HAL Id: hal-02979812
https://hal.archives-ouvertes.fr/hal-02979812
Submitted on 27 Oct 2020
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
WEAK DISPERSION FOR THE DIRAC EQUATION
ON CURVED SPACE-TIME
Federico Cacciafesta, Anne-Sophie de Suzzoni
To cite this version:
Federico Cacciafesta, Anne-Sophie de Suzzoni. WEAK DISPERSION FOR THE DIRAC EQUATION ON CURVED SPACE-TIME. Discrete & Continuous Dynamical Systems - A, 2019. �hal-02979812�
SPACE-TIME
FEDERICO CACCIAFESTA AND ANNE-SOPHIE DE SUZZONI
Abstract. In this paper we prove local smoothing estimates for the Dirac equa-tion on some non-flat manifolds; in particular, we will consider asymptotically flat and warped products metrics. The strategy of the proofs relies on the multiplier method.
1. Introduction
The Dirac equation on R1+3is a constant coefficient, hyperbolic system of the form
(1.1) iut+ Du + mβu = 0
where u : Rt× R3x → C4, m ≥ 0 is called the mass, the Dirac operator is defined as
D = i−1 n X k=1 αk ∂ ∂xk = i−1(α · 5), and the 4 × 4 Dirac matrices can be written as
(1.2) αk = 0 σk σk 0 , k = 1, 2, 3, β = I2 0 0 −I2
in terms of the Pauli matrices
(1.3) σ1 = 0 1 1 0 , σ2 = 0 −i i 0 , σ3 = 1 0 0 −1 . The α matrices satisfy the following relations
αjαk+ αkαj = 2δjkI4, 1 ≤ j, k ≤ 3,
αjβ + βαj = 0, j = 1, 2, 3,
β2 = I4;
as a consequence, the following identity holds
(1.4) (i∂t− D − mβ)(i∂t+ D + mβ)u = (∆ − m2− ∂tt2)I4u.
This identity allows us to study the free Dirac equation through a system of decoupled Klein-Gordon (or wave, in the mass-less case) equations. Therefore, it is not a
2010 Mathematics Subject Classification. 35J10, 35B99.
difficult task to deduce dispersive estimates (time-decay, Strichartz...) for the Dirac flow from the corresponding ones of their more celebrated Klein-Gordon or wave counterparts. Moreover, this fact stresses the substantial difference between the mass-less and the massive cases in equation (1.1). Of course, when perturbative terms appear in equation (1.1), as potentials or nonlinear terms, the argument above needs to be handled with a lot of additional care, and in particular is going to fail in low regularity settings, when the structures of the single terms play crucial roles. The study of dispersive estimates for the Dirac equation with potentials has already been dealt with in literature: we mention at least the papers [4, 6, 8, 12, 13, 10] in which various sets of estimates are discussed for electric and magnetic perturbations of equation (1.1).
In the last years, a lot of effort has been spent in order to investigate higher order perturbations of dispersive partial differential equations: in particular, the problem of understanding how variable coefficients perturbations affect the wave and Schr¨odinger flows has attracted increasing interest in the community. The interest for this kind of problems is of geometric nature, as it is indeed natural to interpret the variable coefficients as a ”change of metrics”, and therefore to recast the problem as the study of dispersive dynamics on non-flat manifolds. It turns out that in this contest a crucial role is played by the so called non-trapping condition on the coefficients that, roughly speaking, is a condition that prevents geodesic flows to be confined in compact sets for large times: the failure of such a condition is indeed understood to be an obstacle for dispersion. Such a condition is in fact guaranteed in case of ”small perturbations” of the flat metric. On the subject of dispersion for Klein-Gordon and wave equations, we mention, in a non exhaustive way, [1,2,3,16,
20, 21, 22].
The aim of this manuscript is to provide some first results in this framework for the Dirac equation for which, to the best of our knowledge, nothing is known; in particular we here aim to prove weak dispersive estimates for its flow under some different assumptions on the geometry. We stress the fact that, due to to the rich algebraic structure of the Dirac operator, its generalization to curved spaces is sig-nificantly more delicate than the one of the Laplacian; we dedicate section 3 to this issue. On the other hand, once the equation is correctly settled, it is possible to rely on the the squaring trick (1.4) as in the free case to reduce to a suitable vari-able coefficients wave equation with a lower order term, for which the full power of the multiplier technique can be exploited. Therefore, in the present paper we will essentially be mixing the strategy developed in [4] to prove dispersive estimates for the magnetic Dirac equation, with [7, 9], in which the same method is adapted to deal with the more involved variable coefficients setting for the Schr¨odinger and Helmholtz equations.
We will show in section3that the general form of the Dirac operator on a manifold with a given metric gµν is the following
(1.5) D = iγaeµ
aDµ
where the matrices γ0 = β and γj = γ0αj for j = 1, 2, 3, eµ is a vierbein (i.e. a
matrix that, roughly speaking, connects the curved spacetime to the Minkowski one) and Dµ defines the covariant derivative for fermionic fields.
In what follows, we shall restrict to metrics gµν having the following structure
(1.6) gµν = φ−2(t) if µ = ν = 0 0 if µν = 0 and µ 6= ν −hµν(−→x ) otherwise.
In other words, time and space are decoupled. What is more, we assume that the manifold is complete: this ensures that the Dirac operator is self-adjoint (see [11]), a property that is crucial in order to guarantee a unitary dynamics, and that we use for the conservation of energy or for estimates of norms in terms of this operator. The same assumption is made in [16]. The function φ is assumed to be strictly positive for all t. Let us remark that after a change of variable on time, one may take φ equal to 1. Within this setting, we will show that equation (D + m)u = 0 can be written in the more convenient form
(1.7) iφ∂tu − Hu = 0
where H is an operator such that H2 = −∆
h + 14Rh + m2, and ∆h and Rh are
respectively the Laplace-Beltrami operator and the scalar curvature associated to the spatial metrics h. As a consequence, it can be proved that if u solves equation (1.7) then u also solves the equation
(1.8) − (φ∂t)2u + 4hu −
1
4Rhu − m
2u = 0.
We should stress the analogy with the free case; the scalar curvature term that appears in the equation above vanishes when reducing to the Minkowski metric.
Our first main result concerns the case of manifolds which are asymptotically flat; let us explain precisely the assumptions in this case. First of all, we require for h(x) = [hjk(x)]nj,k=1 the following natural matrix-type bounds to hold for every x,
ξ ∈ R3
(1.9) ν|ξ|2 ≤ hjk(x)ξjξk ≤ N |ξ|2
where hinv := [hjk(x)]nj,k=1 is the inverse of the matrix h(x), which is equivalent to
Notice that a consequence of (1.9) is that there exist constants Ci,ν,N, i = 1, 2, 3
depending on ν, N , such that for all v ∈ R3,
|hinvv|2 ≤ C1,ν,N|hv|2 ≤ C2,ν,N|v|2 ≤ C3,ν,N|hinvv|2
Moreover, (1.9) implies
(1.11) N−3/2≤pdet(h(x)) ≤ ν−3/2, ∀ x ∈ R3
where det(h(x)) = det[hjk(x)]nj,k=1. Then, we impose an asymptotically flatness
condition in the form
(1.12) |hinv(x) − I| ≤ CIhxi−σ, CI < 1
and
(1.13) |h0inv(x)| + |x||h00inv(x)| + |x|2|h000inv(x)| ≤ Chhxi−1−σ, σ ∈ (0, 1),
where we are denoting by |h(x)| the operator norm of the matrix h(x) and where |h0| = X |α|=1 |∂αh(x)|, |h00| = X |α|=2 |∂αh(x)| and |h000| = X |α|=3 |∂αh(x)|. Note also that these assumptions imply
(1.14) khxi1+α5 (p
det(h(x)))kL∞+ khxi1+α5 (
p
det(h(x))hjk(x))kL∞ ≤ C5
for some α = α(σ) ∈ (0, 1) and some constant C5 (the constant C5 might be
explic-itly written in terms of Ch, ν and N , but here we prefer to introduce another constant
to keep notations lighter). These are often referred to as long range perturbations of the euclidean metrics.
