Universidad de Chile, Casilla 653, Santiago, Chile. Universidad de Santiago de Chile. Casilla 307 Correo 2, Santiago, Chile. May 10, 2017.

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1 A Laplace transorm approach to linear equations with ininitely many derivatives and zeta-nonlocal ield equations arxiv: v2 [math-ph] 9 May 2017 A. Chávez 1, H. Prado 2, and E.G. Reyes 3 1 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile 2,3 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile Casilla 307 Correo 2, Santiago, Chile May 10, 2017 Abstract We study existence, uniqueness and regularity o solutions or linear equations in ininitely many derivatives. We develop a natural ramework based on Laplace transorm as a correspondence between appropriate L p and Hardy spaces: this point o view allows us to interpret rigorously operators o the orm ( t ) where is an analytic unction such as (the analytic continuation o) the Riemann zeta unction. We ind the most general solution to the equation ( t )φ = J(t), t 0, in a convenient class o unctions, we deine and solve its corresponding initial value problem, and we state conditions under which the solution is o class C k, k 0. More speciically, we prove that i some a priori inormation is speciied, then the initial value problem is well-posed and it can be solved using alancallayuc@gmail.com humberto.prado@usach.cl enrique.reyes@usach.cl ; e g reyes@yahoo.ca 1

2 only a inite number o local initial data. Also, motivated by some intriguing work by Dragovich and Are eva-volovich on cosmology, we solve explicitly ield equations o the orm ζ( t + h)φ = J(t), t 0, in which ζ is the Riemann zeta unction and h > 1. Finally, we remark that the L 2 case o our general theory allows us to give a precise meaning to the oten-used interpretation o ( t ) as an operator deined by a power series in the dierential operator t. 1 Introduction Equations with an ininite number o derivatives have appeared recently as ield equations o motion in particle physics [35], string theory [9, 22, 23, 24, 33, 41, 43, 44, 46], and (quantum) gravity and cosmology [1, 2, 3, 4, 5, 6, 10, 12, 13, 32, 34]. For instance, an important equation in this class is p a 2 t φ = φ p, a > 0, (1) where p is a prime number. Equation (1) describes the dynamics o the open p-adic string or the scalar tachyon ield (see [1, 6, 21, 34, 43, 44] and reerences therein) and it can be understood, at least ormally, as an equation in an ininite number o derivatives i we expand the entire unction (called below the symbol o the equation) (s) := p as2 = e as2 log(p) as a power series around zero and we replace s or t. This equation has been studied by Vladimirov via integral equations o convolution type in [43, 44] (see also [1, 34]), and it has been also noted that in the limit p 1, Equation (1) becomes the local logarithmic Klein-Gordon equation [7, 25]. Moreover, Dragovich, see [21], has used (the Lagrangian ormulation o) Equation (1) and the Euler product or the Riemann zeta unction ζ(s) := n=1 1 n s, Re(s) > 1, see [31], to deduce a string theory ield equation o the orm ( ζ 1 ) m 2 2 t + h φ = U(φ), (2) 2

3 in which m, h are constant real numbers and U is a nonlinear unction o φ. Equation (2) is called a zeta-nonlocal ield equation in [21]. In this paper we ocus on analytic properties o linear equations o the orm ( t )φ = J(t), t 0, (3) in which is an analytic unction and J belongs to an adequate class o unctions to be speciied in Section 2. We also investigate careully the ormulation, existence and solution o initial value problems, and we explain precisely in what sense Equation (3) really is an ordinary dierential equation in ininitely many derivatives. Motivated mainly by Dragovich s work (see also [2]), we endeavor to develop a theory or ( t ) able to deal with the case in which is, or instance, the Riemann zeta unction. We think this lexibility is important: it does not appear to be straightorward to use the previous works [26, 27] or equations o the orm ζ( t + h)φ = J, as we explain in Section 2. We remark that linear equations in ininitely many derivatives appeared in mathematics already in the inal years o the XIX Century, see or instance [11, 15] and urther reerences in [5]. It appears to us, however, that a truly undamental stimulus or their study has been the realization that linear and nonlinear nonlocal equations play an important role in contemporary physical theories, as witnessed by [39] and the more recent papers cited above. Now, a serious problem or the development o a rigorous yet lexible enough to allow or explicit computations theory or nonlocal equations has been the diiculties inherent in the understanding o the initial value problem or equations such as (1) and (3). An interpretative diiculty considered already in the classical paper [24] by Eliezer and Woodard is the requently stated argument (see or instance [3, 34] and the rigorous approach o [16]) that i an nth order ordinary dierential equation requires n initial conditions, then the ininite order equation F (t, q, q,, q (n), ) = 0 requires ininitely many initial conditions, and thereore the solution q would be determined a priori (via power series) without reerence to the actual equation. In order to deal with this diiculty, we investigate initial value problems rom scratch. Our approach is to emphasize the role played by the Laplace transorm, in the spirit o [26, 27] and the interesting papers [1, 2, 5, 6]. Our solution (Section 3.2 below) is that i an a priori data directly connected with our interpretation o ( t ) is speciied (Equation (42) below) then the initial value problem is well-posed and it requires only a inite number o initial conditions. Due to the preeminence o Laplace transorm in our theory, it is natural or us to speak indistinctly o 3

4 nonlocal equations or equations in an ininite number o derivatives, and we do so hereater. We inish this Introduction with the remark that our work is not necessarily part o the classical theory o pseudo-dierential operators or two reasons. First, the symbol appearing in Equation (3) is (in principle) an arbitrary analytic unction, and thereore the operator ( t ) may be beyond the reach o classical tools such as the ones appearing in [30]. Second, we aim at solving nonlocal equations on the semi-axis t 0, not on the real line where perhaps we could use Fourier transorms as in the intriguing papers [2, 21], and/or classical pseudo-dierential analysis [30]; motivation or the study o nonlocal equations on a semi-axis comes, or example, rom cosmology [1, 36]: classical versions o Big Bang cosmological models contain singularities at the beginning o time, and thereore the time variable appearing in the ield equations should vary over a hal line. This paper is organized as ollows. In Section 2 we consider a rigorous interpretation o nonlocal ordinary dierential equations via Laplace transorm considered as an operator between appropriate Banach spaces. In Section 3 we investigate (and propose a method or the solution o) initial value problems or linear equations o the orm (3). In Section 4 we solve a class o zeta nonlocal equations and in Section 5 we provide an L 2 -theory or operators ( t ). Finally in Section 6 we discuss some problems which are beyond the reach o the theory presented here; they will be considered in the companion article [17]. 2 Linear nonlocal equations In this section we introduce a rigorous and computationally useul ramework or the analytic study o nonlocal equations, including the possibility o setting up meaningul initial value problems. We consider linear nonlocal equations o the orm ( t )φ(t) J(t) = 0, t 0, (4) in which is an analytic unction. 2.1 Preliminaries We begin by ixing our notation and stating some preliminary acts; our main source is the classical treatise [19] by G. Doetsch. 4

