On the Fourier analysis of the Einstein-Klein-Gordon system: Growth and Decay of the Fourier constants

We consider the $(1 + 3)$-dimensional Einstein equations with negative cosmological constant coupled to a spherically-symmetric, massless scalar field and study perturbations around the Anti-de Sitter spacetime. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator as well as on some oscillatory integrals.

Now, the AdS solution takes the form g AdS (t, x, ω) = l 2 cos 2 (x) −dt 2 + dx 2 + sin 2 (x)dω 2 implying that it is conformal to half of the Einstein static universe. One can see that null geodesics reach the conformal spatial infinity I := {x = π 2 } in finite time even though the spatial distance from any point (t, x) with 0 ≤ x < π 2 to I is infinite. The particular characteristic of the AdS solution as well as of all the asymptotically AdS spacetimes (aAdS) (that is spacetimes that approach the AdS solution at infinity fast enough and share the same conformal boundary) is that the conformal spatial infinity is a time-like cylinder R × S d−1 . Consequently, the AdS metric is not globally hyperbolic and in order to study the evolution of the field φ on the underline manifold ( M, g AdS ) one has to prescribe boundary conditions also on I in addition to the initial data on the {t = 0} slice.
It is well know that the Minkowski space is a ground state among asymptotically flat spacetimes [63]. The AdS spacetime also enjoys a similar variational characterization due to the positive energy theorem which states that for solutions to the Einstein's equations with matter    R αβ − 1 2 g αβ R + Λg αβ = 8π ∂ α φ∂ β φ − 1 2 g αβ (∂φ) 2 , g φ = 0 which are globally regular and satisfy a reasonable energy condition, the AdS space is a ground state among asymptotically AdS spacetimes [35,66]. As far as the initial-boundary value problem is concerned, Smulevici-Holzegel [38] (for Dirichlet boundary conditions) and Warnick-Holzegel [39] (for more general boundary conditions) proved its local well-posedness.
Once the local well-posedness is established, an important question for the AdS solution (as for any ground state) is whether it is stable or not, meaning whether small perturbations of the solution on the {t = 0} slice remain small for all future times or not. For the Minkowski spacetime such a question has been answered by Christodoulou-Klainerman [25] and for the de-Sitter spacetime by Friedrich [34], who proved its stability.
The main mechanism responsible for the stability of the Minkowski spacetime is the dissipation of energy by dispersion. In the case of AdS solution, such a mechanism is no longer present. For "reflective" boundary conditions on the conformal infinity, waves which start at any point inside the region {0 ≤ x < π 2 } and propagate outwards are reflected on I and return back to into the region from where they started [22]. Such boundary conditions are confining enough forcing the AdS solution to act as a closed universe (in terms of its fields inside). Horowitz [40] relates this fact to the singularity theorem of Hawking-Penrose [37] (which states that closed universes are generically singular) suggesting that the AdS solution should be singular.
Although the conjecture on the instability of the AdS spacetime was first announced by Dafermos [27] and Dafermos-Holzegel [50] in 2006, the first work in this direction was a numerical study of Bizoń-Rostworowski [59]. In particular, Bizoń-Rostworowski [59] considered the spherically symmetric Einstein massless scalar field equations with negative cosmological constant in (1 + 3)−dimensions and estabilshed strong numerical (as well as analytical) results which show that the AdS solution to the Einstein equations (although linearly stable) is nonlinearly unstable against the formation of a black hole under arbitrarily small and generic perturbations. In their work [59], Bizoń-Rostworowski used specific Gaussian-type initial data and concluded that such initial data evolve to a wave which, as it propagates in time, collapses quickly and an apparent horizon appears. Furthermore, Dias-Horowitz-Santos [30] considered pure gravity with a negative cosmological constant and provided additional support strengthening the evidence that the AdS spacetime might be nonlinearly unstable. Similar results have been obtained by Jałmużna-Rostworowski-Bizoń [41] and Buchel-Lehner-Liebling [2] for higher dimensions. Furthermore, Choptuik [24] also studied the mechanism of the spherically symmetric collapse of a scalar field with a general time and radial spatial dependent metric and for several families of initial data.
In addition, Bizoń-Rostworowski [59] also conjectured that there may exist specific initial data (islands of stability) for which the evolution of small perturbations around the AdS solution remains globally regular in time. Furthermore, Maliborski-Rostworowski [51] considered the spherically symmetric Einstein-massless scalar field equations with negative cosmological constant in d + 1 dimensions with d ≥ 2 and provided reliable numerical evidence indicating that in fact time-periodic solutions may exist for non-generic initial data. They were able to construct these solutions using both nonlinear perturbative expansions and fully nonlinear numerical methods. Similar conjectures were made by Dias-Horowitz-Marolf-Santos [29] who argued that many aAdS solutions are nonlinearly stable (including geons, boson stars, and black holes) and by Buchel-Liebling-Lehner [1] who considered boson stars in global AdS spacetime and study their stability. Furthermore, rigorous proof of the instability of the AdS solution was given by Mochidis who considered the Einstein-null dust system [58] and the Einstein massless Vlasov system [57].
Finally, the AdS spacetime as well as aAdS spacetimes play an important role in theoretical physics due to the celebrated AdS/CFT correspondence [62] which was brought to light by Maldacena [52,53]. Such a duality relates events that occur within a universe with a negative cosmological constant (AdS) to events in conformal field theories (CFT) and has important applications [2,36,55,62].