We are now ready to state our first Theorem.
Theorem 1.1. Let u be a solution to (1.7) with initial condition u0, with g satisfying
(1.6), and assume that h satisfies (1.9), (1.12) and (1.13) with the constants involved small enough. Then for η1, η2 > 0, there exists Cη1,η2 > 0 independent from u such
that (1.15) khxi−3/2−η1uk L2 φL2x + khxi −1/2−η2 5 uk L2 φL2x ≤ Cη1,η2kHu0k 2 L2 Mh. The norm k · kL2(M h) is the L
2 norm on the manifold M
h, that is kf k2 L2(M h) = Z D(h) |f (x)|2p det(h(x))dx
where D(h) is the set where h is defined. In this case, this set is assumed to be R3.
Remark 1.1. Our asymptotically flatness assumptions listed above are fairly standard in this setting (compare e.g. with [9]). The main example we have in mind is given by the choice hjk = (1 + εhxi−σ)δjk with σ ∈ (0, 1) and for some ε sufficiently
further particular case we can think hjk to be a small and regular enough compactly
supported perturbation of the flat metric.
Remark 1.2. In fact, we can prove under assumptions of Theorem 1.1 a slightly stronger version of estimate (1.15), namely the following
(1.16) kuk2 XL2 φ+ k 5 uk 2 Y L2 φ ≤ Cν,N,σkHu0k 2 L2 Mh
where the Campanato-type norms X and Y are defined in subsection 2.1, by the equations (2.3), (2.4). These spaces represent indeed somehow the natural setting when dealing with the multiplier method (see e.g. [4, 9]); nevertheless, we prefer to state our Theorem in this form for the sake of symmetry with the next result. We stress anyway that estimate (1.16), which is the one that we will prove, implies (1.15).
Remark 1.3. In the flat case, scaling is a useful tool to single out the type of spaces one may hope to obtain inequalities for. Here, however, scaling is not obvious, even in the massless case, because of the geometry: indeed, 4hu(λx) is not in general
equal to λ24hu. It would if h is homogenous but given the type of perturbation we
consider, this is not what we have.
Remark 1.4. As done in [9] for the Helmholtz equation, our proof allows us, after carefully following all the constants produced by the various estimates, to provide some explicit sufficient conditions that guarantee local smoothing estimate (1.15): we indeed quantify the closeness to a flat metric which we require by giving out explicit inequalities that the constants in Assumptions (1.12)-(1.13) must satisfy to get the result. Explicitly, these conditions are given in forthcoming subsection (5.4), by requiring the positivity of the constants M1 and M2 which reflects in smallness
requirements on the constants CI and Ch in (1.12)-(1.13). This fact, as mentioned,
is strictly connected to the geometrical assumption of non-trapping on the metric gjk; therefore our strategy of proof gives, in a way, some explicit sufficient conditions
that ensure the metric g to be non-trapping.
Remark 1.5. In Minkowski space-time, the influence of a magnetic potential in equa-tion (1.1) is reflected in the change of the covariant derivative, that is the substitution
5 → 5A := 5 − iA
where
A = A(x) = (A1(x), A2(x), A3(x)) : R3 → R3
is the magnetic potential. This phenomenon can be generalized to equation (1.7): the presence of a magnetic potential has indeed essentially the effect of changing the covariant derivative Dµ. In particular, by squaring the magnetic Dirac equation on
a space with a metric gjk with the structure (1.6) one obtains the following
Klein-Gordon type equation
−(φ∂t)2u + ˜4hu −
1
4Rhu − m
2u −1
2Fjk[γi, γk]u = 0.
where ˜4h is the magnetic Laplace-Beltrami operator and Fjk = ∂jAk− ∂kAj is the
electromagnetic field tensor. The strategy of the present paper allows to deal with this more general situation: anyway, we prefer not to include magnetic potentials in order to keep our presentation more readable. We refer the interested reader to [9], in which the electromagnetic Helmholtz equation is discussed with the same techniques as here.
Remark 1.6. The problem of proving Strichartz estimates for solutions to equation (1.7) seems significantly more difficult: variable coefficients perturbations indeed prevent the direct use of the standard Duhamel formula to handle the additional terms (see e.g. [4]) and requires a completely different approach involving phase space analysis and parametrices construction. We stress the fact, anyway, that proving Morawetz-type estimates (or local energy decay in the case of the wave equation) still represents a crucial step in this more involved setting, as they indeed provide a convenient space to place the errors of the parametrix. The interested reader should see [17] and references therein. We also mention the fact that one could mimic the argument presented in [5], where it is proved that global in time Strichartz estimates for solutions to the wave equation on a non flat background, and also outside of a compact obstacle, are implied by a suitable local smoothing estimate, provided the metric is assumed to be flat outside some ball and the solutions to (1.8) which are compactly supported in space are known to satisfy local in time Strichartz estimates. This strategy seems to apply to our case, at least in order to obtain homogenous estimates, meaning that it could be adapted to equation (1.8), that presents an additional zero order term, and thus to (1.7). This would give, at least, a conditional result. We intend to deal with all these problems in forthcoming papers.
Next, we consider the specific case of the so called warped products, that is metrics of the form (1.6) with the additional structure
(1.17) h11= 1, h1i = hi1 = 0 if i 6= 1, hij = d(x1)κij(x2, x3)
where κ is a 2 × 2 metric. We denote the scalar curvature of κ by Rκ = Rκ(x2, x3).
We will use the more comfortable (and intuitive) notation r = x1. In the case of
the flat metric of R3, d(r) = r2 and κ is the metric of the sphere S2. In all that
follows, we assume that κ is smooth enough (C2) and that since h should be positive,
that κ is a positive matrix.
Theorem 1.2. Let u be a solution to (1.7) with initial condition u0, with g satisfying
(1.6) and h as in (1.17). Then the following results hold.
• (Hyperbolic-type metrics). Take d(r) = er/2 in (1.17) and assume that for all
(x2, x3)
Rκ(x2, x3) > 0, m2 >
3 32.
Let η1, η2 > 0. There exists Cη1,η2 > 0 such that for all u solution of the linear
Dirac equation, we have (1.18) ke−r/4hri−(1+η1)uk2 L2(M g)+ ke −r/4hri−(1/2+η2)5 huk2L2(M g) ≤ Cη1,η2kHu0kL2(Mh).
• (Flat-type metrics). Take d(r) = r2 in (1.17) and assume that for all (x2, x3),
Rκ ≥ 2, m > 0.
Let η1, η2 > 0. There exists Cη1,η2 > 0 such that for all u solution of the linear
Dirac equation, we have (1.19) khri−(3/2+η1)uk2 L2(M g)+ khri −(1/2+η2)5 huk2L2(M g)≤ Cη1,η2kHu0kL2(Mh).
• (Sub-flat type metrics) Take d(r) = rn in (1.17) with n ∈]2 −√2, 4/3]. There
exists Cn > 0 such that if for all (x2, x3), Rκ ≥ Cn, then for all η1, η2 >
0, there exists Cη1,η2,n > 0 such that for all u solution of the linear Dirac
equation, we have (1.20) khri−(3/2+η1)uk2 L2(M g)+ khri −(1/2+η2)5 huk2L2(M g) ≤ Cη1,η2,nkHu0kL2(Mh). The norm k · kL2(M g) is the L
2 norm on the manifold M
g, that is kf k2 L2(M g) = Z R×D(h) |f (t, x)|2p det(g(t, x))dxdt
where D(h) is the set where h is defined and where 5hf · 5h means the operator
hij∂if ∂j. Due to the structure of g (1.6), we have g(t, x) = −φ−2(t)h(x), which yields
kf k2 L2(Mg) = Z R kf (t, ·)k2 L2(M h)φ −1 (t)dt.