5 We recall that a unction g belongs to L 1 loc (R +), in which R + is the interval [0, ), i and only i exists or any compact set K R +. K g(t) dt Deinition 2.1. A unction g : R C in L 1 loc (R +) belongs to the class T a i and only i g(t) is identically zero or t < 0 and the integral converges absolutely or Re(s) > a. 0 e s t g(t) dt T a is simply a vector space; we do not need to endow it with a topology. The Laplace transorm o a unction g : R C in T a is the integral L(g)(s) = 0 e s t g(t) dt, Re(s) > a. As proven in [19, Theorem 3.1], i g T a, then L(g)(s) converges absolutely and uniormly or Re(s) x 0 > a (a iner statement is in [19, Theorem 23.1]). It is also known (see [19, Theorem 6.1]) that the Laplace transorm L(g) is an analytic unction or Re(s) > a. We also record the ollowing: Proposition Let us ix a unction g T a. I the derivative D k g(t) belongs to T a or 0 k n 1, then D n g(t) belongs to T a and L(D n g)(s) = s n L(g)(s) n (D j 1 g)(0)s n j. 2. I g T a is a continuous unction, then or any σ > a we have the identity or all t 0. g(t) = 1 2πi σ+i σ i j=1 e s t L(g)(s) ds (5) The second part o this proposition is in [19, Theorem 24.4]. The class T a appears to be the simplest class o unctions on which the Laplace transorm is deined and or which there exists an inversion ormula (5). The inverse Laplace transorm is denoted by L 1. 5

6 We recall that we are interested in a theory able to deal with (4) in which, or instance, (s) = ζ(s+h). This is not a trivial requirement: let us suppose that we are interested in applying Theorem 3.1 o [27] on classical initial value problems to the equation ζ( t + h)φ(t) = J(t). Then, we would need to check that L(J)(s)/ζ(s + h) belongs to the Widder space { CW (ω, ) = r : (ω, ) C / r W = sup sup n N 0 s>ω } (s ω) n+1 r (n) (s) <, in which ω > 0 and r (n) (s) denotes the nth derivative o r(s). Certainly, such a check does not look straightorward. It is important thereore to develop an alternative approach to nonlocal equations. The class T a appears to be a good starting point. Regretully, because o reasons to be explained in Section 4, it is not adequate or deining and solving initial value problems. We have ound that a better alternative is to consider Laplace transorm as a correspondence rom Lebesgue spaces L p (R + ) into Hardy spaces H q (C + ), which we now deine. We write C + or the right hal-plane {s C : Re(s) > 0}. The space L p (R + ), 1 p <, is the Lebesgue space o measurable unctions φ on [0, ) such that ( φ L p (R + ) := 0 ) 1 φ(x) p p dx <, and the qth Hardy space H q (C + ) is the space o all unctions Φ which are analytic on C + and such that the integral µ q (Φ, x) given by µ q (Φ, x) := ( 1 ) 1 Φ(x + iy) q q dy 2π, is uniormly bounded or x > 0. We note that H q (C + ) becomes a Banach space with the norm Φ H q (C + ) := sup x>0 µ q (Φ, x). The ollowing classic Representation theorem was irst presented by Doetsch in [20, pp. 276 and 279]: Theorem 2.2. (Doetsch s Representation theorem) (i) I φ L p (R + ), where 1 < p 2, and Φ = L(φ), then Φ H p (C + ) with 1 p + 1 p = 1. Moreover i x > 0 there exists a positive constant C(p) such that: ( ) 1 µ p (Φ, x) C(p) e pxt φ(t) p p dt. 0 6

7 (ii) I Φ H p (C + ), where 1 < p 2, then there exists φ L p (0, ) with 1 p + 1 p = 1 suth that Φ = L(φ). The unction φ is given by the inversion ormula 1 φ(t) := lim v 2π v v e (σ+iη)t Φ(σ + iη)dη, σ 0, in which the limit is understood in L p (R + ), and or x > 0 there exists a positive constant K(p) such that ( 0 ) 1 e p xt φ(t) p p dt K(p)µ p (Φ, x). We remark that the restrictions on p appearing in both parts o the theorem imply that neither (i) is the converse o (ii) nor (ii) is the converse o (i), except in the case p = p = 2. This act relects itsel in the enunciates o our main theorems (Theorems 2.5 and 3.3). I p = p = 2, Doetsch s Representation theorem is the important Paley-Wiener theorem, which states that the Laplace transorm is a unitary isomorphism between L 2 (R + ) and H 2 (C + ); a precise statement is in Section 5, ater [28, 47]. We also note that there exist generalizations o Theorem 2.2: it has been observed that the Laplace transorm determine correspondences between appropriate weighted Lebesgue and Hardy spaces, see [8, 40] and reerences therein, and there also exists a representation theorem or unctions Φ which decay to zero in any closed hal-plane Re(s) δ > a, a R and satisy the additional requirement that Φ(σ + i( )) L 1 (R) (roughly speaking, an L 1 -case o Theorem 2.2, see [18, 19]). Interestingly, as we explain in Section 3, in this L 1 -situation we can ormulate theorems on existence and uniqueness o solutions or nonlocal linear equations but regretully, we cannot develop a meaningul theory o initial value problems. The Doetsch Representation theorem is our main tool or the understanding o nonlocal equations. 2.2 The operator ( t ) : L p (R + ) H q (C + ) As a motivation, we calculate ( t )φ ormally: we take a unction analytic around zero, and suppose that φ L p (0, ) is smooth. Let us write ( t )φ = (n) (0) n t φ. We note that this expansion is actually a rigorous result i extends to an entire unction and φ is a -analytic vector or t on the space o square integrable unctions 7

8 L 2 (R + ) (see Section 5 below); also, this expansion is a ormal theorem i is an entire unction and φ is an entire unction o exponential type (see [16]; a generalization o this result appears in the companion paper [17]). Standard properties o the Laplace transorm (see Proposition 2.1 or [19]) yield, ormally, L(( t )φ)(s) = = = (n) (0) (n) (0) (n) (0) L( n t φ) (s n L(φ) s n 1 φ(0) s n 2 φ (0) φ (n 1) (0)) ( = (s)l(φ)(s) I we deine the ormal series r(s) = s n L(φ) n n=1 j=1 n n=1 j=1 ) n s n j φ (j 1) (0) j=1 (n) (0) (n) (0) s n j φ (j 1) (0). (6) d j 1 s n j, (7) in which d = {d j : j 0} is a sequence o complex numbers, then we can write (6) as where r(s) is given by (7) with d j = φ (j) (0). L(( t )φ)(s) = (s)l(φ)(s) r(s), (8) The ollowing lemma asserts that or some choice o the sequence {d j 1 } j 1, the ormal series deined in (7) is in act an analytic unction. Lemma 2.3. Let R 1 > 1 be the maximum radius o convergence o the Taylor series T (s) = (n) (0) s n. Set 0 < R < 1 and suppose that the series j=1 d j 1 1 s j (9) is uniormly convergent on compact sets or s > R. Then the series (7) is analytic on the disk s < R 1. Proo. Let us write the series (7) in the ollowing orm r(s) = n=1 (n) (0) s n n d j 1 s j, j=1 8

