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On higher order estimates in quantum electrodynamics

We propose a new method to derive certain higher order estimates in quantum electrodynamics. Our method is particularly convenient in the application to the non-local semi-relativistic models of quantum electrodynamics as it avoids the use of iterated commutator expansions. We re-derive higher order estimates obtained earlier by Fr\"ohlich, Griesemer, and Schlein and prove new estimates for a non-local molecular no-pair operator.


Introduction
The main objective of this paper is to present a new method to derive higher order estimates in quantum electrodynamics (QED) of the form for all n ∈ AE, where C > 0 is sufficiently large. In these bounds H f denotes the radiation field energy of the quantized photon field and H is the full Hamiltonian generating the time evolution of an interacting electron-photon system. For instance, estimates of this type serve as one of the main technical ingredients in the mathematical analysis of Rayleigh scattering. In this context, (1.1) has been proven by Fröhlich et al. in the case where H is the non-or semirelativistic Pauli-Fierz Hamiltonian [4]; a slightly weaker version of (1.2) has been obtained in [4] for all even values of n. Higher order estimates of the form (1.1) also turn out to be useful in the study of the existence of ground states in a no-pair model of QED [8]. In fact, they imply that every eigenvector of the Hamiltonian H or spectral subspaces of H corresponding to some bounded interval are contained in the domains of higher powers of H f . This information is very helpful in order to overcome numerous technical difficulties which are caused by the non-locality of the no-pair operator. In these applications it is actually necessary to have some control on the norms in (1.1) and (1.2) when the operator H gets modified. To this end we shall give rough bounds on the right hand sides of (1.1) and (1.2) in terms of the ground state energy and integrals involving the form factor and the dispersion relation.
Various types of higher order estimates have actually been employed in the mathematical analysis of quantum field theories since a very long time. Here we only mention the classical works [5,11] on P (φ) 2 models and the more recent articles [2] again on a P (φ) 2 model and [1] on the Nelson model.
In what follows we briefly describe the organization and the content of the present article. In Section 2 we develop the main idea behind our techniques in a general setting. By the criterion established there the proof of the higher order estimates is essentially boiled down to the verification of certain form bounds on the commutator between H and a regularized version of H n/2 f . After that, in Section 3, we introduce some of the most important operators appearing in QED and establish some useful norm bounds on certain commutators involving them. These commutator estimates provide the main ingredients necessary to apply the general criterion of Section 2 to the QED models treated in this article. Their derivation is essentially based on the pull-through formula which is always employed either way to derive higher order estimates in quantum field theories [1,2,4,5,11]; compare Lemma 3.2 below. In Sections 4, 5, and 6 the general strategy from Section 2 is applied to the non-and semi-relativistic Pauli-Fierz operators and to the no-pair operator, respectively. The latter operators are introduced in detail in these sections. Apart from the fact that our estimate (1.2) is slightly stronger than the corresponding one of [4] the results of Sections 4 and 5 are not new and have been obtained earlier in [4]. However, in order to prove the higher order estimate (1.1) for the no-pair operator we virtually have to re-derive it for the semi-relativistic Pauli-Fierz operator by our own method anyway. Moreover, we think that the arguments employed in Sections 4 and 5 are more convenient and less involved than the procedure carried through in [4]. The main text is followed by an appendix where we show that the semi-relativistic Pauli-Fierz operator for a molecular system with static nuclei is semi-bounded below, provided that all Coulomb coupling constants are less than or equal to 2/π. Moreover, we prove the same result for a molecular no-pair operator assuming that all Coulomb coupling constants are strictly less than the critical coupling constant of the Brown-Ravenhall model [3]. The results of the appendix are based on corresponding estimates for hydrogen-like atoms obtained in [10]. (We remark that the considerably stronger stability of matter of the second kind has been proven for a molecular no-pair operator in [9] under more restrictive assumptions on the involved physical parameters.) No restrictions on the values of the fine-structure constant or on the ultra-violet cut-off are imposed in the present article.
The main new results of this paper are Theorem 2.1 and its corollaries which provide general criteria for the validity of higher order estimates and Theorem 6.1 where higher order estimates for the no-pair operator are established.