Spherical symmetric ansatz.
To make the problem (1.1) trackable we assume spherically symmetric metrics. However, by Birkhoff's theorem, spherically symmetric solutions to the Einstein equations in vacuum are static and therefore we add matter to generate dynamics. We consider the Einstein-Klein-Gordon equation for a self-gravitating massless scalar field, that is the wave equation coupled to the Einstein equations with matter,    R αβ − 1 2 g αβ R + Λg αβ = 8π ∂ α φ∂ β φ − 1 2 g αβ (∂φ) 2 , g φ = 0.
For simplicity, we fix the spatial dimension d = 3. Following the work of Bizoń-Rostworowski [59] we parametrize the spacetime metric g by the spherical symmetric ansatz for (t, x, ω) ∈ M. Under this ansatz the wave equation becomes We transform the second order partial differential equation (1.3) for φ into a first order system by setting Φ(t, x) = ∂ x φ(t, x), Π(t, x) = 1 A(t, x)e −δ(t,x) ∂ t φ(t, x). (1 − A(t, x))e −δ(t,x) = cos 3 (x) sin(x) x 0 e −δ(t,y) Φ 2 (t, y) + Π 2 (t, y) (tan(y)) 2 dy, and now (1.3) can be written as is the operator which governs linearized perturbations of AdS solution. The solutions to the eigenvalue problem L[f ] = ω 2 f subject to Dirichlet boundary conditions on the conformal boundary I = {x = π 2 } fall into the hypergeometric class and hence can be found explicitly. For a rigorous definition of the spectrum, see the Appendix in the work of Bachelot [4]. Specifically, the eigenvalues read ω 2 j := (3 + 2j) 2 , j = 1, 2, . . . and eigenfunctions are weighted Jacobi polynomials, For the definition, basic properties and an introduction to the Jacobi polynomials P α,β j , see Chapter 4, page 48 in Szegö's book [64]. In addition, the linearized operator L is self-adjoint with respect to the weighted inner product For the definition of the domain in which the linearized operator is self-adjoint, see also the Appendix in the work of Bachelot [4]. Finally, note that the eigenvalues are strictly positive and hence the linear problem is stable.

Main result and preliminaries
We consider the spherically symmetric Einstein-massless scalar field equations with negative cosmological constant under the spherically symmetric ansatz (1.2), and we are mainly interested in the asymptotic behaviour of the Fourier constants which appear in the analysis of perturbations around the AdS solution (Φ, Π, A, δ) = (0, 0, 1, 0).