9 and deine r N (s) as the partial sum r N (s) := N n=1 (n) (0) s n n d j 1 s j. Let K {s : R < s < R 1 } be a compact set; there exists a positive constant L K such that or any s K we have: d j 1 s j L K ; j=1 j=1 also, given ɛ > 0, there exist N 0 = N 0 (ɛ) such that (n) (0) s n < ɛ ; n=n 0 +1 rom these inequalities we deduce that r(s) r N (s) = n=n+1 (n) (0) s n n j=1 d j 1 s j < ɛl K, or any N N 0 and uniormly or s K. Thereore, the partial sums r N (s) converge uniormly to r(s) on K; since the compact subset K is arbitrary, we have that r(s) is analytic in {s : R < s < R 1 }. Now we prove that r(s) is analytic in s R. We write r(s) as in which, P n (s) := r(s) = n=1 (n) (0) P n (s), n d j 1 s n j = d 0 S n 1 + d 1 S n 2 + d n 1 S 0, j=1 where S n = n k=1 sk. Then, the convergence o the series j=1 d j 1 and the inequalities R < 1 < R 1 imply that the ollowing three assertions hold: n=1 (n) (0) (n+1) (0) (n + 1)! lim n (n) (0) = 0, <, P n (s) is uniormly bounded on s R. The result now ollows rom [29, Theorem 5.1.8]. 9

10 As a corollary we recover a previously known lemma, see [26, Lemma 2.1]: Corollary 2.4. Let be an entire unction. Set R < 1 and suppose that the series j=1 d j 1 1 s j, is convergent or s > R. Then the series (7) is an entire unction. Remark. There exists a large class o series satisying conditions o Lemma 2.3. Indeed, let r > 0 and denote by Exp r (C) the space o entire unctions o exponential type τ < r (see [16, 42] and reerences therein). It is well know that i φ Exp r (C) and it is o exponential type τ < r, then its Borel transorm B(φ)(s) = j=0 φ j (0) s j+1 converges uniormly on s > τ (see again [16, 42]). In our case, i φ Exp 1 (C), its Borel transorm B(φ)(s) is precisely (9) or d j = φ j (0). We will come back to the class o unctions o exponential type in our orthcoming paper [17]. Motivated by the previous computations and Doetsch s representation theorem, we make the ollowing deinition, generalizing [26, 27]: Deinition 2.2. Let be an analytic unction on a region which contains the halplane {s C : Re(s) > 0}, and let H be the space o all C-valued unctions on C which are analytic on (regions o) C. We ix p and p such that 1 < p 2 and 1 p + 1 p = 1, and we consider the subspace D o L p (0, ) H consisting o all the pairs (φ, r) such that (φ, r) = L(φ) r (10) belongs to the class H p (C + ). The domain o ( t ) as a linear operator rom the product L p (0, ) H into L p (0, ) is D. I (φ, r) D then we deine ( t ) (φ, r) = L 1 ( (φ, r) ) = L 1 ( L(φ) r). (11) We note that this deinition will be urther generalized in [17] by means o the Borel transorm, in order to capture an even larger class o symbols. However, as we will see below, the oregoing interpretation or ( t ) already gives us a satisactory way to deal with the initial value problem or linear nonlocal equations in many interesting cases. 10

11 2.3 Linear nonlocal equations In this subsection we solve the nonlocal equation ( t )(φ, r) = J, J L p (0, ), (12) in which we are using the above interpretation o ( t ). We assume hereater that a suitable unction r H has been ixed; consequently, we understand Equation (12) as an equation or φ L p (0, ) such that (φ, r) D. We simply write ( t )φ = J instead o (12). First o all, we ormalize what we mean by a solution: Deinition 2.3. Let us ix a unction r H. We say that φ L p (0, ) is a solution to the equation ( t )φ = J i and only i 1. φ = L(φ) r H p (C + ) ; ( i.e., (φ, r) D ); 2. ( t )(φ) = L 1 ( (φ, r)) = L 1 ( L(φ) r) = J. Our main theorem on existence and uniqueness o the solution to the linear problem (12) is the ollowing abstract result: Theorem 2.5. Let us ix a unction which is analytic in a region D which contains the hal-plane {s C : Re(s) > 0}. We also ix p and p such that 1 < p 2 and = 1, and we consider a unction J L p (R p p + ) such that L(J) H p (C + ). We assume that the unction (L(J) + r)/ is in the space H p (C + ). Then, the linear equation ( t )φ = J (13) can be uniquely solved on L p (0, ). Moreover, the solution is given by the explicit ormula φ = L 1 ( L(J) + r ). (14) Proo. We set φ = L 1 ((L(J) + r)/ ). Since L(J) H p (C + ), it ollows that the pair (φ, r) is in the domain D o the operator ( t ): indeed, an easy calculation using Theorem 2.2 shows that φ = L(J), which is an element o H p (C + ) by hypothesis. We can then check (using Theorem 2.2 again) that φ deined by (14) is a solution o (13). We prove uniqueness using Deinition 2.3: let us assume that φ and ψ are solutions to Equation (13). Then, item 2 o Deinition 2.3 implies L(φ ψ) = 0 on {s C : Re(s) > a}. Set h = L(φ ψ) and suppose that h(s 0 ) 0 or s 0 in the hal-plane just deined. By analyticity, h(s) 0 in a suitable neighborhood U o s 0. But then = 0 in U, so that (again by analyticity) is identically zero. 11

12 Remark. Interestingly, the above theorem on solutions to Equation (13) is a ully rigorous version o theorems stated long ago, see or instance the classical papers [11, 15] by Bourlet and Carmichael, and also the recent work [5] in which urther reerences appear. We recover a theorem proven by Carmichael in [15] in Corollary 2.7 below. In Section 3 we impose urther conditions on J and which assure us that (14) is smooth at t = 0, and we use these conditions to study the initial value problem or (13). We recall that we have ixed an analytic unction r. Hereater we assume the natural decay condition r(s) C (15) (s) s q or s suiciently large and some real number q > 0 (see or instance [19]), and we examine three special cases o Theorem 2.5: (a) the unction r/ has no poles; (b) the unction r/ has a inite number o poles; (c) the unction r/ has an ininite number o poles. Corollary 2.6. Assume that the hypotheses o Theorem 2.5 hold, that L(J)/ is in H p (C + ), and that r/ is an entire unction ( ) such that (15) holds. Then, solution (14) to Equation (13) is simply φ = L 1 L(J). Proo. The proo is analogous to the proo o Corollary 2.1 o [27]. Corollary 2.7. Assume that the hypotheses o Theorem 2.5 hold, and that r/ has a inite number o poles ω i (i = 1,..., N) o order r i to the let o Re(s) = 0. Suppose also that L(J)/ is in H p (C + ), and that the growth condition (15) holds. Then, the solution φ L p (R + ) given by (14) can be represented in the orm φ(t) = 1 2πi σ+i σ i ( ) L(J) e s t (s) ds + in which P i (t) are polynomials o degree r i 1. N P i (t) e ω it i=1, σ > 0, (16) Proo. We irst notice that the quotient r/ H p (C + ), since (L(J) + r)/ and L(J)/ are in H p (C + ), by assumption. Thus, the general solution (14) can be computed via the inversion ormula or the Laplace transorm appearing in Theorem 2.2 (see also [19]). We obtain the ollowing equation in L p (R + ): φ(t) = 1 σ+i ( ) L(J) e s t (s)ds + 1 σ+i ( ) r e s t (s) ds, σ > 0. (17) 2πi 2πi σ i 12 σ i