Remark 1.7. The hypothesis on the mass comes from the fact that the infimum of the curvature inf Rh is not positive (it is negative in the hyperbolic case, and 0 in
the other ones). This issue arises when one estimates the H1 norm of u with the
energy, the mass is then used to compensate this negative curvature as it is chosen such that inf Rh+ 4m > 0. Note that proving Hardy’s inequality for any κ would
Remark 1.8. There are different conditions for the curvature of κ. One reason is specified in the previous remark : we need the curvature not to be infinitely small, such that the energy controls the H1 norm. This explains R
κ ≥ 0 in the subflat case
and Rκ ≥ 2 in the flat case. What is more, we use the curvature term to compensate
losses due to the bi-Laplace-Beltrami term in the virial identity in the hyperbolic and subflat cases. This is why we have more constraining hypothesis.
Remark 1.9. There is a case to do a change of variable to retrieve the flat Dirac equation. This would probably work for passing from a radial subflat Dirac equation to a radial flat Dirac equation. When κ is the metrics of the sphere of dimension 2, one may restrict the Dirac operator to a subspace of L2 to make it radial. This
is not obvious for a generic κ. In any case, it would not encompass the full Dirac operator, which is not radial.
Remark 1.10. As in the asymptotically flat case, there is an issue with scaling. Writ-ing the Laplace-Beltrami operator in this context, we get
4hu = d0(r) d(r)∂ru + ∂ 2 ru + 1 d(r) 4κu. If we plug in uλ(x) = u(λx) in the hyperbolic case, we get d
0(r) d(r)∂ruλ = λ d0(r) d(r)∂ru λ , while ∂2 ruλ = λ2(∂r2u)λ and d(r)1 4κuλ = e(λ −1−1)r
(d(r)1 4M u)λ. Thus, the first two
terms do not scale in the same way, while the third is problematic. In the subflat case, the first two terms scale in the same way but not the third. In the flat case though, everything scales, so it is not suprising that we recover the usual inequalities. Remark 1.11. By mixing the techniques used to prove Theorem 1.1 and Theorem
1.2, one should expect to be able to prove local smoothing for the Dirac equation on metrics that are asymptotically like the warped products we presented. It is also reasonable to expect, simply by mimicking techniques, the same results to hold for a more generic warped product. In the cases, we present, we always take a multiplier of the same form. The computations for a generic d are somewhat tedious, but if one wishes to repeat the argument for another d, one natural assumption is dd(r)0(r) . hri.
The rest would be finding sufficient hypothesis on 42
hr and 42hr2.
The strategy for proving these results relies on the multiplier method: using some standard integration by parts techniques we will be able to build a proper virial identity for equation (1.7) (see Proposition4.2) which, by choosing suitable multiplier functions, will allow us to prove local smoothing estimates. We stress the fact that such a method does not seem to apply in lower dimensions due to the difficulty of finding proper multipliers, and to the best of our knowledge we are not aware of similar results in dimensions 1 or 2. On the other hand, the method would be
well adapted to deal with high dimensional frameworks: anyway, the extension of the Dirac equation to high dimensions is not quite as straightforward as, e.g, the Scrh¨odinger or wave ones, and it would require some additional work and a fair amount of technicalities that we prefer not to deal with here. Also, to the best of our knowledge, the high dimensional cases present a relatively scarce relevance in the applications. This is the main reason behind our restriction to dimension n = 3. The plan of the paper is the following: in Section 2 we build our setup, fixing the notations and providing some preliminary inequalities that will be needed in the following, in Section 3 we review the theory of Dirac operators on curved spaces, showing how to properly build a dynamical equation, in Section4we prove the virial identity that is the crucial stepping stone for local smoothing with the use of the multiplier method, while Sections 5 and 6are devoted, respectively, to the proofs of Theorems 1.1 and 1.2.
Acknowledgments. The authors are grateful to Paolo Antonini and Gianluca Panati for useful discussions on the topic. The first named author is supported by the FIRB 2012 ”Dispersive dynamics, Fourier analysis and variational methods” funded by MIUR (Italy) and by National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2015 semester. We are also grateful to the referee for his-her careful reading of the manuscript and for providing several comments and suggestions that helped improved our presentation.
2. Preliminaries
We dedicate this section to fix notations and expose some preliminary useful re-sults.
2.1. Notations. We start by recalling some classical notations from differential ge-ometry that we will use in the rest of the paper. Let h = h(x) be a 3 × 3, positive definite, real matrix that defines, in a standard way, a metric tensor. We recall that the scalar curvature can be written as
(2.1) Rh = hjk ∂ ∂xi Λijk − ∂ ∂xk Λiji+ Λ`jkΛii`+ Λ`jiΛik` , where Λi
jk denote the standard Christoffel symbols (we use Γ for g).
In what follows we will use the compact notation for the matrices h = h(x) = [hjk(x)]3j,k=1 hinv = hinv(x) = [hjk(x)]3j,k=1.
We will need the quantities ˆ
where we are using the standard conventions for implicit summation and ˆx = x/|x|. Notice that, as h(x) is assumed to be positive definite,
0 ≤ ˆh(x) ≤ h(x) for every x. Also, we will use the compact notation
˜
hjk =pdet(h)hjk.
Straightforward computations show that, for every sufficiently regular radial function ψ, ∆hψ(x) = ˆhψ00+ h − ˆh |x| ψ 0 + 1 pdet(h)∂j(˜h jk)ˆx kψ0 (2.2)
where 0 denotes the radial derivative and we are slightly abusing notations by iden-tifying the functions ψ(x) and ψ(|x|).
The natural scalar product induced by h is given by hf, gih = Z Mh f g = Z R3 f (x)g(x)pdet(h)d3x. The corresponding k · kL2(M
h) norm is naturally defined.
We now introduce the functional spaces that will represent the setting for our estimates in the asymptotically flat case. We define the Campanato-type norms (note that hRi =√1 + R2) as
(2.3) kvk2 X := sup R>0 1 hRi2 Z Mh∩SR |v|2dS = sup R>0 1 hRi2 Z SR |v|2p det(h)dS where dS denotes the surface measure on the surface of the ball {|x| = R}, and
(2.4) kvk2 Y := sup R>0 1 hRi Z Mh∩BR |v|2dx = sup R>0 1 hRi Z BR |v|2p det(h)dx
where we are denoting by SR and BR, respectively, the surface and the interior of
the sphere of radius R centred in the origin. Notice that k · kY is equivalent to the
norm whose square is
(2.5) sup R≥1 1 R Z Mh∩BR |v|2.
The norm with respect to time will be given by kuk2 L2 φ,T = Z T 0 |u(t)|2 φ(t) dt, kuk 2 L2 φ = Z +∞ 0 |u(t)|2 φ(t) dt
where φ is the positive function given in the definition of g. In particular, when φ = 1 these norms recover the standard L2
T (resp. L2) ones, and we will simply denote with
L2
T = L21,T.
We mean by RM
h the integral over the manifold with the volume unit of the
manifold, that is Z Mh f = Z D(h) f (x)pdet(h(x))dx where D(h) is the set where h is defined.
Finally, we recall that by 5hf · 5h we mean the operator hij∂if ∂j.
2.2. Useful inequalities. The following Hardy-type estimate will be needed in what follows. The precise definition of the operator H will be given in the next section, it is such that the Dirac equation is written i∂tu = Hu, it satisfies m + D = iγ0φ∂t− γ0H,
and −H2 = m − 4h+ 14Rh.
Proposition 2.1. Let m ≥ 0 and assume that for every x |Rh| ≤ K|x|R2 for some
a ∈ (0, 1) and KR < ν4 with ν as in (1.9). Then for any f such that Hf ∈ L2Mh the
following inequality holds
(2.6) m2 Z Mh |f |2+ ν 4− K R 4 Z Mh |f |2 |x|2 ≤ Z Mh |Hf |2.