2.1.
Statement of the main result. Specifically, we consider two types of perturbations.
On the one hand, in light of recent work Maliborski-Rostworowski [51], although the series may not converge, we seek a solution of the form where ψ 2λ+1 ,σ 2λ+1 ,ξ 2λ and ζ 2λ are all periodic in time. Here, ψ 1 , σ 1 , ξ 2 , ζ 2 , ω γ,0 , γ will be chosen later. Furthermore, with a slight abuse of notation, we use the same letters to denote the variables with respect to the (τ, x) and (t, x). On the other hand, we still assume that (Φ, Π, A, δ) are all close to the AdS solution (0, 0, 1, 0) but expand them using a finite sum (2.14) for some error terms Ψ, Σ, B, Θ, η γ where Φ 1 , Π 1 , A 2 , δ 2 are all explicit periodic expressions in time and will be chosen later together with γ. We formulate our main result.
Proof. For the first part, we make use of the facts where x = x(z) = 1 2 arccos(z) and j = 0, 1, . . . . These identities can be found in Chapter 4, page 60, equation (4.1.8) and Chapter 4, page 63, equation (4.21.7) in Szegö's book [64]. Then, the closed formula for e j follows by the chain rule. Indeed, we define z = cos(2x) and compute and so Finally, since j is an integer, we conclude The closed formula for e ′ j follows by differentiating the closed formula for e j . Using these formulas, the orthogonality properties are straightforward. For the fact that the set {e j : j = 0, 1, 2, . . . } forms a basis for L 2 0, π 2 with respect to the weighted inner product (1.4), see the Appendix in the work of Bachelot [4]. In order to show that { e ′ j ω j : j = 0, 1, 2, . . . } also forms a basis for the same function space, one has to prove that f e ′ j ω j = 0, ∀j = 0, 1, 2, · · · =⇒ f = 0. 8 To this end, we define and use the fact that j e j together with integration by parts to compute for all j = 0, 1, 2, . . . . Now, we use the fact that {e j : j = 0, 1, 2, . . . } forms a basis to get F = 0 which in turn implies f = 0.
Remark 2.5. We find the leading order terms as i −→ ∞ and for all x ∈ 0, π 2 . These estimates are uniform with respect to the weighted Indeed, for large i, we estimate The proofs of (2.15) and (2.16) are given in the Appendix, Lemma A.1.
Next, we prove L ∞ −bounds for quantities related to the eigenfunctions.
Finally, we establish the asymptotic behaviour of specific oscillatory integrals which will appear later. Lemma 2.8 (Oscillatory integrals). For any N ∈ N, we have as a −→ ∞ with a ∈ N. Here, c := 1 π − π + 2 1 0 sin(2πy) y dy ≃ 0.01302. 13 Proof. For the first integral, we use Lemma 2.7 to infer The second integral follows similarly. Lemma 2.7 implies for all a ≥ 3, since Next, for the third integral, Lemma 2.7 also yields for all a ≥ 3, since 14 Finally, we conclude with the fourth integral. First, observe that and therefore, for any ǫ > 0, Second, we set ǫ = π a and get cos(2aǫ) = 1, sin(2aǫ) = 0. Now, Lemma 2.7 shows that valid for all a ≥ 1, and we change variables y = a π x to infer π 2 π a cos(2ax) Remark 2.9. A straightforward computation yields and hence the third integral of Lemma 2.8 also gives for a −→ ∞.
Finally, we will also use standard integration by parts.
Lemma 2.10 (Integration by parts). Let a ∈ N and N ∈ N. Assume that F is differentiable in 0, π 2 and continuous in 0, π 2 . Then, Proof. The proof is a straight forward application of integration by parts.
The following result is a direct consequence of Lemma 2.10.
Lemma 2.11. Let b ∈ N and N ∈ N. Assume that F is differentiable in 0, π 2 , continuous in 0, π 2 with uniformly bounded derivatives in 0, π 2 . Then, Proof. All these asymptotic expansions follow from Lemma 2.10 just by computing the boundary terms. Note that if a is even, namely a = 2b for some b ∈ N, then sin(a π 2 ) = 0 and cos(a π 2 ) = (−1) b whereas if a is odd, namely a = 2b + 1 for some b ∈ N, then sin(a π 2 ) = (−1) b and cos(a π 2 ) = 0. For example, Lemma 2.10 yields The other asymptotic expansions follow similarly.
With these auxiliary results at hand we proceed to the main analysis of the Fourier constants.
3.1. Definition of the Fourier constants. First, we compute the density where, for all λ = 0, 1, 2, . . . , Next, we substitute these expressions into (2.1)-(2.2)-(2.3)-(2.4), collect terms of the same order in ǫ and obtain a hierarchy of equations and From the set of all eigenvalues {e i } ∞ i=0 to the linear operator, we choose a dominant mode e γ for some γ ∈ {0, 1, 2, . . . } and pick Now, for each λ = 0, 1, 2, . . . , we expand the coefficients ψ 2λ+1 , σ 2λ+1 , ξ 2λ , ζ 2λ in terms of the eigenvalues of the linearized operator, namely substitute these expressions into the recurrence relations above, take the inner product (·|e ′ m ) for the first equation, the (·|e m ) for all the other equations and use their orthogonality properties (e ′ n |e ′ m ) = ω 2 n δ nm , (e n , e m ) = δ nm . Using the notation where all the interactions with respect to the spatial variable x ∈ 0, π 2 are included into the following Fourier constants x e m (y) sin(y) cos(y)dy tan 2 (x)dx, , x e m (y) sin(y) cos(y)dy tan 2 (x)dx.
We also find and use Lemma 2.4 to compute for all m = 0, 1, 2, . . . . In addition, we differentiate the first equation with respect to τ and use the second to obtain the harmonic oscillator equation where the source term is given by Finally, we make use of the variation constants formula to solve (3.1) and find In conclusion, we get for all m = 0, 1, 2, . . . the following recurrence relations. For all
Then, the integral produces a non-periodic term since Such secular terms are also produced when there exists an m = 0, 1, 2, . . . such that In other words, 2λ+1 (τ ) contains non-periodic terms. Maliborski and Rostworowski [51] were able to numerically cancel these secular terms by prescribing the initial data (f (0)). To explain their approach, we take f  Then, they observed that the fixed index γ belongs in N λ for all λ = 0, 1, 2, . . . and there is only one secular term in f (γ) 2λ+1 (τ ) which can be removed by choosing the frequency shift ω γ,2(λ−1) . Furthermore, for all m ∈ N λ \ {γ}, there are some secular terms which cancel by the structure of the equations, some secular terms cancel by choosing some initial data but some initial data remain free variables at this stage. They choose these free variables together with ω γ,2λ to cancel the secular terms in the f 2(λ+1)+1 (τ ). For more details see [51].
However, there is no proof based on rigorous arguments ensuring that this procedure works for all λ.