13 At this level o generality, all we can say about the irst summand is that it belongs to L p (R + ). On the other hand, the growth condition on r/ implies that we can evaluate the second summand via calculus o residues, see [19, Theorem 25.1 and Section 26]. The second integral in the right hand side o (17) is 1 2πi σ+i σ i e s t ( r ) (s) ds = N res i (t), in which res i (t) denotes the residue o r(s) (s) at ω i. In order to compute res i (t) we use the Laurent expansion o r/ around the pole ω i, that is r(s) (s) = a 1,i (s ω i ) + a 2,i (s ω i ) a r i,i (s ω i ) r i + h i(s), (18) where h i is an analytic unction inside a closed curve around ω i. We multiply (18) by e ts /2πi and use Cauchy s integral ormula. We obtain (c. [19, loc. cit.]) res i (t) = P i (t) e ω it, where P i (t) is the polynomial o degree r i 1 given by P i (t) = a 1,i + a 2,i i=1 t 1! + + a t r i 1 r i,i (r i 1)!, (19) and the result ollows. We note that res i (t) is indeed in L p (R + ) since the poles ω i lie to the let o C +. Formula (16) or the solution φ(t) is precisely Carmichael s ormula appearing in [15], as quoted in [5]. Explicit ormulae such as (16) are crucial or the study o the initial value problem we carry out in Section 3. The last special case o Theorem 2.5 is when the unction r(s)/(s) has an ininite number o isolated poles ω i located to the let o Re(s) = a, and such that ω 0 ω 1 ω 2. Corollary 2.8. Assume that the hypotheses o Theorem 2.5 hold, that L(J)/ is in H p (C + ), that the quotient r/ has an ininite number o poles ω n o order r n to the let o Re(s) = 0, and that ω n ω n+1 or n 1. Suppose that there exist curves σ n in the hal-plane Re(s) 0 satisying the ollowing: The curves σ n connect points +ib n and ib n, where the numbers b n are such that the closed curve ormed by σ n together with the segment o the line Re(s) = 0 between the points +ib n and ib n, encloses exactly the irst n poles o r(s)/(s) and, moreover, lim n b n =. 13

14 I the condition lim e st n σ n ( ) r (s)ds = 0 (almost everywhere in t) holds, then the solution φ L p (R + ) given by (14), can be represented in the orm φ(t) = 1 2πi σ+i σ i ( ) L(J) e s t (s) ds + in which P n (t) are polynomials o degree r n 1. P n (t) e ωnt, σ > 0, (20) Proo. As in the previous corollary, we can write solution (14) in the orm φ(t) = 1 2πi +i i n=1 ( ) L(J) e s t (s)ds + 1 2πi +i i e s t ( r ) (s) ds, (21) as an equation in L p (R + ). Since the integral in the second summand converges in L p (R + ), there exists an increasing sequence o real numbers b n such that lim n 1 +ibn e s t 2πi ib n ( ) r (s) ds = L 1 ( r )(t), t 0, almost everywhere in t, in which lim is being taken in the metric topology o C. Following Doetsch [19, p. 170], we compute using residues: 1 +ibn e s t 2πi ib n ( ) r (s) ds + 1 e s t 2πi σ n ( ) r (s) ds = n res j (t). These residues are calculated as in the previous corollary; we skip the details. Taking limits and using the condition appearing in the hypotheses we obtain L 1 ( r )(t) = P n (t) e ωnt, n=1 almost everywhere in t, in which P n (t) are polynomials o degree r n 1. Since we know that L 1 (r/) L p (R + ), so is the above series, and we obtain (20) as an equality in L p (R + ). j=1 Remark. The solution (14) is not necessarily dierentiable, see or instance the example appearing in [27, p. 9]. This means, in particular, that in complete generality we cannot even ormulate initial value problem or equations o the orm ( t )φ = J. 14

15 3 The initial value problem In this section we discuss the initial value problem or equations o the orm ( t )φ = J, t 0, (22) in which is an (in principle arbitrary) analytic unction, in the context o the theory developed in Section Generalized initial conditions First o all we note that our abstract ormula (14) or the solution φ to Equation (22) tells us that expanding the analytic unction r appearing in (14) as a power series (or considering r = r d where r d is the series (7)) φ depends in principle on an ininite number o arbitrary constants. However, this act does not mean that the equation itsel is superluous as sometimes argued in the literature (see or instance [24]), as ormula (14) or φ depends essentially on and J. As remarked in [27], we may think o r as a generalized initial condition : Deinition 3.1. A generalized initial condition or the equation ( t )φ = J (23) is an analytic unction r 0 such that (φ, r 0 ) D or some φ L p (0, ). A generalized initial value problem is an equation such as (23) together with a generalized initial condition r 0. A solution to a given generalized initial value problem {(23), r 0 } is a unction φ satisying the conditions o Deinition 2.3 with r = r 0. Thus, given a generalized initial condition, we ind a unique solution or (22) using (14), much in the same way as given one initial condition we ind a unique solution to a irst order linear ODE. As remarked ater Corollary 2.8, there is no reason to believe that (or a given r) the unique solution (14) to (23) will be analytic: within our general context, we can only conclude that the solution belongs to the class L p (0, ) or some p > 1. It ollows that classical initial value problems do not make sense in ull generality: Deinition 3.1 is what replaces them in the ramework o our theory. Nonetheless, in the next subsection we show that provided and J satisy some technical conditions we can deine initial value problems subject to a inite number o a priori local data. 15