Proof. We assumed that g was complete, such that the Dirac operator γ0D is
self-adjoint. We refer to [11]. Since γ0D = iφ∂
t− H − γ0m, we get that H is essentially
self-adjoint on C0∞.
We write, as the operator H is self-adjoint with respect to the inner product defined by h, (2.7) Z Mh |Hf |2 = −hH2f, f iMh = Z Mh (m2− ∆h)f f + 1 4 Z Mh Rh|f |2 = I + II.
Notice now that
(2.8) I = Z Mh 5hf · 5hf + m2 Z Mh |f |2 Assumption (1.9) implies Z Mh 5hf · 5hf ≥ ν5/2 Z R3 | 5 f |2
which, by the application of standard Hardy’s inequality Z R3 |f |2 |x|2 ≤ 4 Z R3 | 5 f |2
gives I ≥ ν 4 4 Z Mh |f |2 |x|2 + m 2 Z Mh |f |2.
For the second term, by relying on the assumption on the curvature, we write
(2.9) II ≤ KR 4 Z Mh |f |2 |x|2.
Putting all together, we thus have Z Mh |Hf |2 ≥ ν4− KR 4 Z Mh |f |2 |x|2
which concludes the proof.
Remark 2.1. In what follows we will make also use of the following estimate, which holds for any ε ∈ (0, 1),
(2.10) m2 Z Mh |f |2+ (1 − ε)ν 4− K R 4 Z Mh |f |2 |x|2 + ε Z Mh | 5 f |2 ≤ Z Mh |Hf |2 that can be obtained by combining (2.6) with the obvious inequality
ε Z Mh | 5 f |2+ (1 − ε) 4 ν 4 Z Mh |f |2 |x|2 ≤ Z Mh | 5 f |2.
Remark 2.2. Notice that in our asymptotically flatness assumptions on h (1.9), (1.12) and (1.13) the condition required on the curvature is fulfilled provided the constants CI and Ch are small enough. We estimate indeed each of the term as follows (notice
that (1.13) holds for the matrix h as well, due to the well known relation ∂xjh inv =
−hinv(∂ xjh)h
inv, with the modified constant ν−2C h): |hinv(∂hinv)∂h| ≤ N ν2 C2 h hxi2+2σ, |hinv(∂∂h)| ≤ N2 ν4 Ch |x|hxi1+1σ, |hinv(hinv∂h)2| ≤ N4 ν4 Ch hxi2+2σ
which in particular imply the assumption of Proposition 2.1 for Ch small enough.
In what follows, we will also make use of a number of weighted inequalities that we collect in the following
Proposition 2.2. For any σ ∈ (0, 1) and any v ∈ C0∞(Rn) the following estimates hold (2.11) Z Mh |u|2 hxi1+σ ≤ 8σ −1 CN,νkuk2Y, (2.12) sup R>1 Z Mh∩BRc R2 |x|5|u| 2 ≤ C N,νkuk2X, (2.13) Z Mh∩Bc1 |u|2 |x|2hxi1+σ ≤ 2σ −1 CN,νkuk2X. (2.14) kuk2 X ≤ CN,ν 4 sup R>1 1 R2 Z Mh∩SR |u|2 + 13k 5 uk2 Y (2.15) kuk2 Y ≤ 3CN,ν 2k 5 uk2Y + kuk 2 X (2.16) Z Mh |u|2 |x|2 ≤ CN,ν Z Mh | 5 u|2,
where the constant CN,ν =
ν N
3/2 .
Proof. For the proof we refer to [9] section 3: the generalization from the Euclidean case to our perturbative setting is straightforward under assumptions (1.9)-(1.11). 3. From vierbein to dreibein
Our aim in this subsection is first of all to give some (basic) motivations and backgrounds that lead to the study of the Dirac equation on a non-flat setting, and then to describe the Dirac equation on curved space-time in the case of a metric which dissociates time and space. We prove indeed that it can be written as
iφ∂tu − Hu = 0
with φ a function appearing in the metric and H an operator such that H2 = − 4h+
1
4Rh+ m
2
where h is the space metric, 4h the Laplace-Beltrami operator associated to this
3.1. Motivation and first construction: vierbein. The study of the Dirac equa-tion in curved space-time is part of the more general subject of Quantum Field Theory (later referred to as QFT) in curved space-time. One of the successes of this theory is the description of entropic black holes. Its field of investigation is the description of elementary particles at energy below the Planck constant. The Dirac equation is part of it as it models the dynamics of relativistic electrons. What is underlying this approximation or reconciliation between QFT and general relativity is that the effects of the dynamics of particles should either be included in the model for the metrics via appropriate couplings or simply neglected in which case the metrics is described outside the studied system of particles.
The idea of QFT in curved space-time is the following. As in classical QFT (by which we mean QFT in a Minkowski space-time), the laws of physics should be independent from the choice of coordinates, that is the equations derived from this theory should be covariant. To derive appropriate models for elementary particles, Cartan’s formalism is used. In other words, one introduces a n-bein, or in our 1 + 3 dimensional case a vierbein. A vierbein may be seen as matrices ea
µ(x) depending on
a point x of the manifold M = (R1+3, g). They satisfy
eµa(x)gµν(x)eνb(x) = ηab ⇔ eµa(x)η abeν
b(x) = g µν(x)
where η is the Minkowski metrics. i.e. ηab = 1 if a = b = 0 −1 if a = b 6= 0 0 otherwise.
Note that eaµ is not uniquely defined as (e0)aµ = Lab(x)ebµ(x) where Lab ∈ SO(1, 3) also satisfies the same equations. The vierbein is what links M to the flat space-time or more precisely their tangeant space. Indeed, by writing ya = eaµ(x0)zµ, one
gets yaηabyb = zµgµν(x0)zν, hence eµa sends the tangent space T M to a Minkowski
space-time in a way that preserves the inner product. Note that the matrix e(x) should be reversible. Changing e into e0 induces a change of variable that satisfies eµ
a(x0)∂µ= (e0)µa(x0)∂µ0 and hence preserves the inner product. Conversely, a change
of variables that preserves the inner product may be seen as a change of vierbein. Note that this change depends on the point x0 of the manifold, as the matrix Lab
depends on the point of the manifold. Hence, a change of variable that preserves the inner product of the tangent spaces is described by a matrix field La
b ∈ SO(1, 3) and
eventually a translation, making it a Lorentz group.
Now that we have introduced this point of view for changes of variable, or more precisely covariance, we can introduce the notion of spinors. Let u be a vector field over M. Assume that we have chosen some coordinates such that u = u(x). We want to pass from the set of coordinates xµ to the set of coordinates (x0)µ, where
the change of variable is induced by L (and perhaps a translation). For vectors, we get u0(x0) = u(x). But in the case of spinors ψ, this transform works such that the equations satisfied by ψ are independent from the choice of coordinates. In general, this transform writes ψ0(x0) = S(L(x))ψ(x) where S is a group representation of SO(1, 3). If S is trivial we retrieve the change of variable for vectors. This is analogous to what we have in the case of classical QFT.
We define now the notion of covariant derivative for spinors. Let us recall that the covariant derivative of a vector field u is given by
Dµuν = ∂µuν + Γσµνuσ
and is by definition independent from the choice of coordinates, that is Dµ0u0ν = ∂(x
0)ρ
∂xµ Dρuν.
In the case of a spinor ψ, we write
Dµψ = ∂µψ + Bµ(x)ψ
where Bµ is a field over M which lies in the Lie algebra generated by S and which
is to be determined. We require that Dµ is independent from the choice of variable,
that is Dµ0ψ0 = ∂(x 0)ρ ∂xµ S(L)DρΨ. Write F (L) = ψ 7→ D0µψ0− ∂(x 0)ρ ∂xµ S(L)DρΨ.
