · · · p ν k k , where p k is the k ’th prime number and ν k ∈ N , then d ( n ) = ( ν + 1) · · · ( ν k + 1) . For α ∈ R , we set D α := H w for theweight w n = 1 /d ( n ) α . As with the weights of the previous two examples, it is veryirregular, since highly composite numbers and prime numbers may occur side by side among the natural numbers. Still, it follows by Ramanujan [28] and Wilson[38] that there exists a constant C > such that (cid:88) n ≤ x w n ∼ Cx (log x ) − α − . Hence, by theorems 1, 2, 3 and 4, the space D α behaves locally like D − − α ( C / ) .The previous example is surprising as one would expect the local behavior ofthe space D α to correspond to the classical space D α ( C / ) . We remark that inthis case, the embedding for α < was first observed by Seip [31]. Example 5 (The space D ∞ ) . In the previous example, as α → ∞ , it becomesmore difficult for functions of a given norm to have coefficients corresponding tocomposite numbers. So, as a limit space as α → ∞ , we suggest D ∞ = (cid:40) (cid:88) p prime a p p − s : (cid:88) p prime | a p | < ∞ (cid:41) . In other words, we make the choice of weights w n = (cid:26) if n is prime , else . By the Bohr correspondence, this space is identified with the subspace of H spanned by monomials, i.e. (cid:40)(cid:88) n ∈ N a n z n : (cid:88) n ∈ N | a n | < ∞ (cid:41) . By the prime number theorem (cid:88) n ≤ x w n ∼ x log x , whence we conclude that the space D ∞ behaves locally like the space D ( C / ) inthe sense of theorems 1, 2, 3 and 4.To better see the connection between two previous examples, we consider Figure2. We observe that as the parameter α of example 4 goes to ∞ then what we cancall the smoothness parameter − − α goes asymptotically to . This correspondsto the choice of D ∞ as an endpoint space, and its local connection to the space D ( C / ) appears natural. (See also Section 8.)Next, we move on to an example on the countably infinite dimensional ball B ∞ = (cid:26) ( z , z , · · · ) : (cid:88) i ∈ N | z i | < (cid:27) . In this context, we invoke Ikehara’s tauberian theorem explicitly. As indicated inthe introduction, it allows one to deduce the behavior of the growth of a sequence
OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 11 − − β β Figure 1. by considering functional theoretic properties of a related Dirichlet series. Theversion we state is due to Delange [6].
Theorem (Ikehara-Delange) . Let A ( x ) be a non-decreasing function with supportin (0 , ∞ ) , and for which the function F ( s ) = (cid:82) ∞ A ( x ) x − s − d x converges for σ >σ ≥ . Suppose that F ( s ) is holomorphic on a neighborhood of the puncturedhalf-plane C σ \{ σ } , and that for β < it holds in this neighborhood that F ( s ) = g ( s )( s − σ ) − β + h ( s ) , (7) for functions g, h analytic on a neighborhood of C σ with g ( σ ) (cid:54) = 0 . Then as x → ∞ it follows that A ( x ) ∼ c β x σ log β x . For β = 1 , the same conclusion holds when (7) is replaced by F ( s ) = g ( s ) log 1 s − σ + h ( s ) . We remark that this result can be stated in greater generality (see [17]). Alsonote that we wish to apply this theorem for σ (cid:54) = 1 , in which case the functionsin the space H w will be analytic on C σ / . We can still apply theorems 1 through4 by considering the shift F ( s − / σ / . This is tantamount to replacing theweight ( w n ) n ∈ N by ( n − σ w n ) n ∈ N . Example 6 (The Dirichlet-Besov-Sobolev spaces B γ ) . For γ ≥ the classicalBesov-Sobolev space on the countably infinite dimensional ball is given as B γ ( B ∞ ) = (cid:40)(cid:88) ν a ν z ν : (cid:88) ν | a ν | (cid:0) γ + | ν | ν (cid:1) < ∞ . (cid:41) . Here the multinomial coefficient is defined by (cid:18) | ν | + γν (cid:19) = γ ( γ + 1) · · · ( | ν | + γ − ν ! ν ! . . . , when γ > and with ( | ν | − as the denominator in the case that γ = 0 . Thesignificance of these coefficients is that the reproducing kernel is given by K γ ( z, w ) = (cid:88) ν (cid:18) | ν | + γν (cid:19) z ν w ν = (cid:26) (1 − (cid:80) z j ¯ w j ) − γ if γ > , − log(1 − (cid:80) z j ¯ w j ) if γ = 0 . Applying the Bohr correspondence, the space B γ ( B ∞ ) is seen to be isometricallyisomorphic to the space of Dirichlet series B γ with the reproducing kernel k γ ( s, ξ ) = (cid:26) (1 − ζ P ( s + ¯ ξ )) − γ if γ > , − log(1 − ζ P ( s + ¯ ξ )) if γ = 0 , where ζ P ( s ) = (cid:80) p prime p − s . Let ρ > be the unique number for which ζ P ( ρ ) = 1 .Clearly, ζ P is analytic on a neighborhood of C ρ , and has a simple zero at s = ρ .So, by the Ikehara-Delange theorem in combination with theorems 1, 3 and 4, thespace B γ behaves locally like the space D − γ ( C ρ/ ) . In particular we note thatthe space B , the Dirichlet-Drury-Arveson space, behaves locally like H ( C ρ/ ) .We mention one last example that appeared in [21]. It was introduced as anexample of a Hilbert space of Dirichlet series having a complete Pick kernel. Func-tion spaces with this property have attracted interest in the last decade or so. Forthe definition and related results see e.g. [1, 30, 33]. (The spaces in the previousexample have the complete Pick property when γ ∈ [0 , .) Example 7 (McCarthy’s space) . Let F ( n ) be the number of non-trivial ways tofactor the number n , counting order. E.g., F (10) = 3 since , × and × arethe non-trivial ways to factor this number. With this, McCarthy’s space is H w with the weight w n = 1 /F ( n ) . It is straight-forward to check that the reproducingkernel for this space is given by translates of the function (cid:88) n ∈ N F ( n ) n − s = 12 − ζ ( s ) . From this relation, it follows that H w has the complete Pick property. If ρ isthe number satisfying ζ ( ρ ) = 2 , it follows as in the previous example, that H w behaves locally as H ( C ρ / ) . Note that it is possible to compute weights anddetermine the local behavior for an entire scale of these spaces by taking thereproducing kernels to some power γ > , or by using the logarithm.3. preliminaries on sampling sequences and measures In this section we give the necessary background on sampling sequences andmeasures.
OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 13
Sobolev spaces.
To discuss the boundary behavior of functions in spaceswhich locally behave like D α ( C / ) , we need to introduce the restricted Sobolevspaces W α ( I ) and their dual spaces W − α ( I ) .Denote the space of tempered distributions by S (cid:48) ( R ) . For α ∈ R , we first definethe unrestricted Sobolev space W α ( R ) = (cid:26) u ∈ S (cid:48) ( R ) : ˆ u ∈ L and (cid:107) u (cid:107) W α := (cid:90) R | ˆ u ( ξ ) | (1 + | ξ | ) α d ξ < ∞ (cid:27) . For an open and (possibly unbounded) interval I ⊂ R , we let W α ( I ) be thesubspace of W α ( R ) that consists of distributions having support in I . By a scalingand mollifying argument one easily checks that this subspace coincides with theclosure of C ∞ ( I ) in the norm of W α ( R ) . With this, we define the Sobolev space W α ( I ) := W α ( R ) /W α ( R \ ¯ I C ) . In other words, the quotient space W α ( I ) contains the restrictions of distributionsin W α ( R ) to the interval I with the norm (cid:107) u (cid:107) W α ( I ) = inf v ∈ W α ( R ) v | I = u (cid:107) v (cid:107) W α ( R ) . Under the natural pairing ( u, v ) = (cid:82) R ˆ u ( ξ )ˆ v ( ξ )d ξ , the dual space of W α ( I ) isisometric to W − α ( I ) , as is readily verified. It is well-known that the functionsin the spaces D α ( C / ) have distributional boundary values that belong to theSobolev spaces W α/ ( I ) on bounded and open intervals I ⊂ R .3.2. Sampling sequences and measures for the Paley-Wiener space.
Let H be a Hilbert space of functions on some set Ω , with the property that for every µ ∈ Ω there exists a reproducing kernel k µ ∈ H . A sequence Λ = ( λ j ) is a samplingsequence for H if for all f ∈ H it holds that (cid:88) λ ∈ Λ | f ( λ ) | k λ ( λ ) (cid:39) (cid:107) f (cid:107) H . For a bounded interval I ⊂ R , we let L ( I ) denote the subspace of L ( R ) consistingof functions with support in I . The Paley-Wiener space PW( I ) may then bedefined as the image of L ( I ) under the Fourier transform. If the interval I iscentered at the origin, this space may also be described as the space of entirefunctions of exponential type | I | / that are square summable on R .The sampling sequences for PW( I ) on the real line have been given a partialdescription by Seip [32] and Jaffard [15], following an idea of Beurling [3, p. 345],in terms of the density D − (Λ) = lim r →∞ inf ξ ∈ R | ( ξ, ξ + r ) ∩ Λ | r . The result can be stated as follows.
Theorem (Beurling-Jaffard-Seip) . Let Λ ⊂ R be a sequence of distinct numbers. (i) If Λ is sampling for PW( I ) , then D − (Λ) ≥ | I | π .(ii) If D − (Λ) > | I | π , then Λ is sampling for PW( I ) . Following Ortega-Cerda [26], we call a positive measure µ on R a samplingmeasure for PW( I ) if there exists constants such that for all g ∈ PW( I ) it holdsthat (cid:90) R | g ( ξ ) | d ξ (cid:46) (cid:90) R | g ( ξ ) | d µ ( ξ ) (cid:46) (cid:90) R | g ( ξ ) | d ξ. (8)Note that if the right-hand inequality holds, we say that µ satisfies the Carleson-type inequality in (8), or that µ is a Carleson measure for P W ( I ) .Next, given any r > and (cid:15) > we define Λ µ ( r, (cid:15) ) = { k : µ ([ rk, r ( k + 1)]) ≥ (cid:15) } . The following result [27, Proposition 1] completely characterizes the samplingmeasures for
PW( I ) in terms of sampling sequences. Theorem (Ortega-Cerda and Seip 2002) . Let I ⊂ R be an interval centered at .A positive Borel measure µ is a sampling measure for PW( I ) if and only if:(i) There exists a constant C > such that µ ([ ξ, ξ + 1)) ≤ C for all ξ ∈ R .(ii) For all sufficiently small r > there exists a δ = δ ( r ) > such that Λ µ ( r, δ ) is sampling for PW( I ) .Remark . Condition ( i ) of the theorem alone is equivalent to the Carleson-typeinequality in (8) (see e.g. [26]).Combined with Beurling’s density result on sampling sequences, this theoremgives a condition on when a measure is a sampling measure (see Corollary 1 below).3.3. Sampling for weighted spaces.
We now combine some of the above resultsto extract a simple condition for sampling in the spaces F W β ( I ) := { ˆ f : f ∈ W β ( I ) } . To formulate it, we say that a positive Borel measure µ that satisfies (cid:90) R | ˆ f ( ξ ) | (1 + ξ ) β d ξ (cid:46) (cid:90) R | ˆ f ( ξ ) | (1 + ξ ) β d µ ( ξ ) (cid:46) (cid:90) R | ˆ f ( ξ ) | (1 + ξ ) β d ξ, (9)for all f ∈ W β ( I (cid:15) ) , is a sampling measure for F W β ( I ) . Proposition 1.
Let I be some bounded interval, and for (cid:15) > , denote by I (cid:15) theinterval co-centric with I such that | I \ I (cid:15) | = 2 (cid:15) . If µ is a sampling measure for F W β ( I ) for some β ∈ R , then for any (cid:15) > and β ∈ R it holds that µ is asampling measure for F W β ( I (cid:15) ) Proof.