Some frequently used notation. For a, b ∈ Ê, we write a ∧ b := min{a, b} and a ∨ b := max{a, b}. D(T ) denotes the domain of some operator T acting in some Hilbert space and Q(T ) its form domain, when T is semi-bounded below. C(a, b, . . .), C ′ (a, b, . . .), etc. denote constants that depend only on the quantities a, b, . . . and whose value might change from one estimate to another.

Higher order estimates: a general criterion
The following theorem and its succeeding corollaries present the key idea behind of our method. They essentially reduce the derivation of the higher order estimates to the verification of a certain sequence of form bounds. These form bounds can be verified easily without any further induction argument in the QED models treated in this paper.
Theorem 2.1. Let H and F ε , ε > 0, be self-adjoint operators in some Hilbert space K such that H 1, F ε 0, and each F ε is bounded. Let m ∈ AE ∪ {∞}, let D be a form core for H, and assume that the following conditions are fulfilled: (a) For every ε > 0, F ε maps D into Q(H) and there is some c ε ∈ (0, ∞) such that (b) There is some c ∈ [1, ∞) such that, for all ε > 0, (c) For every n ∈ AE, n < m, there is some c n ∈ [1, ∞) such that, for all Then it follows that, for every n ∈ AE, n < m + 1, (An empty product equals 1 by definition.) Proof. We define T ε (n) is well-defined and bounded because of the closed graph theorem and Condition (a), which implies that F ε ∈ L (Q(H)), where Q(H) = D(H 1/2 ) is equipped with the form norm. We shall prove the following sequence of assertions by induction on n ∈ AE, n < m + 1.
A(n) :⇔ The bound (2.1) holds true and, if n > 3, we have For n = 1, the bound (2.1) is fulfilled with C 1 = c on account of Condition (b).
Next, assume that n ∈ AE, n < m, and that A(1), . . . , A(n) hold true. To Proof. Let U : K → L 2 (Ω, µ) be a unitary transformation such that a = U A U * is a maximal operator of multiplication with some non-negative measurable function -again called a -on some measure space (Ω, A, µ). We pick some ψ ∈ K , set φ n := U H −n/2 ψ, and apply the monotone convergence theorem to conclude that for every n ∈ AE, n < m + 1, which implies the first assertion. Now, assume that H and F ε , ε > 0, fulfill Condition (c') of Corollary 2.2. Applying the dominated convergence theorem in the spectral representation introduced above we see that F n ε ψ → A κ n ψ, for every ψ ∈ D(A κ n ). Hence, (2.7) and Ran(H −n/2 ) ⊂ D(A κ n ) imply, for n < m and φ, ψ ∈ D(H), In particular, A κ n H −n/2 ψ ∈ D(H * ) = D(H), for all ψ ∈ D(H), and the second asserted bound holds true.

Commutator estimates
In this section we derive operator norm bounds on commutators involving the quantized vector potential, A, the radiation field energy, H f , and the Dirac operator, D A . The underlying Hilbert space is where the bosonic Fock space, F b , is modeled over the one-photon Hilbert space With regards to the applications in [8] we define A := {k ∈ Ê 3 : |k| m}, for some m 0. We thus have where S n = S 2 n = S * n is given by (S n ψ (n) )(k 1 , . . . , k n ) := 1 n! π∈Sn ψ (n) (k π(1) , . . . , k π(n) ), ψ (n) ∈ L 2 (A × 2 ) n , S n denoting the group of permutations of {1, . . . , n}. The vector potential is determined by a certain vector-valued function, G, called the form factor.
x (k, λ), for almost every k and all x ∈ Ê 3 Example. In the physical applications the form factor is often given as for (x, k) ∈ Ê 3 × (Ê 3 × 2 ) with k = 0. Here the physical units are chosen such that energies are measured in units of the rest energy of the electron. Length are measured in units of one Compton wave length divided by 2π. The parameter Λ > 0 is an ultraviolet cut-off and the square of the elementary charge, e > 0, equals Sommerfeld's fine-structure constant in these units; we have e 2 ≈ 1/137 in nature. The polarization vectors, ε(k, λ), λ ∈ 2 , are homogeneous of degree zero in k such that {k, ε(k, 0), ε(k, 1)} is an orthonormal basis of Ê 3 , for everẙ k ∈ S 2 . This corresponds to the Coulomb gauge for ∇ x · G e,Λ = 0. We remark that the vector fields S 2 ∋k → ε(k, λ) are necessarily discontinuous. ⋄ It is useful to work with more general form factors fulfilling Hypothesis 3.1 since in the study of the existence of ground states in QED one usually encounters truncated and discretized versions of the physical choice G e,Λ . For the applications in [8] it is necessary to know that the higher order estimates established here hold true uniformly in the involved parameters and Hypothesis 3.1 is convenient way to handle this.