Note that F (L) belongs to the Lie algebra induced by the representation S. Because SO(1, 3) is connected, it is sufficient to have that Bµ satisfies F (Id) = 0 and the
differential of F at the identity is 0, to have that F is identically 0. In other words, write La
b = δba+ εab(x) where δ is the Kronecker symbol, write F (L) at first order in
ε and chose B such that this first order is null for all suitable variations ε. We do not wish to repeat the computations of [18] but one may find them at pages 221-229. These computations yield
Bµ0(x) = Bµ(x) + iεab(x)[Σab, Bµ(x)] − i∂µεabΣab
where Σabare generators of the Lie algebra induced by S and [·, ·] is the commutator.
At first order S(1 + εab) = 1 + iεabΣab. Because Bµ belongs to this Lie algebra, we
have
Bµ(x) = Bµab(x)Σab.
The previous equation on Bµ yields a system of equation on Bµab which is solved by
Babµ = iωµab where ω is the spin connection : ωµab = eaν∂µeνb+ eaνΓ
ν µσe
σ b
where Γ is the affine connection given by Γνµσ = 1
2g
νλ(∂
µgλσ+ ∂σgµλ− ∂λgµσ).
Let us focus on the Dirac equation. It is written in analogy with the Dirac equation in the Minkowski space-time
(DγaeµaDµ− m)ψ = 0
where γa are the usual Dirac matrices, i.e. γ0 = β and γi = γ0αi for i ∈ {1, 2, 3}. In
particular [γa, γb] = ηab. Writing γµ= eµaγa, the Dirac equation writes (γµDµ− m)ψ = 0
with [γµ, γν] = gµν. It remains to specify S or Σ
ab such that this equation is
inde-pendent from the choice of coordinates. We still work at first order and we get that for the independence to be satisfied, we require
S(L)γaS−1(L)Lba = γb.
By replacing Lba by 1 + εba and S(L) by 1 + iεabΣab we get a system of equations on
Σ which is solved by Σab= −8i[γa, γb].
In the end, we get that the covariant derivative for a Dirac spinor is given by Dµ= ∂µ+
1 8ω
ab
µ[γa, γb]
and that the Dirac equation built in such a way is independent from the choice of variables, or in other words, relativistically invariant. One important fact to be noticed is that, by construction, we have
D2 = −g−
1 4Rg
where Rgis the scalar curvature associated to the metric g and gis the d’Alembertian
associated to the metric g, that is, g = (φ∂0)2− 4h.
3.2. Dreibein. In this subsection, we prove that if time and space are decorrelated in the metrics, then they also are in the Dirac equation. The idea is then the following, having a metrics g of the form
g =1 0
0 −h
where h is positive, we write the dreibein, that is the connection between h and a Euclidean space of dimension 3, the affine connection, the spin connection and the covariant derivatives relative to h, and explain how they relate to the vierbein, affine connection, the spin connection and the covariant derivative relative to g. Then,
we write the Dirac equation with the help of the information on h, which helps us disconnect time and space as in
iφ∂tu = Hu
with H = −γ0(ifµ
aγaDµ+ m) where faµ is a dreibein for h, γaare the standard Dirac
matrices, and Dµ is the covariant derivative for spinors in R3, h. We must say that
the result is the natural one, and that this subsection is preeminently a technical one.
We consider a metric g of the following form gµν = φ−2(x0) if µ = ν = 0 0 if µν = 0 and µ 6= ν −hµν(−→x ) otherwise. where −→x = (x1, x2, x3).
Note that in the sequel we will use the latin letters a,b, etc... for the Minkowski space R1+3, η for the Euclidean space R3, the latin letters i,j, etc... for the space
D(h), h (where D(h) is the space where h is defined) and the greek letters α, M, etc... for the space, R × D(h), g. WHAT DOES THIS MEAN?I WOULD MAYBE ERASE THIS
Let fai be a so-called dreibein hence satisfying hij = faiδabfbj
where δ here denotes the Kronecker symbol. In this sum, a and b are taken only between 1 and 3. Note that we can and do choose f independent from x0.
In the sequel, we write ea
µ a vierbein for g, Γσµν the affine connection for g, while
Λk
ij is the affine connection for h, ωµab is the spin connection for g, and αiab is the one
for h. Proposition 3.1. Write eµa = φ(x0) if µ = a = 0 0 if µa = 0 and µ 6= a fµ a otherwise.
The matrix eµa is a vierbein for g.
Proof. The issue is to prove that eµaηabebν = gµν. We start with µ = ν = 0. We have
e0aηabe0b = φ2δa0ηabδb0 = φ2η00 = g00. What is more, with i 6= 0 (µ = 0, ν = i 6= 0),
e0aηabeib = φei0 = 0 = g0i and for the same reason eiaηabe0b = gi0.
And finally, with ij 6= 0,
eaiηabejb = faiηabfbj = fai(−δab)fbj = −hij = gij. This makes eµ
a a suitable vierbein for g.
Let us see how the Christoffel symbol is changed. Proposition 3.2. Let Λkij = 1 2h kl(∂ ihlj+ ∂jhil− ∂lhij). We have Γσµν = −φ−1φ0 if µ = ν = σ = 0 Λσ µν if µνσ 6= 0 0 otherwise . Proof. We have Γ0µν = 1 2g 0λ(∂ µgλν + ∂νgµλ− ∂λgµν). Since g0λ = 0 if λ 6= 0, we get Γ0µν = 1 2φ 2(∂ µg0ν+ ∂νgµ0− ∂0gµν).
Assume ν 6= 0. We have that g0ν = gν0 = 0. Since g00 depends only on x0, and
gµ0 = 0 if µ 6= 0, we have ∂νgµ0 = 0. Since h does not depend on x0, we have
∂0gµν = 0. This yields Γ0µν = 0 if ν 6= 0 and by symmetry, Γ0µν = 0 if µ 6= 0. Besides, Γ000 = 1 2φ 2∂ 0g00= −φ−1φ0.
We have considered all the cases when σ = 0. Now we assume σ 6= 0, and we consider all the cases when µ = 0. We have
Γσ0ν = 1 2g
σλ(∂
0gλν+ ∂νg0λ− ∂λg0ν).
Since σ 6= 0, the sum over λ is only for λ ∈ {1, 2, 3}. Hence, replacing λ by i Γσ0ν = −1
2h
σi
(∂0giν+ ∂νg0i− ∂ig0ν).
Now it appears that ∂νg0i = ∂ig0ν = 0. Therefore,
Γσ0ν = −1 2h
σi∂ 0giν.
If ν = 0, giν = 0 and otherwise it does not depend on x0, hence
Γσ0ν = 0 and by symmetry
Γσµ0 = 0.
Finally, if µνσ 6= 0, we may write µ = i, ν = j, σ = k. We have Γkij = 1
2g
kλ(∂
igjλ+ ∂jgiλ− ∂λgij).
Because of the form of g, the sum over λ is taken only for λ ∈ {1, 2, 3}, we replace λ by l, and get
Γkij = 1 2h
kl(∂
ihjl+ ∂jhil− ∂lhij) = Λkij.
Therefore, we retrieve the result : Γσµν = −φ−1φ0 if µ = ν = σ = 0 Λσµν if µνσ 6= 0 0 otherwise . Let us see how the spin connection is changed.
Proposition 3.3. Let αabi = fja∂ifjb+ fjaΛ j ikf kb. We have ωµab = α ab µ if µab 6= 0 0 otherwise .
Proof. Indeed, we have
ω0ab = eaν∂0eνb+ eaνΓ ν 0σe
σb.
Because Γν
0σ = 0 if σ + ν 6= 0 we have eaνΓν0σeσb = ea0Γ000e0b which, since e00 = η0ae0a=
φ, is equal to −φφ0 if a = b = 0 and to 0 otherwise.
If ν + b 6= 0, then either eνb = e0b = 0 since b 6= 0, or eνb = eν0 = 0 or eνb = fνb
and does not depend on x0.