Clearly, if (9) holds for a given β ∈ R , we can apply the derivative toextend it to β + 2 N . Similarly, using integration, N can be replaced with Z .However, to conclude for a general β , let h ∈ C ∞ ( − (cid:15), (cid:15) ) be a function with areal and positive Fourier transform, and set ψ β = h · F − { (1 + ξ ) ( β − β ) / } . For f ∈ C ∞ ( I (cid:15) ) it is now readily checked that the function f ∗ ψ β is in C ∞ ( I ) and thatthe estimate | (cid:92) f ∗ ψ β ( ξ ) | (cid:39) | ˆ f ( ξ ) | (1 + ξ ) β − β holds. (cid:3) OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 15
Remark . As follows from the proof, Proposition 1 also holds if we only considerthe Carleson-type inequality in (9).By combining the previous results, necessary and sufficient conditions for when ameasure is a sampling measure for F W β ( I ) now follow immediately. To formulatethem, we define Λ βµ ( r, δ ) = { k : ν ( rk, r ( k + 1)) ≥ δ (1 + ( rk ) ) β } . Corollary 1.
Suppose that ν is a sampling measure for F W β ( I ) . Then ν ( ξ, ξ +1) ≤ C (1 + ξ ) β , and given (cid:15) > by choosing sufficiently small r > , there exists δ = δ ( r ) > such that D − (Λ βµ ( r, δ )) ≥ | I (cid:15) | / π .Conversely, suppose that ν ( ξ, ξ + 1) ≤ C (1 + ξ ) β and that for sufficiently small r > there exists δ = δ ( r ) such that D − (Λ βµ ( r, δ )) > | I | / π . Then given (cid:15) > themeasure ν is sampling for F W β ( I (cid:15) ) .Remark . By combining remarks 1 and 2, it follows that a positive Borel measure ν on R is a Carleson measure on F W β ( I ) if and only if there exists some C > such that ν ( ξ, ξ + 1) ≤ C (1 + ξ ) β for all ξ ∈ R .3.4. Measures continuous at infinity.
The previous discussion simplifies formeasures µ which are continuous in the sense that for every (cid:15) > there exists an R < ∞ and h > such that µ ([ x, x + h ]) ≤ (cid:15) (1 + ξ ) β for all | x | ≥ R . We say thatsuch a measure is β -continuous at infinity.The following theorem is due to Ya. Lin [19]. Theorem (Lin) . Suppose that the positive Borel measure µ on R is -continuousat infinity. Then the measure µ is sampling for P W ( I ) , for every bounded interval I ⊂ R , if and only if, for some L > it holds that inf x ∈ R { µ ([ x − L, x + L ]) } > . By Proposition 1, the above theorem has the following immediate consequence.
Corollary 2.
Suppose that the positive Borel measure µ on R is β -continuous atinfinity. Then the measure µ is sampling for F W β ( I ) , for every finite interval I ⊂ R , if and only if we have inf ξ ∈ R { µ ([ ξ − L, ξ + L ])(1 + ξ ) − β } > for some L > . Proof of Theorem 1
We begin with an elementary lemma that will be used in several argumentsbelow.
Lemma 2.
Let ( w n ) be sequence of non-negative numbers, and suppose that α ∈ R .Then ∃ η ∈ (0 , s.t. (cid:88) n ∈ ( ηx,x ) w n (cid:46) x (log x ) α ⇐⇒ (cid:88) n ∈ (0 ,x ) w n (cid:46) x (log x ) α . Moreover, suppose that the upper Chebyshev-type inequality holds, then ∃ η ∈ (0 , s.t. (cid:88) n ∈ ( ηx,x ) w n (cid:38) x (log x ) α ⇐⇒ (cid:88) n ∈ (0 ,x ) w n (cid:38) x (log x ) α . Proof.
It is clear that for each statement, one implication is trivial. As for the‘ = ⇒ ’ part of the first statement, note that (cid:88) n ≤ e ξ w n = (cid:88) k ≤ ξ (cid:88) n ∈ (e k − , e k ) w n ≤ C (cid:88) k ≤ ξ e k k − α = C e ξ ξ − α (cid:88) k ≤ ξ e k − ξ (cid:18) kξ (cid:19) − α . This gives the desired conclusion since, by a simple calculation, the sum on theright-hand side is bounded by a constant.Finally, ‘ ⇐ ’ part of the second statement follows from an argument by contra-diction. Indeed, assume it holds for no η > , and set ψ ( x ) = x (log x ) − α . Thenthere exist sequences x k → ∞ and η k → for which (cid:88) n ∈ ( η k x k ,x k ) w n ≤ ψ ( x k ) k . Then, this, and the upper Chebyshev-type inequality, imply (cid:88) n ∈ (1 ,x k ) w n ≤ (cid:88) n ∈ (1 ,η k x k ) w n + (cid:88) n ∈ ( η k x k ,x k ) w n (cid:46) ψ ( η k x k ) + ψ ( x k ) k . Applying the lower Chebyshev inequality to the left-hand side now yields a con-tradiction, since the quotient ψ ( η n x n ) /ψ ( x n ) goes to zero as k → ∞ . (cid:3) Proof. ( a ) ⇒ ( b ) : In the introduction we have already proved this for the case α = 0 . For α (cid:54) = 0 , the argument holds with minor modifications. Namely, if wemultiply and divide by (cid:112) log α nw n on the right-hand side of (2), then for σ > / we obtain (cid:90) I | F ( σ +i t ) | d t ≤ (cid:32) N (cid:88) n =1 | a n | w n n ) α n σ − (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) ( ∗ ) sup g ∈ L ( I ) (cid:107) g (cid:107) =1 N (cid:88) n =1 | ˆ g (log n ) | (log n ) α w n n (cid:124) (cid:123)(cid:122) (cid:125) ( ∗∗ ) . The factor ( ∗∗ ) can be dealt with exactly as before, using the compact support ofthe functions g , to yield ( ∗∗ ) ≤ C . For α < , we use this to evaluate (cid:90) / (cid:90) I | F ( σ + i t ) | (cid:18) σ − (cid:19) − α − d t d σ ≤ C N (cid:88) n =1 | a n | w n n ) α (cid:90) / n − (2 σ − (cid:18) σ − (cid:19) − α − d σ (cid:39) (cid:107) F (cid:107) H w . A similar argument holds for α ∈ (0 , . OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 17 ( b ) ⇒ ( a ) : Define the function g k ( s ) := (cid:80) n ∈ (e k , e k +1 ) w n n − s . Suppose that α < .Then the local embedding of H w into D α ( C / ) implies that for any δ > thereexists a constant C > such that (cid:90) / (cid:90) δ − δ | g k ( s ) | (cid:18) σ − (cid:19) − − α d σ d t ≤ C (cid:88) n ∈ (e k , e k +1 ) w n . By expanding | g k ( s ) | , we find that the left-hand side of the above expression isequal to δ (cid:88) n,m ∈ (e k , e k +1 ) w n w m sin δ ln( n/m ) δ ln( n/m ) (cid:90) / ( nm ) − σ (cid:18) σ − (cid:19) − − α d σ. We fix δ > small enough so that δ ln( n/m ) ≤ π/ . By evaluating the integralwith respect to σ , then up to a constant the previous expression is seen to begreater than or equal to (cid:88) n,m ∈ (e k , e k +1 ) w n w m (log nm ) α √ nm ≥ (2 k ) α e k +1 (cid:88) n ∈ (e k , e k +1 ) w n . By combining the above estimates, we obtain (cid:88) n ∈ (e k , e k +1 ) w n (cid:46) e k k α . By Lemma 2, this implies the desired conclusion. The cases α = 0 and < α < are treated in the same way. (cid:3) In the paper [24], a more operator theoretic perspective was taken. This madeit possible to study the span of the boundary values of functions in the Dirichlet-Hardy space H , as well as the more general spaces introduced by McCarthy(see Example 1). In the next section, we utilise this point of view to study theboundary spaces of the spaces H w . To prepare for this, we introduce the denselydefined embedding operator E I : N (cid:88) n =1 a n n − s (cid:55)−→ (cid:32) N (cid:88) n =1 a n n − − i t (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) t ∈ I , and establish the following result. Lemma 3.
Let ( w n ) be a sequence of non-negative numbers, and α ∈ ( −∞ , .Then the conditions of the previous theorem are equivalent to either of the state-ments:(a’) For intervals I of fixed length, there exists a constant such that for all f ∈ W − α/ ( I ) then (cid:88) n ∈ N | ˆ f (log n ) | n w n (cid:46) (cid:107) f (cid:107) W − α/ ( I ) . (b’) The operator E I is bounded from H w to W α/ ( I ) .Proof. The equivalence ( a (cid:48) ) ⇔ ( b (cid:48) ) is obtained exactly as in [24]. One simplycomputes that the adjoint operator of E I with respect to the natural non-weightedpairings is given by E ∗ I : g ∈ W − α/ ( I ) (cid:55)−→ (cid:88) n ∈ N ˆ g (log n ) √ n n − s ∈ H /w . To establish ( a (cid:48) ) ⇔ ( a ) , we observe that using the measure ν = (cid:88) n ∈ N δ log n w n n , it is clear that ( a (cid:48) ) is equivalent to the inequality (cid:90) | ˆ f ( ξ ) | d ν (cid:46) (cid:90) | ˆ f ( ξ ) | (1 + ξ ) − α/ d ξ. By Remark 3, this is in turn equivalent to ν ( ξ, ξ + 1) (cid:46) (1 + ξ ) − α/ . It is plain tosee that this is equivalent to (cid:88) n ∈ (e ξ − , e ξ ) w n (cid:46) e ξ ξ − α , ∀ ξ ≥ . By Lemma 2, this gives the desired conclusion. (cid:3) Proof of Theorem 2
Continuing in the spirit of the previous lemma, we take the same approach asin [24]. Namely, inspired by the embedding operator E I , which we considered inthe previous section, we define an operator R I : ( a n ) n ∈ Z ∗ (cid:55)−→ (cid:32)(cid:88) n ∈ N a n n − i t + a − n n i t √ n (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I . Essentially a direct sum of two embedding operators, it allows us the flexibility tostudy the real parts of functions in H w . To this end, we define the space (cid:96) w ( Z ∗ ) = { ( a n ) n ∈ Z ∗ : (cid:88) n ∈ N ( | a n | + | a − n | ) /w n < ∞} . With this, we establish the following lemma.
Lemma 4.
Let ( w n ) be a sequence of non-negative numbers and α ∈ R , andsuppose that the upper Chebyshev-type inequality for ( w n ) holds with this α . Thenthe following are equivalent:(a’) For intervals I sufficiently small, there exists constants such that A (cid:107) f (cid:107) W − α/ ( I ) ≤ (cid:88) n ∈ N | ˆ f (log n ) | + | ˆ f ( − log n ) | n w n . OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 19 (b’) For intervals I sufficiently small, the operator R I : (cid:96) w ( Z ∗ ) → W α/ ( I ) isbounded and onto.(c’) For intervals I sufficiently small, then for every f ∈ D α ( C / ) there exists an F ∈ H w such that the real part of f − F vanishes on / I , and therefore f − F has an analytic continuation across this segment.Proof. In light of Lemma 3, the equivalence of ( a (cid:48) ) ⇔ ( b (cid:48) ) , as well as the implication ( b (cid:48) ) ⇒ ( c (cid:48) ) is proved more or less exactly as in [24]. The implication ( c (cid:48) ) ⇒ ( b (cid:48) ) follows by the same line of reasoning. Indeed, Let h ( t ) ∈ W α ( I ) be given, andwrite h = u + i v . By hypothesis, there exists F, G ∈ H w such that Re F = u and Re G = v on / I . If we write F = (cid:88) ( α n + i β n ) n − s and G = (cid:88) ( γ n + i δ n ) n − s , it follows by considering real parts and imaginary parts, that R I ( c n ) (cid:55)−→ (cid:88) n ∈ N c n n − i t + c − n n i t √ n = h, where c n = α n + γ n δ n + β n ,c n = α n − γ n δ n − β n . (cid:3) We are now ready to prove the theorem.