We recall the definition of the creation and the annihilation operators of a photon state f ∈ F n=0 ∈ F b such that the right hand sides again define elements of F b . a † (f ) and a(f ) are formal adjoints of each other on the dense domain Then the quantized vector potential is the triplet of operators given by The radiation field energy is the direct sum where dΓ (0) (ω) := 0, and dΓ (n) (ω) denotes the maximal multiplication operator in F (n) b associated with the symmetric function (k 1 , . . . , k n ) → ω(k 1 ) + · · · + ω(k n ). By the permutation symmetry and Fubini's theorem we thus have where we use the notation valid for almost every k, is called the pull-through formula. Finally, we let α 1 , α 2 , α 3 , and β := α 0 denote hermitian four times four matrices that fulfill the Clifford algebra relations They act on the second tensor factor in L 2 (Ê 3 As a consequence of (3.5) and the C * -equality we have A standard exercise using the inequality in (3.6), the Cauchy-Schwarz inequality, and the canonical commutation relations, (Here and in the following we identify H f ≡ ½⊗H f , etc.) These relative bounds imply that α · A is symmetric on the domain D(H 1/2 f ). The operators whose norms are estimated in (3.9) and the following lemmata are always well-definedà priori on the following dense subspace of H , Given some E 1 we set in the sequel. We already know from [10] that, for every ν 0, there is some constant, C ν ∈ (0, ∞), such that In our first lemma we derive a generalization of (3.9). Its proof resembles the one of (3.9) given in [10]. Since we shall encounter many similar but slightly different commutators in the applications it makes sense to introduce the numerous parameters that obscure its statement (but simplify its proof).
Proof. We notice that all operatorsȞ s f and f s ε (H f ) leave the dense subspace D invariant and that α · a ♯ (G) maps D into D(Ȟ s f ), for every s ∈ Ê. Now, let For almost every k, the pull-through formula yields the following representation, , for t 0. We compute .
Using that f ε is increasing in t 0 and that it is elementary to verify that for all t 0 and k. We deduce that the term in (3.12) can be estimated as In the last step we used δ + τ 1/2 and applied (3.4). (3.15) immediately gives a bound on the term in (3.13), too. For we have which after the replacements (ν, κ, γ, δ, σ, τ ) → (−ν, −κ, δ, γ, τ, σ) and ϕ → −ψ is precisely the term we just have treated.
Lemma 3.2 provides all the information needed to apply Corollary 2.3 to nonrelativistic QED. For the application of Corollary 2.3 to the non-local semirelativistic models of QED it is necessary to study commutators that involve resolvents and sign functions of the Dirac operator, An application of Nelson's commutator theorem with test operator −∆+H f +1 shows that D A is essentially self-adjoint on D. The spectrum of its unique closed extension, again denoted by the same symbol, is contained in the union of two half-lines, and the spectral calculus yields The next lemma is a straightforward extension of [ Proof. Without loss of generality we may assume that ε > 0 for otherwise we could simply replace ν by ν + κ and f κ 0 by f 0 0 = 1. First, we assume in addition that ν 0. We observe that are bounded (here we use that ν 0 and ε > 0), this identity implies . Taking the adjoint of the previous identity and replacing y by −y we obtaiň ) . In view of the norm bounds on T 1 and T 2 we see that (3.16) and (3.17 provided that E is sufficiently large, depending only on κ, ν, d 1 , and d ρ , such that the Neumann series converge. Now, let ν < 0. Then we write T 0 on the domain D as by a computation analogous to (3.18). Taking the adjoint of this identity with y replaced by −y we get Next, we invert ½ + R A (iy) T * 0 by means of the same Neumann series as above. As a result we obtaiň where the definition of Υ κ,ν and Υ κ,ν has been extended to negative ν. It follows that ) into itself and that (3.16) and (3.17) still hold true when ν is negative.