Hence, ∂0eνb = 0 if ν + b 6= 0 and we have ∂0e00 = ∂0φ = φ0 we get eaν∂0eνb = φφ0
if a = b = 0 and to 0 otherwise. Therefore, ωab0 = 0.
We have considered all the cases where µ = 0. We assume µ 6= 0. We deal with the case a = 0. We have ω0bµ = e0ν∂µeνb+ e0νΓ ν µσe σb = φ∂ µe0b+ φΓ0µσe σb.
Since µ 6= 0, Γ0
µσ = 0. Since e0b depends only on x0 and µ 6= 0, ∂µe0b= 0.
Since ωab
µ = −ωµba, we have that ωa0µ = 0.
We have dealt with all the cases where either a, b or µ is equal to 0. We now treat the case µab 6= 0.
We can replace the sums on the greek letters by sums on latin letters, this yields ωiab = fja∂ifjb+ fjaΓ j ikf kb = fa j∂ifjb+ fjaΛ j ikf kb = αab i .
This gives the result.
3.3. Covariant derivative and Dirac operator. The covariant derivative is given by D0 = ∂0, Di = ∂i+ 1 8α ab i [γa, γb].
Therefore, the Dirac operator can be written D = iγ0φ∂
0+ iγafajDj.
Let
H = iγafajDj and H = −γ0(H + m).
Proposition 3.4. With these notations, we have
(3.1) H2 = m2− 4h+
1 4Rh. Proof. First, we prove that H2 = 4h− 14Rh.
We have
D2 = H2+ (iγ0φ∂
0)2+ (iγ0φ∂0H + Hiγ0φ∂0).
Since γ0 commutes with φ and ∂
0 and (γ0)2 = 1, we have (iγ0φ∂0)2 = −(φ∂0)2.
Since φ and ∂0 commute with H and γ0, we have
(iγ0φ∂0H + Hiγ0φ∂0) = iφ∂0(γ0H + Hγ0).
Given the Dirac matrices (1.2) (recall, again that γ0 = β and γi = γ0α
i) , we have
for all a > 0, γ0γ
a = −γaγ0. Hence, γ0 commutes with [γa, γb] and thus with Dj.
Therefore, we get (γ0H + Hγ0) = i(γ0γ a+ γaγ0)fajDj = 0. We get D2 = H2− (φ∂ 0)2. We recall that D2 = −
g−14Rgand given the metric g = (φ∂0)2−4hand Rg = Rh.
Therefore,
H2 = 4 h−
1 4Rh.
Finally, we have
H2 = γ0(H + m)γ0(H + m) and since m commutes with γ0 and (γ0)2 = Id we get
H2 = (γ0Hγ0+ m)(H + m)
and since H anti-commutes with γ0 and commutes with m, we get
H2 = (−H + m)(H + m) = m2 − H2 = m2− 4 h+ 1 4Rh. Besides m + D = γ0(iφ∂0− H) and (iφ∂0+ H)(iφ∂0− H) = −(φ∂0)2− H2 = −(φ∂0)2+ 4h− 1 4Rh+ m 2.
Remark 3.1. We shall use freely either ∂0 or ∂t for the time derivative.
Corollary 3.5. If u solves the Dirac equation
(3.2) iφ∂tu − Hu = 0
then u satisfies also
(3.3) − (φ∂t)2u + 4hu −
1
4Rhu − m
2u = 0.
Remark 3.2. After a change of variable, we can replace φ by φ = 1, and we get the more simple expression,
m + D = γ0(i∂t− H), and (i∂t+ H)(i∂t− H) = −∂t2+ 4h− 1 4Rh− m 2. 4. Virial Identity We consider the linear equation
(4.1) iφ∂tu − Hu = 0.
We have seen that if u solves (4.1), then u also solves
(4.2) (φ∂t)2u + Lu = 0
with L = H2 = 14Rh + m2− 4h. Note that L is self-adjoint for the inner product
h·, ·ih.
We define
where ψ is a real valued function of space.
Let us be more precise about the spaces for ψ and for u. For ψ, we choose a C2
map such that 4hψ is bounded. For u, let us mention that the mass and cuvature
have been chosen in all the cases such that the L2 norm of Hu controls the H1 norm of u. In other terms, we always assume that m2+ 14inf Rh is positive, which makes
L a positive operator such that
hu, Luih ≥ hu, − 4hui.
What is more, in the cases we consider, Rh may have a singularity in 0. Hence
L is well defined on the space of the maps C∞ with compact support included in Mhr {0}. We denote this space C0∞. Write for all f ∈ C
∞
0 and s ≥ 0,
kf ks = k(1 + L)s/2f kL2(M h).
This defines a norm which is a priori conserved by the flow of the Dirac equation. Let Hs be the completion of C∞
0 with respect to this norm.
Assume u0 ∈ Hsand let u be the solution of the Dirac equation with initial datum
u0, we have u ∈ C(R, Hs) (this is correlated to the fact that the manifold is complete
and thus H self-adjoint). Assume s is big enough, for instance s = 10.
At some point in the computation we take the bilaplacian (or bi-Laplace-Beltrami operator) of ψ. This is defined in terms of distributions as (in the computations we get R Mh4 2 hψ|u|2) h42 hψ, |u|2i = h4hψ, 4h|u|2i
which is well-defined, since 4h|u|2 ∈ L1(Mh) and 4hψ is continuous and bounded.
To get the result in the energy space, it is sufficient to proceed by density.
To conclude this subsection, we compute φ∂tΘ and (φ∂t)2Θ when u solves (4.1).
The computation is the same as in the case of a flat metric and is mainly based on the self-adjointness of L, and one gets the following
Proposition 4.1. Let u be a solution of (4.1). We have that Θ satisfies
φ∂tΘ = Reh[L, ψ]u, φ∂tuih, (4.3) (φ∂t)2Θ = − 1 2Reh[L, [L, ψ]]u, uih. (4.4)
4.1. Commutators. We compute explicit formulae in terms of u and h of φ∂tΘ and
Proposition 4.2. Let u be a solution of (4.1). The explicit expressions of φ∂tΘ and
(φ∂t)2Θ in terms of u and h are
φ∂tΘ = −Re Z Mh (4hψ)uφ∂tu + 2 Z Mh 5hψ · 5huφ∂tu (4.5) (φ∂t)2Θ = Z Mh 1 2(4 2 hψ) + 1 4 5hψ · 5hRh |u|2+ 2 Z Mh (∂ju∂iu)D2(ψ)ij where D2(ψ)ij = hilhkj∂
l∂kψ − Λk,ij∂kψ, from which we deduce the virial identity
(4.6) − Z Mh 1 2(4 2 hψ) + 1 4 5hψ · 5hRh |u|2+ 2 Z Mh (∂ju∂iu)D2(ψ)ij = −(φ∂t)Re Z Mh (4hψ)uφ∂tu + 2 Z Mh 5hψ · 5huφ∂tu . Proof. First, the commutator between L = 14Rh+ m2− 4h and ψ is given by
[L, ψ] = [−4h, ψ] = − 4hψ − 2 5h ψ · 5h
where 5hψ · 5h is the operator given by
5hψ · 5hϕ = hij∂iψ∂jϕ.
We deduce from that φ∂tΘ = −Re Z Mh (4hψ)uφ∂tu + 2 Z Mh 5hψ · 5huφ∂tu . We now have that, since m2 commutes with everything,
[L, [L, ψ]] = [L, − 4hψ] + [
1
4Rh, −2 5hψ · 5h] + [4h, 2 5hψ · 5h] = M1+ M2+ M3. We have, in terms of distributions
M1 = (4h)2ψ + 2 5h(4hψ) · 5h , M2 = 1 25hψ · 5hRh and hence hM1u, uih = Z Mh (4h)2ψ|u|2+ Z Mh 2 5h(4hψ) · 5huu and hM2u, uih = + 1 2 Z Mh 5hψ · 5hRh|u|2.