Proof of Theorem 2.
By the previous lemma, it suffices to show ( a ) ⇔ ( a (cid:48) ) , ( c (cid:48) ) ⇒ ( b ) and ( b ) ⇒ ( b (cid:48) ) . ( c (cid:48) ) ⇒ ( b ) : In order to get the analytic continuation, the idea is to expressthe difference f − F by using a Szegő-type integral. The norm estimates thenfollow in a straight-forward manner. For the Dirichlet-Hardy space, this is provenin detail as a part of [24, Theorem 1]. For more general weights satisfying theChebyshev-type inequalities for α = 0 , this proof can be repeated word by word.For α (cid:54) = 0 , the necessary adjustments to the argument are outlined in the proofof [24, Theorem 5]. ( a ) ⇔ ( a (cid:48) ) : We define the measure ν = (cid:88) n ∈ N (cid:16) δ log n + δ − log n (cid:17) w n n . In Lemma 3, we established that the upper Chebyshev inequality is equivalent tothe Carleson-type inequality, so we may assume that it holds. As a consequence, ( a (cid:48) ) holds if and only if ν is a sampling measure for F W − α/ ( I ) . Suppose that ν is a sampling measure for F W − α/ ( I ) . By Corollary 1, it followsthat for some r > and δ > , then Λ r,δ has positive density. In particular, for m > and sufficiently large ξ , we have | Λ ∩ ( ξ − rm, ξ ) | ≥ C > . So, for sufficiently large ξ there exists k ∈ N with kr ∈ ( ξ − ( m + 1) r, ξ − r ) forwhich ν ( kr, ( k + 1) r ) ≥ δ ( kr ) − α/ . This yields (cid:88) n ∈ (1 , e ξ ) w n (cid:38) e kr (cid:88) n ∈ (e kr , e ( k +1) r ) w n n ≥ e kr ( kr ) − α/ (cid:39) e ξ ξ − α/ . We turn to the converse. By Lemma 2, if the lower Chebyshev inequality holds,then there exists η ∈ (0 , so that for large enough x we have (cid:88) n ∈ ( ηx,x ) w n (cid:38) x (log x ) − α . By setting η = e − q and x = e ξ , this implies that there exists C > such that for ξ large enough we have ν ( ξ − q, ξ ) ≥ Cξ − α . In other words, the set Λ = { k : ν ( kq, ( k + 1) q ) ≥ C (1 + ( kr ) ) − α/ } has density D − (Λ) = 1 /q . By basic considerations, it also follows that for r < q ,the same holds when the constant C is suitably reduced. Hence, by Corollary 1,it follows that ν is a sampling measure for F W − α/ ( J ) , whenever | J | ≤ π/q . ( b ) ⇒ ( b (cid:48) ) : We use the following basic lemma from operator theory. We referthe reader to e.g. [39, Lemma 4, p. 182] for a proof. Lemma 5.
Suppose that
X, Y are Banach spaces, and that Z : X → Y is a closedlinear operator. Let B X and B Y denote the unit balls of X and Y , respectively. Ifthere exists a number M > such that for every y ∈ B Y there exists x ∈ M B X for which (cid:107) Zx − y (cid:107) ≤ / , then Z is surjective. Suppose that the interval J is such that R J satisfies part ( b ) of Theorem 2. Weshow that ( b (cid:48) ) holds for sufficiently small co-centric intervals I ⊂ J . To this end,suppose that h in the unit ball of W α/ ( I ) is given, and let f, g ∈ D α ( C / ) befunctions such that f + ¯ g has h as its boundary distribution on / I . Since h isthe restriction of a compactly supported distribution, it is not hard to use Laplacetransforms to show that f, g can be chosen so that both (cid:107) f (cid:107) D α and (cid:107) g (cid:107) D α aresmaller than or equal to some constant B , independent of h . We then apply part ( b ) of Theorem 2 to the disc Γ which has the segment / I as a diameter. Since Γ has a positive distance to C \ C J , there exists a constant C > , only dependingon Γ and I , and F, G ∈ H w such that both φ := f − F and ψ := g − G extendsanalytically across / I and satisfies sup s ∈ Γ | φ ( s ) + ψ ( s ) | ≤ C (cid:107) h (cid:107) W α/ = C. (10) OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 21
With this, and a slight abuse of notation, we get R I ( F + G ) = h + ( φ + ψ ) | I . By a straightforward computation using (10), we get (cid:107) φ + ψ (cid:107) W α/ ( I ) ≤ C | I | .Choosing I so small that | I | ≤ / C , we invoke Lemma 5 to get the desiredconclusion. (cid:3) We now give two remarks which shows that Theorem 2 cannot be improved.