In order to control the Coulomb singularity 1/|x| in terms of |D A | and H f in the proof of the following corollary, we shall employ the bound [10, Theorem 2.3] which holds true in sense of quadratic forms on Q(|D A |) ∩ Q(H f ). Here k ∈ (0, ∞) is some universal constant. We abbreviate the sign function of the Dirac operator, which can be represented as a strongly convergent principal value [6, Lemma VI.5.6], by We recall from [10, Lemma 3.3] that S A maps D(H ν f ) into itself, for every ν > 0. This can also be read off from the proof of the next corollary. Then we find some C ≡ C(κ, ν, d 1 , d ρ ) ∈ (0, ∞) such that, for all γ, δ, σ, τ 0 with γ + δ + σ + τ 1/2 and all ε 0, E k 1 , on D and employing (3.16), (3.17), and (3.11) we obtain the following estimate, for all ϕ, ψ ∈ D, and E k 1 , This estimate shows that the vector in the right entry of the scalar product in the first line belongs to D((|D A | 1/2 ) * ) = D(|D A | 1/2 ) and that (3.22) holds true. Next, we observe that (3.23) follows from (3.22) and (3.19). Finally, (3.21) fol- . Such a bound on X is, however, an immediate consequence of (3.22) (where we can choose ε = 0) because

Non-relativistic QED
The Pauli-Fierz operator for a molecular system with static nuclei and N ∈ AE electrons interacting with the quantized radiation field is acting in the Hilbert space the tensor product understood in the algebraic sense, by A superscript (i) indicates that the operator below is acting on the pair of variables (x i , ς i ). In fact, the operator defined in (4.2) is a two-fold copy of the usual Pauli-Fierz operator which acts on two-spinors and the energy has been shifted by N in (4.2). For (3.5) implies Here σ = (σ 1 , σ 2 , σ 3 ) is a vector containing the Pauli matrices (when α j , j ∈ {0, 1, 2, 3}, are given in Dirac's standard representation). We write H V nr in the form (4.2) to maintain a unified notation throughout this paper.
We shall only make use of the following properties of the potential V .

Hypothesis 4.1. V can be written as
There exist a ∈ (0, 1) and b ∈ (0, ∞) such that V − a H 0 nr +b in the sense of quadratic forms on D N .
Example. The Coulomb potential generated by K ∈ AE fixed nuclei located at the positions {R 1 , . . . , R K } ⊂ Ê 3 is given as for some e, Z 1 , . . . , Z K > 0 and a.e. X = ( It is well-known that V C is infinitesimally H 0 nr -bounded and that V C fulfills Hypothesis 4.1. ⋄ It follows immediately from Hypothesis 4.1 that H V nr has a self-adjoint Friedrichs extension -henceforth denoted by the same symbol H V nr -and that D N is a form core for H V nr . Moreover, we have for all ℓ ∈ {−1, 0, 1, 2, . . .}. Then, for every n ∈ AE, we have D((

Proof.
We pick the function f ε defined in (3.10) with E = 1 and verify that the operators F n ε := f for all Ψ ∈ D N , which is Condition (b). Next, we observe that F ε maps D N into itself. Employing (4.5) once more and using −V − 0 and the fact that V + 0 and F ε act on different tensor factors we deduce that for every Ψ ∈ D N . Thanks to (3.11) with κ = 1/2, ν = γ = δ = σ = τ = 0, and (4.5) we further find some C ∈ (0, ∞) such that for all Ψ ∈ D N . (4.9) and (4.10) together show that Condition (a) is fulfilled, too. Finally, we verify the bound in (c'). We use is the magnetic field and the j-th entry of the formal vector Σ is −i ǫ jkℓ α k α ℓ , j, k, ℓ ∈ {1, 2, 3}, to write the square of the Dirac operator on the domain D as This yields . Furthermore, we observe that Lemma 3.2 is applicable to Σ · B as well instead of α · A; we simply have to replace the form factor G by ∇ x ∧ G and to notice that Σ · v L ( 4 ) = |v|, v ∈ Ê 3 , in analogy to (3.6). Note that the indices of d ℓ are shifted by 2 because of (4.6). Finally, we observe that Lemma 3.2 is applicable to ∇ x · A, too. To this end we have to replace G by (∇ x · G, 0, 0) and d ℓ by some universal constant times d 2+ℓ because of (4.7). Taking all these remarks into account we arrive at for all Ψ 1 , Ψ 2 ∈ D N . From this estimate, Lemma 3.2, and (4.5) we readily infer that Condition (c') is valid with c n = (|E nr |+1) C ′′ (N, n, a, b, d −1 , . . . , d 5+n ).