The term with M2 is dealt with. We now deal with the term with M1 that does
First, we compute (2 5h ϕ · 5h)∗, the symmetric of 2 5h ϕ · 5h for any ϕ ∈ C0
bounded. We have for any test functions v, w ∈ H10,
h2 5hϕ · 5hv, wi = 2 Z p det(h)hij∂iϕ∂jvwd3x = −2 Z ∂j( p det(h)hij∂iϕ)vwd3x − 2 Z p det(h)hij∂iϕv∂jw = −2hv, (4hϕ)wih− 2hv, 5hϕ · 5hwih in other words, (2 5hϕ · 5h)∗ = −2 4hϕ − 2 5hϕ · 5h.
Remark, when we say symmetric, we always mean for the scalar product h·, ·ih.
This gives in particular, for ϕ = 4hψ,
h2 5h(4hψ) · 5hu, ui = −2 Z Mh (42hψ)|u|2− hu, 2 5h(4hψ) · 5huih and thus (4.7) Reh2 5h(4hψ) · 5hu, ui = − Z Mh (42hψ)|u|2 which yields RehM1u, uih = 0.
We deal with the M3 term by directly taking the inner product.
We have M3 = 4h(2 5hψ · 5h) − (2 5h ψ · 5h)4h and hence
hM3u, uih = h4h(2 5hψ · 5h)u, uih− h(2 5hψ · 5h) 4hu, uih
and then, given the symmetric of 2 5hψ · 5h(and the self-adjointness of the
Laplace-Beltrami operator),
h(2 5hψ · 5h) 4hu, uih = −h4hu, 2(4hψ)uih− hu, 4h(2 5hψ · 5h)uih.
Therefore, we get
RehM3u, uih = 2Reh4h(2 5hψ · 5h)u, uih+ 2Reh4hu, (4hψ)uih.
The second term is given by
h4hu, 4hψuih = −
Z
Mh
5hu · 5h(4hψu)
which decomposes, by the Leibniz rule, into
h4hu, 4hψuih = −h5h(4hψ) · 5hu, ui −
Z
Mh
and thanks to previous computations, (4.7), we get Reh4hu, 4hψuih = − Z Mh 4hψ 5hu · 5hu + 1 2 Z Mh (42hψ)|u|2. By summing up, we get
Reh[L, [L, ψ]]u, ui = Z Mh (42hψ) +1 2 5hψ · 5hRh |u|2 − 2 Z Mh (4hψ) 5h u · 5hu + 2Reh4h(2 5hψ · 5h)u, uih.
It remains to compute ReI, where
I = 2h4h(2 5hψ · 5h)u, uih.
We have, by definition of the Laplace-Beltrami operator, I = 4 Z ∂i p det(h)hij∂j(hkl∂kψ∂lu) ud3x. By integration by parts, we have
I = −4 Z p det(h)hij∂j hkl∂kψ∂lu ∂iud3x
which decomposes, by the Leibniz rule, into I = −4 Z p det(h)hij∂j(hkl∂kψ)∂lu∂iud3x − 4 Z p det(h)hijhkl∂kψ∂l∂ju∂iud3x.
Let II be the second term of the right hand side, that is, II = −4
Z p
det(h)hijhkl∂kψ∂l∂ju∂iud3x.
By integration by parts, we have II = 4 Z ∂l p det(h)hijhkl∂kψ∂iu ∂jud3x
which decomposes into II = 4 Z ∂l p det(h)hijhkl∂kψ ∂iu∂jud3x − II. Hence, ReII = 2 Z ∂l p det(h)hijhkl∂kψ ∂iu∂jud3x. Therefore, ReI = −4Re Z p det(h)hij∂j(hkl∂kψ)∂lu∂iud3x
+2 Z ∂l p det(h)hijhkl∂kψ ∂iu∂jud3x that is ReI = Re Z Mh (∂ju∂iu)D(ψ)ij with D(ψ)ij = 2 pdet(h)∂l p det(h)hijhkl∂kψ − 4hil∂l(hkj∂kψ). Since ∂l p det(h)hijhkl∂kψ = hij∂l p det(h)hkl∂kψ + (∂lhij) p det(h)hkl∂kψ , we have D(ψ)ij = 2hij 4 ψ + 2hkl∂kψ∂lhij − 4hil∂l(hkj∂kψ).
To sum up, we get (φ∂t)2Θ = − Z Mh 1 2(4 2 hψ) + 1 4 5hψ · 5hRh |u|2+ Z Mh (4hψ) 5hu · 5hu − 1 2Re Z Mh (∂ju∂iu)D(ψ)ij. We get (φ∂t)2Θ = − Z Mh 1 2(4 2 hψ) + 1 45hψ · 5hRh |u|2+ 1 2Re Z Mh (∂ju∂iu)D1(ψ)ij with D1(ψ)ij = 2 4hψhij − D(ψ)ij
that is, by definition of D(ψ)
D1(ψ)ij = −2hkl∂kψ∂lhij + 4hil∂l(hkj∂kψ)
and by the Leibniz rule :
D1(ψ)ij = 4hilhkj∂l∂kψ + 2∂kψ(−hkl∂lhij + 2hil∂lhkj).
Thanks to the real part, we have a symmetry in i and j. Indeed, Re ∂ju∂iuD1(ψ)ij
= Re∂iu∂juD1(ψ)ij
. Thus, we can replace D1(ψ)ij by 12
D1(ψ)ij + D1(ψ)ji , which yields 1 2Re Z Mh (∂ju∂iu)D1(ψ)ij = 2 Z Mh (∂ju∂iu)D2(ψ)ij
with D2(ψ)ij = hilhkj∂l∂kψ + 1 2∂kψ(−h kl∂ lhij + hil∂lhkj + hjl∂lhki).
We recognize the affine connection or Christoffel symbol Λk,ij = hilhjmΛklm = 1 2 hkl∂lhij − hil∂lhkj− hjl∂lhki , which yields (4.8) D2(ψ)ij = hilhkj∂l∂kψ − Λk,ij∂kψ.
Finally, we get the virial identity, − Z Mh 1 2(4 2 hψ) + 1 4 5hψ · 5hRh |u|2+ 2 Z Mh (∂ju∂iu)D2(ψ)ij = −(φ∂t)Re Z Mh (4hψ)uφ∂tu + 2 Z Mh 5hψ · 5huφ∂tu . 5. The asymptotically flat case
The proof of Theorem (1.1) is fairly classical in this setting (see e.g. [4, 9]); nevertheless, before getting into details, let us give a brief sketch of it in order to make the various steps easier to be followed. The idea is to rely on virial identity (4.6), plug in it a proper choice of the multiplier (with a fixed R > 0) that we will define in the next subsection, integrate in time and carefully estimate all the terms. By multiplier we mean the function ψ. We will start from the Right Hand Side: making use of the modified Hardy inequality (2.6) will allow us to estimate from above with some energy-type terms at some fixed times 0 and T . Then, we will have to bound the Left Hand Side from below, which will be significantly more technical. Here we will make heavy use of our asymptotic-flatness (and smallness) assumptions to prove estimates of the different terms and, roughly speaking, treat the non-flat ones as perturbations. To absorb them, it will be necessary to take the sup in R > 0: this will prevent us from exchanging the time and space norms in the Left Hand Side of (1.15). Eventually, we will take the sup in time and use conservation of energy.