Remark . There exist sequences w n which satisfy both an upper and lower Cheby-shev inequality for the same α , but for which H w only reproduces boundaryfunctions on small intervals. Indeed, for each k ∈ N , choose n k ∈ N such that log n ∈ ( k − / , k + 1 / . For n ∈ N , set w n = n if n = n k for some k , and w n = 0 otherwise. We extend this to negative indices by the rule w − n = w n . Onthe one hand, it now follows by Kadec’s / theorem (see e.g. [39, p. 36, Theorem14]) that ν = (cid:80) δ log n w n /n is a sampling measure for PW( − π, π ) . On the otherhand, by the Beurling-Jaffard-Seip density theorem above, this sequence cannotbe sampling for PW( − π − (cid:15), π + (cid:15) ) for any (cid:15) > . Remark . As an example of a sequence w n for which H w reproduces boundaryvalues, but for which the embedding does not hold. One can choose w n as inthe remark above, and set w (cid:48) n = e n for n (cid:54) = n k for any k . Then the space H w (cid:48)(cid:48) ,where w (cid:48)(cid:48) n = w n + w (cid:48) n reproduces boundary values of PW( − π, π ) since H w ⊂ H w (cid:48)(cid:48) .However, no embedding of the type that we have considered holds.6. Proof of Theorem 3
Proof Theorem 3. ( a ) : Suppose that µ is a Carleson measure for D α ( C / ) withcompact support. Since the space H w is embedded into D α ( C / ) for some α ∈ R ,it holds that for N ∈ N large enough then F ∈ H w implies F ( s ) /s N ∈ D α ( C / ) .Hence, (cid:90) C / | F ( s ) | d µ ( s ) (cid:46) (cid:90) C / (cid:12)(cid:12) F ( s ) /s N (cid:12)(cid:12) d µ ( s ) (cid:46) (cid:107) F/s (cid:107) D α (cid:46) (cid:107) F (cid:107) H w . As for the converse, we use the fact that by Theorem 2, for every f ∈ D α ( C / ) and interval I , there exists and F ∈ H w such that F − f has an analytic extensionacross the segment / I . Hence, by choosing the interval I large enough, weget (cid:90) C / | f | d µ ≤ (cid:90) C / | F | d µ + (cid:90) C / | f − F | d µ (cid:46) (cid:107) F (cid:107) H w + C. The conclusion now follows either by the norm control offered by Theorem 2, orthe closed graph theorem. ( b ) : The following argument is different from the one found in [25], which wasapplied in the case of the weights w n = (log n ) α , as it avoids use of the reproducingkernels beyond their role as norms for point evaluations. Instead, it relies on Lemma 5 and part ( b ) of Theorem 2. One direction is simple, and follows by thesame arguments as in [25]. Indeed, by Theorem 1, the space H w is embeddedlocally into the space D α ( C / ) . As above, it follows that if F ∈ H w , then forsome N ∈ N large enough, we have F/s N ∈ D α ( C / ) . Since we are dealing withbounded interpolating sequences, the problem F ( s n ) = w n has a solution if andonly if F ( s n ) = w n s Nn does. Hence, f ( s ) = F ( s ) /s N is a function in D α ( C / ) thatsolves the problem f ( s n ) = w n .We turn to the converse. Assume that S = ( s n ) n ∈ N is a bounded interpolatingsequence for D α ( C / ) , and write k D α n and k H w n for the reproducing kernels at s n of the respective spaces. This means that the interpolation operator defined by f ∈ D α ( C / ) (cid:55)−→ (cid:18) f ( s n ) (cid:107) k D α n (cid:107) D α (cid:19) n ∈ N ∈ (cid:96) is bounded and onto (cid:96) . Since (cid:107) k D α n (cid:107) D α (cid:39) (cid:107) k H w n (cid:107) H w , as follows from Lemma 1,the same remains true if we replace the weights of the operator by / (cid:107) k H w n (cid:107) H w .Next, without loss of generality, we may assume that the sequence ( s n ) n ∈ N satisfies σ n +1 ≥ σ n . With this in mind we set S N = ( s n ) n ≥ N and define the operators T N : F ∈ H w (cid:55)−→ (cid:18) F ( s n ) (cid:107) k H w n (cid:107) H w (cid:19) n ≥ N . By the same reasoning as in the proof of [25, Thm. 2.1], it follows that if T N issurjective for some N ∈ N , then the operator T is also surjective. The argumentuses Lagrange-type sums of finite products of the type N (cid:89) j =1 (cid:16) − p − ( s − s j ) j (cid:17) , where the p j are prime numbers.Next, we show that T N is onto for large enough N . So, suppose that b j is in theunit ball of (cid:96) , and assume first that there exists f in the unit ball of D α ( C / ) ,such that f ( s j ) = b j (cid:107) k H w j (cid:107) H w for j ∈ N . (In general, it only follows by the openmapping theorem that such an f exists in some dilation of the unit ball.)To apply part ( b ) of Theorem 2, let Γ be an open disk in C / that contains S , and let I ⊂ R be some bounded interval such that sup (cid:8) | Im s | : s ∈ I (cid:9) ≥ (cid:8) | Im s | : s ∈ Γ (cid:9) . Now, since Γ is at a positive distance from C \ C I , thereexist a constant C > , only depending on Γ and I , and F ∈ H w such that φ := f − F extends analytically across / I and satisfies sup s ∈ Γ | φ ( s ) | ≤ C (cid:107) f (cid:107) D α ( C / ) . It now follows, with a slight abuse of notation, that T N F ( n ) = T N f ( n ) + T N ( F − f )( n ) = b n + φ ( s n ) / (cid:107) k H w n (cid:107) H w . OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 23
So, to conclude by Lemma 5, we need to show that for N large enough, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) φ ( s n ) (cid:107) k H w n (cid:107) H w (cid:19) n ≥ N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ≤ / . (11)But this follows immediately as φ ( s n ) is uniformly bounded, and the sequence (1 / (cid:107) k H w n (cid:107) H w ) n ∈ N is square summable. (cid:3) Proof of Theorem 4