16 Example 3.1. In [3, Section 5], Barnaby considers D-brane decay in a background de Sitter space-time. Up to some parameters, the equation o motion is e 2 ( + 1)φ = αφ 2, (24) in which α is a constant. In de Sitter space-time we have = 2 t β t or a constant β, and so Equation (24) becomes e 2( 2 t +β t) ( 2 t + β t 1)φ = αφ 2. (25) We check that φ 0 (s) = 1/α solves (25) by using Equation (6). We set 0 (s) = 2(s2 +βs) e and we take as generalized initial condition the series r 0 given by (7) with d j = φ (j) 0 (0). We have, L( 0 ( t )φ 0 )(s) = 0 (s)l(φ 0 )(s) = 0 (s) 1 αs = 1 αs. n=1 n n=1 j=1 (n) 0 (0) s n 1 1 α (n) 0 (0) s n j φ (j 1) 0 (0) Thus, the let hand side o (25) evaluated at φ 0 equals 1/α, and the claim ollows. We linearize about φ 0, recalling that rigorously the domain o the nonlocal operator appearing in the let hand side o Equation (25) is ormed by pairs (φ, r), see Deinition 2.2. We obtain that i φ = φ 0 + τ ψ and r = r 0 + τ r or small τ, then the deormation ψ satisies the equation e 2( 2 t +β t) ( 2 t + β t 1)ψ + 2ψ = 0. (26) We can solve this equation very easily or ψ L p (0, ). Let us take p > 0 such that 1/p + 1/p = 1; we consider the entire unction (s) = e 2(s2 +βs) (s 2 + βs 1) + 2, and we let r be a generalized initial condition, that is, r is an analytic unction such that r/ belongs to the class H p (C + ). The most general solution to (26) is ψ = L 1 (r/). (27) Remark. 16

17 An equation similar to (24) appears in [36] in the context o Friedmann cosmology, and even more general equations o interest or cosmology have been considered in [12]. In the Friedmann case we have = t 2 3H(t) t, in which H represents the Hubble rate, and Equation (25) must be changed accordingly. I we set (t, s) = e 2(s2 +3H(t)s) (s 2 + 3H(t)s 1) + 2, we see that in this case the symbol (t, s) is beyond the class o symbols we are considering here. We will come back to this example in another publication. Besides the rigorous theory we considered in the previous papers [26, 27] and the one we develop herein, there exists a large body o literature on approximated, numerical, or ormal solutions to nonlocal equations, see or instance [5, 14, 32, 34, 45] and reerences therein. 3.2 Classical initial value problems In this subsection we point out that i we can unravel the abstract ormula (14) as in Corollaries 2.6, 2.7 or 2.8, then we can deine and solve initial value problems depending on a inite number o initial local data, and not on initial unctions. In Corollaries 2.6, 2.7, and 2.8 there are explicit ormulas or solutions to nonlocal equations o the orm (22). We would expect these ormulas to help us in setting up initial value problems. Now, Corollary 2.6 ixes completely the solution using only and J, and thereore it leaves no room or an initial value problem. On the other hand, ormula (20) o Corollary 2.8 depends on an ininite number o parameters and, as explained in [27], this act implies that we cannot ensure dierentiability o the solution (and hence existence o initial value problems) using only conditions on and J. On the other hand, the explicit ormula (16) tells us that a solution or the linear equation (22) is uniquely determined by, J, and a inite number o parameters related to the singularities o the quotient r/. It is thereore not unreasonable to expect that, by using these initely many parameters, we can set up consistent initial value problems, as conjectured in [34]. In order to ensure dierentiability o solutions, we use the ollowing two technical lemmas: Lemma 3.1. Let J be a unction such that L(J) exists and let be an analytic unction. Suppose that there exist an integer M 0 and a real number σ > 0 such that ( ) L(J)(σ + iy) y y n (σ + iy) belongs to L 1 (R) (28) 17

18 or each n = 0,..., M; then the unction is o class C M. σ i t 1 2πi σ+i σ i e s t ( L(J) Proo. Ater a suitable change o variables we have 1 σ+i ( ) L(J) e s t (s) ds = eσt 2πi 2π ) (s) ds (29) e ity ( L(J)(σ + iy) (σ + iy) ) dy. It ollows that the let hand side is o class C M as a unction o t, since (28) implies we can dierentiate the right hand side o the above equation with respect to t at least M times. We remark that Doetsch s representation theorem (Theorem 2.2) implies that it is enough to assume condition (28) or some σ > 0 instead o or every σ > 0. Lemma 3.1 implies the ollowing: Lemma 3.2. Let the unctions and J satisy the conditions o Corollary 2.7, and also that they satisy (28) or some σ > 0. Then, the solution (16) to the nonlocal equation ( t )φ(t) = J(t) (30) is o class C M or t 0, and it satisies the identities ( ) N n φ (n) n (0) = L n + ωi k d n k k dt n k P i (t), t=0 n = 0,..., M, (31) or some numbers L n. to be i=1 k=0 This result is proven in [26, 27]. The numbers L n, or n = 0,..., M, are computed ( L n = dn 1 dt n t=0 2πi σ+i σ i ( ) ) L(J) e st (s) ds. (32) Our main result on initial value problems is the ollowing. (The word generic appearing in the enunciate will be explained in the proo, see paragraph below Equation (33) ). Theorem 3.3. We ix real numbers 1 < p 2 and p > 0 such that 1/p + 1/p = 1, and we also ix an integer N 0. Let be a unction which is analytic in a region D which contains {s C : Re(s) > 0}, and let J be a unction in L p (R + ) 18

19 satisying the conditions L(J) H p (C + ) and L(J)/ H p (C + ). We choose points ω i, i = 1,, N, to the let o Re(s) = 0, and positive integers r i, i = 1,..., N. Set K = N i=1 r i and assume that or some σ > 0 condition (28) holds or each n = 0,..., M, M K. Then, generically, given K values φ 0,..., φ K 1, there exists a unique analytic unction r 0 such that (α) r 0 Hp (C + ) and it has a inite number o poles ω i o order r i, i = 1,..., N; (β) L(J) + r 0 H p (C + ); (γ) r 0 (s) M or some q 1 and s suiciently large. s q ( Moreover, the unique solution φ = L 1 L(J)+r0 to Equation (30) belongs to L p (R + ), is o class C K, and it satisies φ(0) = φ 0,..., φ (K 1) (0) = φ K 1. Proo. We consider the K arbitrary numbers φ n, n = 0, 1,..., K 1. Recalling (31), Lemma 3.1, and Lemma 3.2, we set up the linear system ( ) N n n φ n = L n + ωi k d n k k dt n k P i (t), t=0 n = 0... K 1, (33) i=1 k=0 in which the unknowns are the coeicients o polynomials P i (t) = a 1,i + a 2,i and the numbers L n are given by (32). ) t 1! + + a t r i 1 r i,i (r i 1)!, System (33) can be solved (generically, this is, away rom the variety in R N deined by all the points ω i or which the main determinant o the linear system (33) vanishes) uniquely in terms o the data φ n. We deine r 0 on the hal-plane {s C : Re(s) > 0} as ollows: ( N ) r 0 (s) = (s) L P i (t)e ω it (s). (34) i=1 Then, on this hal-plane we have the identity L 1 (r 0 /)(t) = N P i (t)e ω it i=1 (35) 19