The semi-relativistic Pauli-Fierz operator
The semi-relativistic Pauli-Fierz operator is also acting in the Hilbert space H N introduced in (4.1). It is obtained by substituting the non-local operator We thus define,à priori on the dense domain D N , where V is assumed to fulfill Hypothesis 4.1 with H 0 nr replaced by H 0 sr . To ensure that in the case of the Coulomb potential V C defined in (4.4) this yields a well-defined self-adjoint operator we have to impose appropriate restrictions on the nuclear charges.
Example. In Proposition A.1 we show that H V C sr is semi-bounded below on D N provided that Z k ∈ (0, 2/πe 2 ], for all k ∈ {1, . . . , K}. Its proof is actually a straightforward consequence of (3.19) and a commutator estimate obtained in [10]. If all atomic numbers Z k are strictly less than 2/πe 2 we thus find a ∈ (0, 1) and b ∈ (0, ∞) such that in the sense of quadratic forms on D N . In particular, V C fulfills Hypothesis 4.1 with H 0 nr replaced by H 0 sr as long as Z k ∈ (0, 2/πe 2 ), for k ∈ {1, . . . , K}. ⋄ For potentials V as above H V sr has a self-adjoint Friedrichs extension which we denote again by the same symbol H V sr . Moreover, D N is a form core for H V sr and we have the following analogue of (4.5), In order to apply Corollary 2.3 to the semi-relativistic Pauli-Fierz operator we recall the following special case of [7, Corollary 3.7]: Lemma 5.1. Assume that ω and G fulfill Hypothesis 3.1. Let τ ∈ (0, 1]. Then there exist δ > 0 and C ≡ C(δ, τ, d 1 ) ∈ (0, ∞) such that in the sense of quadratic forms on D.
In the next theorem we re-derive the higher order estimates obtained in [4] for the semi-relativistic Pauli-Fierz operator by means of Corollary 2.3. (The second estimate of Theorem 5.2 is actually slightly stronger than the corresponding one stated in [4].) The estimates of the following proof are also employed in Section 6 where we treat the no-pair operator. We set Proof. Let m ∈ AE. We pick the function f ε defined in (3.10) with E = k 1 ∨ C.
The estimates (4.8) and (4.9) are still valid without any further change when the subscript nr is replaced by sr. Employing (5.3) twice and using (5.2) we obtain the following substitute of (4.10), Altogether we see that Conditions (a) and (b) are satisfied. In order to verify (c') we set By virtue of (3.22) we know that the norms of U A | 1/2 are bounded uniformly in ε > 0 by some constant, C ∈ (0, ∞), that depends only on n, d 1 , and d 3+n . We employ the notation (4.11) and (5.4) to write The previous identity, (5.2), and |D A | 1 permit to get for all Ψ 1 , Ψ 2 ∈ D N and some constant c n = C ′′ (n, a, b, d 1 , d 3+n ) (|E sr | + 1). So (c') is fulfilled also and the assertion follows from Corollary 2.3.

The no-pair operator
We introduce the spectral projections (6.1) A . The no-pair operator acts in the projected Hilbert space and isà priori defined on the dense domain P + A,N D N by Notice that all operators D x, y ∈ Ê 3 , i, j ∈ {1, 2, 3}, commute in the sense that all their spectral projections commute; see the appendix to [9] for more details. (Here we use the assumption that G x (−k, λ) = G x (k, λ).) So the order of the application of the projections P +,(i) A is immaterial. In this section we restrict the discussion to the case where V is given by the Coulomb potential V C defined in (4.4). To have a handy notation we set for all i ∈ {1, . . . , N } and 1 i < j N , respectively. Thanks to [10, Lemma 3.4(ii)], which implies that P + A maps D into D(|D 0 |) ∩ D(H ν f ), for every ν > 0, and Hardy's inequality, we actually know that H V C np is well-defined on D N . In order to apply Corollary 2.3 to H V C np we extend H V C np to a continuously invertible operator on the whole space H N : We pick the complementary projection, and define the operator H npà priori on the domain D N by Evidently, we have [ H np , P + A,N ] = 0 and H np P + A,N = H V C np P + A,N on D N . In Proposition A.2 we show that the quadratic forms of the no-pair operator H V C np and of H np are semi-bounded below on D N provided that the atomic numbers Z 1 , . . . , Z K 0 are less than the critical one of the Brown-Ravenhall model determined in [3], Z np := (2/e 2 )/(2/π + π/2) .