Let I be any bounded interval in R . In light of part ( a (cid:48) ) of Lemma 4 it sufficesto show that there exist constants such that for all f ∈ C ∞ ( I ) we have (cid:107) f (cid:107) W − α/ ( I ) (cid:46) (cid:88) n ∈ N | ˆ f (log n ) | + | ˆ f ( − log n ) | n w n (cid:46) (cid:107) f (cid:107) W − α/ ( I ) . By definition this is equivalent to the measure µ = (cid:88) n ∈ N δ log n + δ − log n n w n being a sampling measure for F W − α/ . To apply Corollary 2, we first need tocheck that µ is ( − α/ -continuous at infinity.For L > and ξ > we get µ [( ξ − L, ξ )] = (cid:88) n ∈ (e ξ − L , e ξ ) w n n (cid:46) e − ξ (cid:18) (cid:88) n ≤ e ξ w n − (cid:88) n ≤ e ξ − L w n (cid:19) . Given (cid:15) > , we choose ξ large enough for (3) to yield ( C + (cid:15) ) ξ − α − ( C − (cid:15) )e − L ( ξ − L ) − α . Clearly, by choosing L > small, and letting ξ be large enough, we obtainthe desired inequality µ [( ξ − L, ξ )] ≤ (cid:15) (1 + ξ ) − α/ . To complete the proof, we need to check that there exists some
L > such that inf ξ ∈ R µ [( ξ − L, ξ )](1 + ξ ) α/ > . But by what was done in the proof of ( a ) ⇔ ( a (cid:48) ) in Theorem 2, there exists L > such that for large enough ξ we have (cid:88) n ∈ (e ξ − L , e ξ ) w n (cid:38) e ξ ( ξ − L ) − α . By Lemma 2, the conclusion now follows. further remarks It is possible to define the spaces A β of example 2 when β > for general p (cid:54) = 2 using the expression (6). By [5], the resulting function spaces of Dirichlet serieshave bounded point evaluations on C / . That the same is true for the spaces D α of example 4, for α < , is less obvious. However, as it is possible to solve theHausdorff moment problem ( n + 1) α = (cid:82) r n +1 d ν α , for some positive measure ν α ,one obtains a radial probability measure on D (see e.g. [37, Chapter III]). Thisyields the required integral expression for the norm on polydisks.By the previous remark, it is not hard to determine the multiplier algebras ofthese spaces. In the language of [5], it is clear that the multipliers of the spaces A β ( D ∞ ) and D α ( D ∞ ) are exactly the elements of the spaces H ∞ for the respectiveinfinite product measures. But as these measures are products of radial probabilitymeasures supported on ¯ D , it was shown in [5, Theorem 11.1] that these spaces aresimply H ∞ ( T ∞ ) . As explained in [11] for the space H , which we identified withthe space H ( T ∞ ) in the introduction, it now follows that the multiplier algebraof both the spaces A β and D α is H ∞ = (cid:110) (cid:88) a n n − s : sup Re s> | (cid:88) a n n − s | < ∞ (cid:111) . The same argument holds for any p ≥ . Recently, similar results were obtainedfor p ∈ (0 , for function spaces on finite polydisks by Harutyunyan and Lusky[9].Our next remark concerns a consequence of an improvement of an inequality ofHardy and Littlewood. Mateljevic [20] showed that the constant C = 1 is bestpossible in the inequality (cid:88) n | a n | ( n + 1) ≤ C (cid:90) T | f (e i t ) | d tπ . (12)We remark that the proof of the latter fact was essentially known in the smoothcase to Carleman, who considered only the finite Blaschke products, and wasproved in full generality by Mateljevic using the same method. Since it seems thathis paper did not become widely known, the same proof was later rediscovered byVukotic [36]. In language of Dirichlet series, Helson [13] exploited this preciseestimate to prove, using a method due to Bayart, that (cid:107) F (cid:107) D − ≤ (cid:107) F (cid:107) H . Our observation is that by following the classical proof of the Riesz-Thorin in-terpolation theorem, it is possible to interpolate between (12) and the Plancherelidentity for p = 2 to obtain (in the notation of example 4) (cid:107) f (cid:107) D − /p ( D ) ≤ (cid:107) f (cid:107) H p ( D ) . (13)By generalising the argument of Bayart and Helson, this yields (cid:107) F (cid:107) D − /p ≤ (cid:107) F (cid:107) H p . OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 25
With respect to Figure 2, this family of inequalities takes place between the twopoints of intersection between the curves which represent the "smoothness" of thespaces of D α ( C / ) and D α I.e., at the points α = − and α = 0 , where the space D α behaves locally as one would expect.In addition to the local embeddings discussed above, others are possible. Forinstance, Seip observed that it follows from (13) and a duality argument that D α is locally embedded into the space H α +1 ( C / ) . Specifically, given a boundedinterval I , then there exists a constant C > such that for f ∈ D α we have sup σ> / (cid:90) I | f ( σ + i t ) | α +1 ≤ C (cid:107) f (cid:107) α +1 D α . We point out the the best possible constant of (13) is not needed for this argument.Finally, we mention that the Helson-Bayart inequality mentioned above is usedin [13] to prove a special case of the Nehari lifting theorem for Hankel forms onthe Hardy space H ( T ∞ ) . A Hankel form in countably infinitely many variablesis defined by ( a j , b j ) := (cid:88) j,k ∈ N a j b k ρ jk , where j and k are multiplied in the index of ρ jk . (Note that the one variabledefinition is retrieved by only summing over indices j = 2 m .) The result of Helsonsays that if the Hankel form is a Hilbert-Schmidt operator, then there exists afunction φ in L ∞ ( T ∞ ) such that ˆ φ ( n ) = ρ n for n ∈ N . The connection is that theHilbert-Schmidt condition is exactly (cid:88) j,k ∈ N | ρ jk | = (cid:88) n ∈ N d ( n ) | ρ n | < ∞ , where d ( n ) is the number of divisors function (see Example 4). By the Helson-Bayart inequality, the solution now follows by a duality argument. In the generalcase, the problem has been settled by Ferguson and Lacey on the bidisk [7] andLacey and Terwilleger on polydisks of finite dimension [18], but it remains openon the infinite dimensional polydisk. (See also [12, p. 54] for a discussion of thisproblem.) Acknowledgements
Parts of this paper is based on research done during the work on the PhDthesis of the author, and he would therefore like to thank his supervisor professorKristian Seip for advice and access to the unpublished note [31]. The author wouldalso like to thank professor Eero Saksman for valuable conversations regarding theproof of Theorem 3, and Anders Olofsson for suggesting example 2.
References
1. Jim Agler and John E. McCarthy,
Pick interpolation and Hilbert function spaces , GraduateStudies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002.