20 or all t R +. Let us prove that r 0 belongs to Hp (C + ). First o all, we have the standard ormula L(bt n e ωjt b )(s) = (s ω j ) n+1 or any n N and b C, and so the unction r 0 is analytic on the hal-plane Re(s) > 0. We show that µ p ( r 0, x) is uniormly bounded on x > 0. For this, it is enough to prove that the unction µ p (L(bt n e ωjt ), x) is uniormly bounded or x > 0. Indeed, since the poles ω j := a j + ib j o r 0 satisy a j = Re(ω j ) < 0, we have x a j > a j > 0 or x > 0. Thereore: L(bt n e ωjt )(x + iy)) p dy = b = b = b < 2 b <. 0 dy x + iy ω j (n+1)p dy ((x a j ) 2 + (y b j ) 2 ) n+1 2 p dξ ((x a j ) 2 + ξ 2 ) n+1 2 p dξ (a 2 j + (36) ξ2 ) n+1 2 p Thus, µ p (L(bt n e ω jt ), x) is uniormly bounded or x > 0. This shows that r 0 belongs to H p (C + ), and (α) ollows. We easily check that r 0 / also satisies conditions (β), and (γ) appearing in the enunciate o the theorem; we omit the details. Now we deine the unction φ(t) = 1 2πi σ+i σ i ( ) L(J) e st (s) ds + N P i (t) e ω i t in L p (R + ). We claim that this unction solves Equation (30) and that it satisies the conditions appearing in the enunciate o the theorem. In act, the oregoing analysis implies that φ(t) = 1 2πi σ+i σ i i=1 ( ) L(J) e st (s) ds + L 1 (r 0 /)(t), and this is precisely the unique solution to (30) appearing in Corollary 2.7 or r = r 0. (37) It remains to show that this solution satisies φ (n) (0) = φ n or n = 0,..., K 1. Indeed, condition (28) tells us that φ(t) is at least o class C K and clearly ( ) N n φ (n) n (0) = L n + ωi k d n k k dt n k P i (t) (38) t=0 i=1 k=0 20

21 in which ( L n = dn 1 dt n t=0 2πi σ+i σ i ( ) ) L(J) e st (s) ds Comparing (33) and (38) we obtain φ (n) (0) = φ n, n = 0,..., K 1.. (39) Remark. The proo o Theorem 3.3 breaks down i p = 1, as we lose the uniorm bound (36). This is the reason why, as advanced in Subsection 2.1, L 1 -correspondence theorems (see [18, Theorem 2] and [19, Theorem 28.2]) do not allow us to obtain a meaningul theory o initial value problems. On the other hand, it is not hard to develop Section 2 within the ramework o [18, 19]. Theorem 3.3 tells us that we can reely chose the irst K derivatives φ (n) (0), n = 0,..., K 1 o the solution φ to Equation (30) but, rom n = K onward, i the derivative φ (n) (0) exists, it is completely determined by (38) and (39). Thus, it does not make sense to ormulate an initial value problem ( t )φ(t) = J(t), φ (n) (0) = φ n, with more than K arbitrary initial conditions. The above proo shows that the a priori given points ω i become the poles o the quotient r 0 /, that the a priori given numbers r i are their respective orders, and that it is essential to give this inormation in order to have meaningul initial value problems. I no points ω i are present, the solution to the nonlocal equation (30) is simply φ = L 1 (L(J)/), a ormula which ixes completely ( or and J satisying (28) ) the values o the derivatives o φ at zero. This discussion motivates the ollowing deinition: Deinition 3.2. A classical initial value problem or nonlocal equations is a triplet ormed by a nonlocal equation ( t )φ = J, (40) a inite set o data: { N 0 ; {ω i C} 1 i N ; {r i Z, r i > 0} 1 i N ; {{φ n } 0 n K 1, K = } N i=1 r i} (41) and the conditions φ(0) = φ 0, φ (0) = φ 1,.φ (K 1) (0) = φ K 1. (42), 21

22 A solution to a classical initial value problem given by (40), (41) and (42) is a pair (φ, r 0 ) D satisying the conditions o Deinition 2.3 with r = r 0 such that φ is dierentiable at zero and (42) holds. Theorem 3.3 implies that this deinition makes sense. From its proo we also deduce the ollowing important corollaries: Corollary 3.4. We ix 1 < p 2 and p > 0 such that 1/p + 1/p = 1. Let be unction which is analytic on the hal-plane {s C : Re(s) > 0}, and ix a unction J in L p (0, ) such that L(J) H p (C + ) and L(J)/ H p (C + ). Then, generically (in the sense o Theorem 3.3), the classical initial value problem (40) (42) has a unique solution which depends smoothly on the initial conditions (42). Proo. The unction (37) is the unique solution to (40) and it satisies (42). Moreover, the coeicients o the polynomials P i appearing in (37) depend smoothly on the data φ n and ω i, since they are solutions to the linear problem (33). Corollary 3.5. We ix 1 < p 2 and p > 0 such that 1/p + 1/p = 1. Let be unction which is analytic on the hal-plane {s C : Re(s) > 0}, and ix J in L p (0, ) such that L(J) H p (C + ) and L(J)/ H p (C + ). Suppose also that there exist a positive integer K such that or some σ > 0 condition (28) holds, and consider a set o complex numbers φ 0, φ 1,, φ K 1. Then, there exists a classical initial value problem (40) (42). Proo. Since K > 0 we can ind positive integers N, r 1,, r N such that K = N i=1 r i. Also, we can choose N complex numbers ω i, 1 i N to the let o Re(s) = 0. Thus, we have constructed the data (41). The nonlocal equation (40) is ( t )φ(t) = J(t), and conditions (42) are determined by the given complex numbers φ 0, φ K 1. Remark. We note that there is at least one natural way to choose the poles ω i appearing in the proo o Corollary 3.5: i the symbol has K zeroes (counting with multiplicities), say {z 1, z 2, z K }, to the let o Re(s) < 0, and condition (28) holds or n = 0,, K, we can obtain a unique solution to the initial value problem (40) (42) i we choose ω i = z i. In this case, the determinant o the linear system (33) is precisely the non-zero determinant o the K K-Vandermonde matrix A K = (a ji ) with a ji = z j 1 i, 1 j, i K. 22