Therefore, both H V C np and H np possess self-adjoint Friedrichs extensions which are again denoted by the same symbols in the sequel. D N is a form core for H np and we have the bound on D N , where |Z | := max{Z 1 , . . . , Z K } < Z np . Moreover, it makes sense to define The semi-relativistic Pauli-Fierz operator H 0 sr has already been treated in the previous section and the bound Hence, it only remains to consider the operator W .
We fix some n ∈ AE, n m. When we verify (a) we can ignore the potentials v i since they are negative.
for every Ψ ∈ D N , where, for all n ∈ AE and ν ∈ Ê, Likewise we have where the first norm in the second line of (6.8) is bounded (uniformly in ε > 0) due to Lemma 6.2. Taking these remarks, v i 0, (6.4), and (6.5) into account we infer that showing that (a) is fulfilled.
For Ψ 1 , Ψ 2 ∈ D N , we thus obtain where we can further estimate (6.11) and, of course, The operator norms in (6.10) can be estimated by means of (6.7) with ν = ±1/2, the one in the last line of (6.11) is bounded by some C(n, d 1 , d 3+n ) ∈ (0, ∞) due to (3.21). In a similar fashion we obtain, for all i, j ∈ {1, . . . , N }, i < j, and Ψ 1 , Ψ 2 ∈ D N , Here we can further estimate Lemma 6.2 below ensures that all operator norms in (6.13) and (6.14) that involve w 1/2 ij are bounded uniformly in ε > 0 by constants depending only on e, n, d 1 , and d 5+n . Furthermore, it is now clear how to treat the terms involving v i or E np . (In order to treat v i just replace P +,(i,j) A by P +,(i) A , w ij by v i , and w 1/2 ij by |v i | 1/2 in (6.13) and (6.14).) Combining (6.9)-(6.14) and their analogues for the remaining operators in W we arrive at for all Ψ 1 , Ψ 2 ∈ D N and some ε-independent C ≡ C(N, n, e, d 1 , d 5+n ) ∈ (0, ∞). Employing successively (3.19), which implies |v i | (πe 2 |Z |/2)(|D
In this appendix we verify that the semi-relativistic Pauli-Fierz and no-pair operators with Coulomb potential are semi-bounded below for all nuclear charges less than the critical charges without radiation fields. We do not attempt to give good lower bounds on their spectra since this is not the topic addressed in this paper. Our aim here is essentially only to ensure that these operators possess self-adjoint Friedrichs extensions. We recall that the stability of matter of the second kind has been proven for the no-pair operator in [9] under certain restrictions on the fine-structure constant, the ultra-violet cut-off, and the nuclear charges. The stability of matter of the second kind is a much stronger property than mere semi-boundedness. It says that the operator is bounded below by some constant which is proportional to the total number of nuclei and electrons and uniform in the nuclear positions. The restrictions imposed on the physical parameters in [9] do, however, not allow for all atomic numbers less than Z np . First, we consider the semi-relativistic Pauli-Fierz operator. The following proposition is a simple generalization of the bound (3.19) proven in [10] to the case of N ∈ AE electrons and K ∈ AE nuclei. We set ̺ := min{|R k − R ℓ | : k = ℓ}/2 and pick a smooth partition of unity on Ê 3 , {χ k } K k=0 , such that χ k ≡ 1 on B ̺/2 (R k ) and supp(χ k ) ⊂ B ̺ (R k ), for k = 1, . . . , K, and such that K k=0 χ 2 k = 1. Then we have the following IMS type localization formula, Next, we turn to the no-pair operator discussed in Section 6. The semiboundedness of the molecular N -electron no-pair operator is essentially a consequence of the following inequality [10, Equation (2.14)], valid for all ω and G fulfilling Hypothesis 3.1, γ ∈ (0, 2/(2/π + π/2)), and δ > 0, we deduce that In the last step we repeatedly applied (3.16). Commuting ζ and D A and using D A R A (iy) 1, R A (iy) 2 (1 + y 2 ) −1 , and the fact that Υ 0,1/2 (iy) is uniformly bounded in y ∈ Ê, we readily deduce that E-mail address: matte@math.lmu.de