2. Frédéric Bayart,
Hardy spaces of Dirichlet series and their composition operators , Monatsh.Math. (2002), no. 3, 203–236.3. Arne Beurling,
The collected works of Arne Beurling. Vol. 2 , Contemporary Mathematicians,Birkhäuser Boston Inc., Boston, MA, 1989, Harmonic analysis, Edited by L. Carleson, P.Malliavin, J. Neuberger and J. Wermer.4. Harald Bohr,
Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorieder Dirichletschen Reihen (cid:80) a n /n s , Nachr. Akad. Wiss. Gˆttingen Math.-Phys. Kl. (1913),441–488.5. Brian J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinitepolydisk algebra , Proc. London Math. Soc. (1986), no. 3, 112–142.6. Hubert Delange, Généralisation du théorème de ikehara , Ann. Sci. Ec. Norm. Sup. (1954),no. 3, 213–242.7. Sarah H. Ferguson and Michael T. Lacey, A characterisation of product BMO by commuta-tors , Acta. Math. (2002), no. 2, 143–160.8. Julia Gordon and Håkan Hedenmalm,
The composition operators on the space of Dirichletseries with square summable coefficients. , Michigan Math. J. (1999), no. 2, 313–329.9. A. V. Harutyunyan and W. Lusky, Bounded operators on the weighted spaces of holomorphicfunctions on the polydiscs , Complex Var. Elliptic Equ. (2009), no. 1, 23–40.10. Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces , GraduateTexts in Mathematics, vol. 199, Springer-Verlag, New York, 2000.11. Håkan Hedenmalm, Peter Lindqvist, and Kristian Seip,
A Hilbert space of Dirichlet seriesand systems of dilated functions in L (0 , , Duke Math. J. (1997), 1–37.12. Henry Helson, Dirichlet Series , Regent press, 2005.13. ,
Hankel forms and sums of random variables , Studia Math. (2006), no. 1, 85–92.14. Aleksandar Ivić,
The Riemann zeta-function , Dover Publications Inc., Mineola, NY, 2003,Theory and applications, Reprint of the 1985 original [Wiley, New York; MR0792089(87d:11062)].15. S. Jaffard,
A density criterion for frames of complex exponentials , Michigan Math. J. (1991), no. 3, 339–348.16. Sergei V. Konyagin and Hervé Queffélec, The translation in the theory of Dirichlet series ,Real Anal. Exchange (2001/02), no. 1, 155–175.17. Jacob Korevaar, Distributional Wiener-Ikehara theorem and twin primes , Indag. Math.(N.S.) (2005), no. 1, 37–49.18. Michael Lacey and Erin Terwilleger, Hankel operators in several complex variables and prod-uct BMO , Houston J. Math. (2009), no. 1, 159–183.19. V. Ja. Lin, On equivalent norms in the space of square integrable entire functions of expo-nential type , Mat. Sb. (N.S.)
67 (109) (1965), 586–608.20. M. Mateljević,
The isoperimetric inequality in the Hardy class H , Mat. Vesnik (1979), no. 2, 169–178.21. John E. McCarthy, Hilbert spaces of Dirichlet series and their multipliers , Trans. Amer.Math. Soc. (2004), no. 3, 881–893.22. Hugh L. Montgomery,
Ten Lectures on the Interface Between Analytic Number Theory andHarmonic Analysis , CBMS Regional Conference Series in Mathematics, vol. 84, AMS, 1994.23. Anders Olofsson,
On the shift semigroup on the hardy space of dirichlet series , Acta Math-ematica Hungarica (2010), no. 3, 265–286.24. Jan-Fredrik Olsen and Eero Saksman,
Some local properties of functions in Hilbert spaces ofDirichlet series , (2010), to appear in J. Reine Angew. Math.25. Jan-Fredrik Olsen and Kristian Seip,
Local interpolation in Hilbert spaces of Dirichlet series ,Proc. Amer. Math. Soc. (2008), 203–212.26. Joaquim Ortega-Cerdà,
Sampling measures , Publ. Mat. (1998), no. 2, 559–566. OCAL PROPERTIES OF HILBERT SPACES OF DIRICHLET SERIES 27
27. Joaquim Ortega-Cerdà and Kristian Seip,
Fourier frames , Ann. of Math. (2) (2002),no. 3, 789–806.28. Srinivasan Ramanujan,
Some formulae in the analytic theory of numbers , Messenger Math. (1916), 81–84.29. Eero Saksman and Kristian Seip, Integral means and boundary limits of Dirichlet series ,Bull. Lond. Math. Soc. (2009), no. 3, 411–422.30. Eric T. Sawyer, Function theory: interpolation and corona problems , Fields Institute Mono-graphs, vol. 25, American Mathematical Society, Providence, RI, 2009.31. Kristian Seip,
Embeddings associated with the divisor function , Unpublished note.32. Kristian Seip,
On the connection between exponential bases and certain related sequences in L ( − π, π ) , J. Funct. Anal. (1995), no. 1, 131–160.33. Kristian Seip, Interpolation and Sampling in Spaces of Analytic Functions , University Lec-ture Series, vol. 33, American Mathematical Society, Providence, R. I., 2004.34. Gérald Tenenbaum,
Introduction to Analytic and Probabilistic Number Theory , english ed.,Cambridge studies in advanced mathematics, vol. 46, Cambridge University Press, 1995.35. Hans von Mangoldt,
Zu Riemanns’s Abhandlung "Über die Anzahl..." , J. Reine Angew.Math. (1895), 255–305.36. Dragan Vukotic,
The isoperimetric inequality and a theorem of Hardy and Littlewood , Amer.Math. Monthly (2003), 532–536.37. David Vernon Widder,
The Laplace Transform , Princeton Mathematical Series, v. 6, Prince-ton University Press, Princeton, N. J., 1941.38. B. M. Wilson,
Proofs of some formulae enunciated by Ramanujan , London M. S. Proc. (1922), no. 2, 235–255.39. Robert M. Young, An Introduction to Nonharmonic Fourier Series , first ed., Academic PressInc., New York, NY, 1980.
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