23 4 Zeta nonlocal equations In this section we apply our previous approach to zeta nonlocal equations o the orm ζ h ( t )φ(t) = J(t), t 0, (43) in which h is a real parameter and the symbol ζ h is the shited Riemann zeta unction ζ h (s) := ζ(h + s) = n=1 1 n s+h. These equations are motivated by the cosmological models appearing in [2, 21, 22, 23]. In turn, it has been called to the authors attention by A. Koshelev that the approach o [2] is at least partially based on [32], in which the author introduces a ormal schemme or analyzing some nonlocal equations o interest or cosmology. We recall some properties o the Riemann zeta unction ζ(s) := 1 n=1 n, s Re(s) > 1, ollowing [31]. It is analytic on its domain o deinition and it has an analytic continuation to the whole complex plane with the exception o the point s = 1, at which it has a simple pole with residue 1. The analytic continuation o the Riemann zeta unction will be also denoted by ζ, and we will reer to it also as the Riemann zeta unction. From standard properties o the Riemann zeta unction (see [31]) we have that the shited Riemann zeta unction ζ h is analytic or Re(s) > 1 h, and uniormly and absolutely convergent or Re(s) σ 0 > 1 h. We also ind (see [31, Chp. I.6]) that ζ h has ininite zeroes at the points { 2n h : n N} (we call them trivial zeroes ) and that it also has nontrivial zeroes in the region h < Re(s) < 1 h (we call this region the critical region o ζ h ). The Euler product expansion or the shited Riemann zeta unction is ζ h (s) = p P n=1 ( 1 1 ) 1, p s+h where P is the set o the prime numbers. Thereore, or Re(s) = σ > 1 h, we have 1 ( ζ h (s) = 1 1 ) p s+h = µ(n) n s+h p P n=1 1 n 1 + dx σ+h x = σ + h σ+h σ + h 1, (44) 1 23

24 where µ( ) is the Möebius unction deined as ollows: µ(1) = 1, µ(n) = 0 i n is divisible by the square o a prime, and µ(n) = ( 1) k i n is the product o k distinct prime numbers, see [31, Chp. II.2]. We study Equation (43) or values o h in the region (1, ), since in this case ζ h is analytic or Re(s) > 0 and the theory developed in the previous sections apply. We start with the ollowing lemma: Lemma 4.1. We ix 1 < p 2 and p > 0 such that 1/p + 1/p = 1. Let us assume that J L p (R + ) and that L(J) is in the space H p (C + ). Then, the unction F = L(J) ζ h belongs to H p (C + ). Proo. We have: 1. The unction F is clearly analytic or every s such that Re(s) > Since L(J) H p (C + ) we have that µ p (L(J), x) is uniormly bounded or x > 0, Now or x > 0 and using inequality (44), we obtain µ p (F, x) = Since the unction x ollows. ( R p 1 L(J)(σ + iy) ζ h (x + iy) ( L(J)(x + iy) p dy R x + h x + h 1 x + h x + h 1 ) 1 p dy ) 1 p <. is uniormly bounded or x 0, the result Now we note that we can replace the general condition (28) or the ollowing assumption on the unction J: (H) For some M 0, or each n = 1, 2, 3,, M and or some σ > 0 we have, t 1 2πi y y n L(J)(σ + iy) L 1 (R). Condition (H) is enough to ensure dierentiability o the unction σ+i ) (s)ds, σ i e st ( L(J) ζ h as asked in the hypotheses o Lemma 3.1. Our main theorem on classical initial value problems or the zeta non-local equation (43) is the ollowing 24

25 Theorem 4.2. We ix 1 < p 2 and p > 0 such that 1/p + 1/p = 1. Let ζ h be the shited Riemann zeta unction, and assume that J L p (R + ) with L(J) H p (C + ). We also ix a number N 0, a inite number o points ω i, i = 1,..., N, to the let o Re(s) = 0, and a inite number o positive integers r i. We set K = N i=1 r i and we assume that condition (H) holds or all n = 0,..., M, M K. Then, generically, given K initial conditions, φ 0,..., φ K 1, there exists a unique analytic unction r 0 such that (α) r 0 ζ h H p (C + ) and it has a inite number o poles ω i o order r i, i = 1,..., N to the let o Re(s) = 0 ; (β) L(J) + r 0 H p (C + ); ζ h (γ) r 0 (s) ζ h M or some q 1 and s suiciently large. s q Moreover, the unique solution φ to Equation (43) given by (14) with r = r 0 is o class C K and it satisies φ(0) = φ 0,..., φ (K 1) (0) = φ K 1. Proo. The proo consists in checking that the hypotheses o Theorem 3.3 hold. In act, ζ h is analytic on {s C : Re(s) > 0}, and Lemma 4.1 tells us that L(J)/ζ h belongs to H p (C + ). 5 An L 2 (R + )-theory or linear nonlocal equations In this section we show that in the p = p = 2 case o the oregoing theory, we can justiy rigorously the interpretation o ( t ) as an operator in ininitely many derivatives on an appropriated domain. Our approach uses a notion o analytic vectors, motivated by Nelson s classical paper [37]. Deinition 5.1. Let A be a linear operator rom a Banach space V to itsel, and let : R C be a complex valued unction, such that (n) (0) exist or all n 0. We say that v V is a -analytic vector or A i v is in the domain o A n or all n 0 and the series deines a vector in V. (n) (0) A n v, As stated in Section 2, the Paley-Wiener theorem (see [28, 47]) is the ollowing special case o Doetsch s representation theorem: 25

26 Theorem 5.1. The ollowing assertions hold: 1) I g L 2 (R + ), then L(g) H 2 (C + ). 2) Let G H 2 (C + ). Then the unction g(t) = 1 2πi σ+i σ i e st G(s)ds, σ > 0, is independent on σ, it belongs to L 2 (R + ) and it satisies G = L(g). Moreover the Laplace transorm L : L 2 (R + ) H 2 (C + ) is a unitary operator. We examine existence o analytic vectors or the operator t on L 2 (R + ). Lemma 5.2. Let be an analytic unction on a region containing zero, and let R 1 be the maximun radius o convergence o the Taylor series T (s) := (n) (0) s n. a) I p is a polynomial on R +, and I a inite interval on R +, then the unction ψ := p χ I is an -analytic vector or t on L 2 (R + ). b) Let R 1 > 1. I ψ C (R + ) L 2 (R + ) such that or all n 1 and some h L 2 (I) or I equal to either R or R +, we have ψ (n) L 2 (R + ) c(n) h L 2 (I), with {c(n)} n N =: c l 1 (N). Then ψ is an -analytic vector or t on L 2 (R + ). } Proo. Part a) is immediate. For b) we have: Since R 1 > 1, the sequence { (n) (0) is bounded; thereore, there is a positive constant C such that (n) (0) C or every n N. Now, by Minkowski inequality we have, n N (n) (0) ψ (n) L 2 (R + ) (n) (0) ψ (n) L 2 (R + ) C ψ (n) L 2 (R + ) C h L 2 (A) c(n) C c l 1 (N) h L 2 (A). Remark. We present two large amilies o unctions ψ which meet conditions b) o Lemma