3.3.
Choice of the initial data. For example, we fix γ = 0 and choose First, we use the recurrence relation above and find periodic expressions for p Based on the discussion above, we get two secular terms in the list {f We choose to ensure the periodicity of f

3.4.
Growth and decay of the Fourier constants. In this section, we focus on the asymptotic behaviour of all the Fourier constants which appear using this approach. We shall use the notation that is summation with respect to all possible combinations of plus and minus. Furthermore, expressions like ω i ± ω j ± ω m stand not only for ω i + ω j + ω m and ω i − ω j − ω m but also for ω i + ω j − ω m and ω i − ω j + ω m , that is considering all possible combinations of plus and minus. Specifically, we prove the following result the proof of which is based on the leading order terms (Remark 2.5) together with the asymptotic behavior of the oscillatory integrals (Lemma 2.8), the orthogonality properties (Lemma 2.4) and the L ∞ −bounds (Lemma 2.6).

Growth and decay estimates for
Proof. First, observe that ω i ± ω j ± ω m are all odd, namely For large values of i, j, m and in the case where all ω i ± ω j ± ω m −→ ∞, we obtain On the other hand, for large values of i, j, m such that some ω i ± ω j ± ω m −→ ∞, Holder's inequality implies Similarly, for large values of i, j, m and in the case where all ω i ± ω j ± ω m −→ ∞, we obtain On the other hand, for large values of i, j, m such that some ω i ± ω j ± ω m −→ ∞, Holder's inequality implies Next, observe that ω i ± ω j ± ω l ± ω m are all even, Now, for large values of i, j, l, m and in the case where all ω i ± ω j ± ω l ± ω m −→ ∞, we obtain whereas, for large values of i, j, l, m such that some ω i ± ω j ± ω l ± ω m −→ ∞, Holder's inequality implies Furthermore, for large values of i, j, l, m and in the case where all ω i ± ω j ± ω l ± ω m −→ ∞, (ω i ± ω j ± ω l ± ω m ) N , 26 whereas, for large values of i, j, l, m such that some ω i ± ω j ± ω l ± ω m −→ ∞, Holder's inequality implies Next, we use once more Remark 2.5 to compute π 2 x e m (y) cos(y) sin(y)dy ≃ 2 √ π π 2 x sin(ω m y) cos 2 (y)dy for m −→ ∞. Hence, for large values of i, j, l, m and in the case where all ω i ±ω j ±ω l ±ω m −→ ∞, x e m (y) sin(y) cos(y)dy tan 2 (x)dx whereas, for large values of i, j, l, m such that some ω i ± ω j ± ω l ± ω m −→ ∞, Holder's inequality implies x e m (y) sin(y) cos(y)dy tan 2 (x)dx x e m (y) sin(y) cos(y)dy dx π 2 · e m (y) sin(y) cos(y)dy Finally, for large values of i, j, l, m and in the case where all x e m (y) sin(y) cos(y)dy tan 2 (x)dx (ω i ± ω j ± ω l ± ω m ) N , 28 whereas, for large values of i, j, l, m such that some ω i ± ω j ± ω l ± ω m −→ ∞, Holder's inequality implies x e m (y) sin(y) cos(y)dy dx π 2 · e m (y) sin(y) cos(y)dy which ends the proof.