27 Let k > 1 be a parameter and consider the unctions ψ(t) := e t k, t 0; then we have ψ (n) (t) = ( 1)n e t k n k, thereore ψ (n) (t) = 1 e t k n k c(n) := 1, we have {c(n)} k n n N l 1 (N). and i we deine We recall that an arbitrary entire unction φ o exponential type τ which is also in L 2 (R), satisies the generalized L 2 -Berstein inequality φ (n) L 2 (R) τ n φ L 2 (R), see [38, Chp.3]. We denote by Exp 2 1(C) the space o entire unctions o exponential type τ < 1 which are L 2 -unctions on R. Then, or any φ Exp 2 1(C) with exponential type τ φ > 0, the unction ψ := χ R+ φ satisies part b) o the lemma with h = φ. In particular, let φ be a smooth unction on R with compact support in [ τ, τ] R with τ < 1. Then its Fourier transorm F(φ)(t) := (2π) 1/2 τ τ e ixt φ(x)dx is an L 2 (R)-unction and it has an extension to an entire unction Φ which is o exponential type τ. The unction ψ := χ R+ Φ satisies part b) o the lemma. Deinition 2.2 o the operator ( t ) restricts to the present L 2 -context. We state it explicitly or the reader s convenience. Deinition 5.2. Let be an analytic unction on a region which contains zero and the hal-plane {s C : Re(s) > 0}, and let H be the space o all unctions which are analytic on regions o C. We consider the subspace D o L 2 (R + ) H consisting o all the pairs (φ, r) such that (φ, r) = L(φ) r (45) belongs to the space H 2 (C + ). The domain o ( t ) as a linear operator rom L 2 (R + ) H to L 2 (R + ) is the set D. I (φ, r) D then ( t ) (φ, r) = L 1 ( (φ, r) ) = L 1 ( L(φ) r). (46) We show that this deinition is not empty, and that in act the domain D is quite large. The act that the Laplace transorm is an unitary operator plays an essential role at this point: Proposition 5.3. Let be a unction which is analytic on a region containing C +, and let R 1 > 1 be the maximum radius o convergence o the Taylor series 27

28 T (s) := (n) (0) s n. Let φ be a smooth -analytic vector or t in L 2 (R + ) and suppose that the sequence {d j = φ (j) (0)} satisies the condition o Lemma 2.3. Then, there exists an analytic unction r e on C + such that (φ, r e ) is in the domain D o ( t ). Proo. From equation (7) and Lemma 2.3 we can deine the ollowing analytic unction ˆφ on {z C : z < R 1 } : ˆφ(s) (n) (0) n := s n (n) (0) L(φ)(s) φ (j 1) (0) s n j n=1 j=1 ( ) (n) (0) n = s n L(φ)(s) φ (j 1) (0) s n j j=1 (n) (0) = L( t n (φ))(s) (n) (0) = e st t n (φ)(t)dt. 0 Using this last equality and the act that the Laplace transorm is an isometric isomorphism rom L 2 (R + ) onto H 2 (C + ), see Theorem 5.1, we have ˆφ(s) = e st 0 n=1 (n) (0) t n (φ)(t)dt. (47) Thereore, on the disk {z C : z < R 1 } we have the equation ( ) (n) (0) L t n (φ)(t) (s) = (s)l(φ)(s) r(s), (48) n=1 where r(s) = n (n) (0) n=1 j=1 φ (j 1) (0) s n j. Now, we stress the act that the right hand side o (47) belongs to the Hardy space H 2 (C + ); it ollows that we can extend ˆφ(s) via analytic continuation to a unction ˆφ e on the hal-plane Re(s) > 0. This unction belongs to the Hardy space H 2 (C + ) by construction. Also, we note that Equation (48) implies r(s) = (s)l(φ)(s) ˆφ e (s) (49) or s < R 1. However, the right hand side o (49) is deined on Re(s) > 0, and thereore it deines an analytic continuation r e o the series r to the hal-plane Re(s) > 0. Thus, on Re(s) > 0 we have the equation: ˆφ e (s) = (s)l(φ)(s) r e (s). Since ˆφ e is in H 2 (C + ), we have that (φ, r e ) D. 28

29 Example 5.1. It ollows rom our discussion on the unction ζ h introduced in Section 4, that ζ h is analytic around zero or h > 1 large enough, and that thereore we have the power series expansion ζ h (s) = a n (h)s n. Again or appropriate h > 1, we can assume that its maximum radius o convergence is R 1 > 1. Then, Lemma 5.2 implies that there exists a large class o ζ h -analytic vectors φ L 2 (R + ) and moreover, the above proposition applies. Thus, ζ h ( t ) is a well-deined operator on (a subspace o) L 2 (R + ) H. An easy corollary o Proposition 5.3 is the ollowing: Corollary 5.4. Let be an entire unction and let φ be a smooth -analytic vector or t in L 2 (R + ). Suppose that the sequence {d j = φ (j) (0)} satisies d j CR j or 0 < R < 1. Then (φ, r) D, in which r is the series deined in the above proo. The proo o Corollary 5.4 consists in noting that the stated hypotheses allow us to apply Lemma 2.3. Proposition 5.3 and Corollary 5.4 imply that (i is entire) the operator ( t ) is an operator in ininitely many derivatives on the space o smooth -analytic vectors. In act, let φ be a -analytic vector or t in L 2 (R + ) and let r(s) = n n=1 j=1 (n) (0) φ (j 1) (0) s n j. I conditions o corollary 5.4 hold, then ( ) (n) (0) L t n (φ) = L(φ) r = L(( t )φ), and thereore ( t )φ = (n) (0) t n (φ). 6 Discussion As mentioned in Section 1, the ollowing nonlocal equation appears naturally in the study o a zeta nonlocal scalar ield model in string theory (See [21, 22, 23]; recall 29

30 that we are using signature so that in the dimensional case the d Alambert operator is 2 t ): ζ( 2 t + h)φ = AC n h φ n, t 0, (50) where AC means analytic continuation. We stress, ater [1], that it is natural to consider the restriction t 0 since classical versions o cosmological models contain singularities at the beginning o time. Equation (50) motivates the study o the ollowing nonlocal linear equations ζ( 2 t + h)φ = J, t 0, (51) or appropriate unctions J. Interestingly, the behavior o the symbol ζ(s 2 + h) is quite dierent to the behavior o the symbol ζ(s + h) appearing in Section 4. We show here that a study o Equation (51) requires a generalization o the theory developed in the above sections. First o all, rom the properties o the Riemann zeta unction, we observe that the symbol ζ(s 2 + h) = 1 n s2 +h is analytic in the region Γ := {s C : Re(s) 2 Im(s) 2 > 1 h}, which is not a hal-plane; also we can note that its poles are the vertices o the hyperbolas Re(s) 2 Im(s) 2 = 1 h and its critical region is the set {s C : h < Re(s) 2 Im(s) 2 < 1 h}. In act, according to the value o h we have: (52) i) For h > 1, Γ is the region limited by the interior o the dark hyperbola Re(s) 2 Im(s) 2 = 1 h containing the real axis: The poles o ζ(s 2 + h) are the vertices o dark hyperbola, indicated by two thick dots. The trivial zeroes o ζ(s 2 + h) are indicated by thin dots on the imaginary axis; and the non-trivial zeroes are located on the darker painted region (critical region). 30