4.1.
Definition of the Fourier constants. As before, we expand the error terms (Ψ, Σ, B, Θ) in terms of the eigenfunctions of the linearized operator as follows After substituting these expressions into the equations above, we take the inner product (·|e ′ j ) (for the equation for Ψ) and (·|e j ) (for the equations for Σ, B and Θ) in both sides. A long but straightforward computation yields that all the interactions with respect the space variable x are included into the following Fourier constants: for the equation for Ψ, for the equation for Σ, for the equation for Θ, and finally x e i (y) sin(y) cos(y)dy tan 2 (x)dx x e i (y) sin(y) cos(y)dy tan 2 (x)dx x e i (y) sin(y) cos(y)dy tan 2 (x)dx x e i (y) sin(y) cos(y)dy tan 2 (x)dx x e i (y) sin(y) cos(y)dy tan 2 (x)dx x e i (y) sin(y) cos(y)dy tan 2 (x)dx, for the equation for B. In addition, the non-linear system for the error terms boils down to Here, the source terms are given explicitly in terms of the Fourier constants by where δ 0,i stands for the Kronecker's delta, whereas the linear and non-linear terms are given by

4.2.
Approximate periodic solution and small divisors. As in the first approach with the infinite sum, the linear and homogeneous part of the ordinary differential equation for (ψ i , σ i ) is simply the equation for the harmonic oscillator and hence we can use the variational 35 constants formula to solve it. We find the fixed-point formulation Furthermore, one can use the trigonometric identities to write ψ F i (s) = K i (ǫ 2 ) sin(s) + Λ i (ǫ 2 ) sin(3s) + M i (ǫ 2 ) sin(5s), σ F i (s) = N i (ǫ 2 ) cos(s) + Ξ i (ǫ 2 ) cos(3s) + T i (ǫ 2 ) cos(5s) 36 where Now, computing the integrals above we obtain provided that we choose η 0 so that for all i = 0, 1, 2, . . . and all 0 < ǫ ≤ ǫ 0 for sufficiently small ǫ 0 > 0. Such a condition for the choice of η 0 is closely related to small divisors which play an important role in KAM theory [5,6,8,9,11,21]. Observe that the identity holds true for all i = 0, 1, . . . and all ǫ > 0. Here, all the constants involved depend explicitly on the Fourier constants defined above and most importantly depend only on one index. Due to this fact, we can compute them (see Lemma B.1 in Appendix B). The fact that these constants are given in closed forms has a numerous advantages. First, we get the asymptotic behaviour for large i and fixed ǫ > 0, Second, we get their asymptotic behaviour for sufficiently small ǫ and fixed i = 0, 1, . . . , We compute √ 10π and hence Consequently, by the structure of the equations, ψ R 3 , ψ N 3 , σ R 3 and σ N 3 cannot blowup as ǫ does to zero. However, we choose θ 0 := 153 4π to ensure that every component of the periodic parts ψ S i (τ ) and σ S i (τ ) of ψ i and σ i respectively are bounded as ǫ goes to zero. This choice coincides with the choice of θ 2 from the first approach as well as with the numerical computations of Rostworowski-Maliborski [51]. Similarly, using the trigonometric identities As before, all the constants involved depend explicitly on the Fourier constants defined above and most importantly depend only on one index. Due to this fact, we can compute them (see Lemma B.1 in Appendix B). We immediately get their asymptotic behaviour for large i,

4.3.
Choice of the initial data. We choose This choice is motivated by the fact that the source terms ψ S i (τ ) and σ S i (τ ) of the solutions ψ i (τ ) and σ i (τ ) would give rise to a periodic term. Indeed, and similarly for Σ, B and Θ.

4.4.
Growth and decay of the Fourier constants. We are interested in the asymptotic behaviour of all the Fourier constants which appear using this approach. To begin with, we split them into five groups as follows As before, we shall use the notation that is summation with respect to all possible combinations of plus and minus and expressions like ω i ± ω j ± ω m stand not only for ω i + ω j + ω m and ω i − ω j − ω m but also for ω i + ω j − ω m and ω i − ω j + ω m , that is considering all possible combinations of plus and minus. We will use the leading order terms (Remark 2.5) together with the asymptotic behavior of the oscillatory integrals (Lemma 2.8), the orthogonality properties (Lemma 2.4), the L ∞ −bounds for quantities related to the eigenfunctions (Lemma 2.6) and the L ∞ −bounds of the weights Γ a (estimate (4.1)).

4.4.1.
Fourier constants in A 1 , A 2 , A 3 and A 4 . First, we focus on the elements of A 1 .
Proposition 4.1 (Fourier constants in A 1 ). The following growth and decay estimates hold.
Growth and decay estimates for the Fourier constants in Proof. All these estimates follow directly from Lemma 2.11 and in particular from as b −→ ∞. However, we illustrate the proof only for the first constant, namely ω i C 13ji . For large values of i, j and in the case where both ω i ± ω j −→ ∞ (equivalently when ω i − ω j −→ ∞), Observe that both ω i + ω j and ω i − ω j are even, If ω i − ω j −→ ∞, then Lemma 2.11 applies and since On the other hand, if ω i − ω j −→ ∞, then we see that In conclusion, Second, we focus on the elements of A 2 .
Proposition 4.2 (Fourier constants in A 2 ). Let N ∈ N. The following growth and decay estimates hold. 43 Growth and decay estimates for the Fourier constants in A 2 as i, j, k −→ +∞ Proof. All these estimates follow directly from Lemma 2.11 and in particular from as b −→ ∞. However, we illustrate the proof only for the first constant, namely ω i F 1jki . For large values of i, j, k and in the case where all ω i ± ω j ± ω k −→ ∞, Observe that all ω i ± ω j ± ω k are odd, We define F (x) := Γ 1 (x) tan(x) . Lemma 2.11 applies and since as i, j, k −→ ∞. Finally, for large values of i, j, k such that some ω i ± ω j ± ω k −→ ∞, Holder's inequality implies as i, j, k −→ ∞. In conclusion, Now, we focus on the elements of A 3 . A 3 ). Let N ∈ N. The following growth and decay estimates hold.

Growth and decay estimates for the Fourier constants in
Proof. All these estimates follow directly from Lemma 2.11 and in particular from as b −→ ∞. However, we illustrate the proof only for the first constant, namely ω 0 ω i F 10ji . For large values of i, j and in the case where ω i − ω j −→ ∞, Observe that both ω i + ω j and ω i − ω j are even, We define F (x) := Γ 1 (x)e ′ 0 (x) and compute Now, Lemma 2.11 yields as i, j −→ ∞. On the other hand, for large values of i, j such that ω i − ω j −→ ∞, Holder's inequality implies as i, j −→ ∞. In conclusion, Finally, we focus on the elements of A 4 . A 4 ). Let N ∈ N. The following growth and decay estimates hold. 47 Growth and decay estimates for the Fourier constants in A 4 as i, j, k −→ +∞

Proposition 4.4 (Fourier constants in
Proof. All these estimates follow directly from Lemma 2.11 and in particular from as b −→ ∞. However, we illustrate the proof only for the first constant, namely I jki . For large i, we have π 2 x e i (y) sin(y) cos(y)dy ≃ π 2 x sin(ω i y) cos 2 (y)dy Now, for large values of i, j, k and in the case where all ω i ± ω j ± ω k −→ ∞, x e i (y) sin(y) cos(y)dy tan 2 (x)dx Observe that all ω i ± ω j ± ω k , We define F (x) := cos 2 (x) and compute F π 2 = F ′ (0) = 0, F ′′ π 2 = 2 = 0.
In conclusion,  Growth and decay estimates for the Fourier constants in Proof. First, observe that are all odd. All results here follow from Lemma 2.8 and Remark 2.9. For large values of i, j, k and in the case where all ω i ± ω j ± ω k −→ ∞, By Remark 2.9, we infer that in this case as i, j, k −→ ∞, whereas, for large values of i, j, k such that some ω i ± ω j ± ω k −→ ∞, Holder's inequality implies Second, for large values of i, j, k and in the case where all ω i ± ω j ± ω k −→ ∞, we have Hence, by Lemma 2.8, we get that in this case as i, j, k −→ ∞. On the other hand, for large values of i, j, k such that some ω i ± ω j ± ω k −→ ∞, Holder's inequality implies Similarly, for large values of i, j, k and in the case where all ω i ± ω j ± ω k −→ ∞, and finally as i, j, k −→ ∞. However, for large values of i, j, k such that some ω i ± ω j ± ω k −→ ∞, Holder's inequality implies Proposition 4.6 (Fourier constants in B 2 ). Let N ∈ N. The following growth and decay estimates hold.