5.1. Choice of the multiplier. We here present the multiplier function ψ that will be used in the main proof. We define the radial function ψ(x) as
ψ0(x) =
Z r
0
where ψ00(r) = r 3, r ≤ 1 1 2− 1 6r2, r > 1
(with a slight abuse we are using the same notation for ψ(x) and ψ(r) where r = |x|). We then define the scaled function
ψR(r) := Rψ0
r R
, R > 0 for which we have
ψ0R(r) = r 3R, r ≤ R 1 2 − R2 6r2, r > R. Moreover, ψ00R(r) = 1 3R, r ≤ R R2 3r3, r > R. ψ000R(r) = −R 2 r41r≥R, ψivR = 4R 2 r51r≥R− 1 R2δ(r − R)
where 1r≥R denotes the characteristic function of the set {x : |x| ≥ R} and δ the
standard Dirac delta distribution. Notice that, for every r ≥ 0, ψR00(r) ≤ 1 2 max{R, r} ≤ 1 2r, ψ 0 R(r) ≤ 1 2. Moreover, notice that
(5.1) ψR00 − ψ 0 R r = 0, r ≤ R − 1 2r2 1 −R 2 r2 , r > R. In the following we shall simply denote with ψ = ψR for a fixed R.
Remark 5.1. Our choice of the multiplier ψ here is classical, and already highly used in several papers to prove smoothing estimates for different dispersive equations, also in some perturbative settings. The function ψ is a mix of the so called virial and Morawetz multipliers, which are respectively given by |x| and |x|2; originally,
the choice of such a function was dictated by the conditions of having a negative bi-Laplacian and a positive Hessian, together with some good decay at infinity. Of course, in a fully variable coefficients setting, this properties are much more difficult
to be fulfilled, and a smart choice of the multiplier, to the best of our knowledge, has never been attempted in this general case. Therefore our choice is motivated by perturbative arguments: the idea of the proof will be that the ”leading” terms in the inequality will mainly recover the ones in the flat case, while the terms involving the variable coefficients will be treated by ”smallness” arguments.
5.2. Estimate of the right hand-side (RHS). We use the Dirac equation (1.7) to rewrite the right hand side of (4.6) as (notice that the mass term vanishes when taking the real part)
(5.2) (φ∂t)Re i Z Mh (∆hψ)Hu u + 2 Z Mh 5hψ · 5hu Hu .
First of all, by the application of Young’s inequality we can write the estimate (5.3) Z Mh (∆hψ)Hu u + 2 5hψ · 5hu Hu ≤ 3 2kHuk 2 L2 Mh + k 5hψ · 5huk 2 L2 Mh + 1 2k∆hψ uk 2 L2 Mh.
Recalling (2.2), (1.11) and (2.6) we then have
k∆hψ uk2L2 Mh ≤ 3N 4 x|−1u 2 L2 Mh + C5 4ν3/2 hxi−1−αu 2 L2 Mh (5.4) ≤ CH 3N 4 + C5N3/2 4 kHukL2 Mh.
Moreover, due to condition (1.13), we have
| 5hψ · 5hu| ≤ N/2| 5 u|,
so that applying estimate (2.10) with the choice ε = 1 −4C(σ)ν4 , we have
(5.5) k 5hψ · 5huk2L2 Mh ≤ 1 2 N 1 −4C(σ)ν4 kHuk2 L2 Mh.
Therefore, multiplying (5.4) times φ−1 and integrating in time between 0 and T we obtain, plugging (5.4) and (5.5) into (5.3),
(5.6) Z T 0 φ−1(t)(φ(t)∂t) Z Mh (∆hψ)Hu u + 2 5hψ · 5hu Hu .ν,N,σ kHu(T )k2L2 Mh + kHu(0)k 2 L2 Mh.
5.3. Estimate of the left hand-side (LHS). We now deal with the left hand side of identity (4.6). We estimate each term separately, and start with the one involving the gradient, namely
(5.7) 2
Z
Mh
(∂ju∂iu)D2(ψ)ij.
Recalling (4.8), we treat separately terms involving derivatives on the coefficients from the others. Concerning Λk,ij we have
|(∂ju∂iu)Λk,ij| ≤ 3|hinv||h0inv|| 5 u| 2|ψ0|
and thus, by our assumptions (1.13) and from the bound on ψ0,
(5.8) |(∂ju∂iu)Λk,ij| ≤
3
2N Chhxi
−1−σ| 5 u|2
.
Turning to the other term, we use the fact that ψ is radial to rewrite it as follows hilhkj∂l∂kψ = hilhkjxˆlxˆk ψ00− ψ 0 |x| + hilhjlψ 0 |x|.
We restrict the quantity above first in the region |x| ≤ R where, notice, ψ00 = |x|ψ0. Therefore, 1|x|≤R(∂ju∂iu)hilhkj∂l∂kψ = 1 3R1|x|≤Rh ilhjl(∂ ju∂iu) ≥ ν 2 3R1|x|≤R| 5 u| 2 (5.9)
where in the last inequality we have used (1.9). In the region |x| > R we have instead (5.10) 1|x|>Rhilhkj∂l∂kψ = 1 2|x|h ilhjl− hilhkjxˆ lxˆk + R2 2|x|3xˆlxˆkh ikhjl− R 2 6|x|3h ilhjl≥ 0
in the sense of matrices (notice that hilhjl− hilhkjxˆ
lxˆk ≥ 0 in the sense of matrices).
We can therefore neglect this term.
We thus multiply (5.7) by φ−1, and integrate in time between 0 and T . Exchanging the integrals and applying (5.8), (5.9) and (5.10) therefore gives (recall (2.11))
(5.11) 2 Z Mh φ−1 Z T 0 (∂ju∂iu)D2ψij ≥ 2ν2 3R Z Mh∩BR k 5 uk2L2 φ,T − CD 2k 5 uk2 Y L2 φ,T
with the constant
(5.12) CD2 =
12ν3/2C h
√ N σ
Now we turn to the bi-Laplacian term, that is (5.13) 1 2 Z Mh (42hψ)|u|2. First of all observe that
∆h(f g) = (∆hf )g + 2 5hf · 5hg + (∆hg)f,
so that we can write, after some manipulations
∆2hψ = ∆h∆hψ = I + II + III + IV with I = ˆh · ∆hψ00+ (h − ˆh)∆ ψ0 |x| , II = Aˆh · ψ00+ A(h − ˆh) · ψ 0 |x|, III = 2 5hˆh · 5hψ00+ 2 5h(h − ˆh) · 5h ψ0 |x|, IV = ∆h 1 pdet(h)∂j(˜h jkxˆ kψ0) ! .
We separate terms involving derivatives on the coefficients of hjk (which will be
of perturbative nature) from the others. After some long winded but not difficult computations (see [9] section 4.4 for further details) one gets
∆2hψ = S(x) + R(x) S(x) =bh2ψiv+ 2bh(h − bh)ψ000 |x| + (h−bh)(h−3bh) |x|2 ψ00− ψ|x|0+ +|x|22[h`mh`m− hbh − 4(|hbx|2− bh2)] ψ00− ψ|x|0+ +|x|4 [|hx|b2− bh2]ψ000− ψ00 |x| + ψ0 |x|2 and p det(h)R(x) =bh∂m(˜h`m)bxmψ 000+ (h − bh)∂ k(˜hjk)xbk ψ00 |x| − ψ0 |x|2 + + [∂j(˜hjk∂k(h`m)bx`bxm) + ∂j(˜h jkh`m)∂ k(xb`xbm)] ψ00− ψ0 |x| +pdet(h)(∆hh) ψ0 |x| + 2 p det(h)hjk∂kh`mxb`xbmxbj ψ000−ψ|x|00+ + 2pdet(h)h(5h, 5ψ0 |x|) +pdet(h)∆h 1 √ det(h)∂j(˜hjkxˆkψ 0) .
In our assumptions on the metric h and noticing that by the definition of ψ we have |ψ0| ≤ |x| 2(R∨|x|), |ψ 00| ≤ n−1 2n(R∨|x|), |ψ 000| ≤ n−1 2(R∨|x|)|x|,
the remainder term R(x) can be estimated as
(5.14) |R(x)| ≤ 36Ch(N + Ch)
|x|hxi1+σmax{R, |x|}.
We regroup the terms in S(x) to write
S(x) = ˆh2ψiv+2hˆh − 6ˆh2+ 4|hˆx|2ψ 000 |x| (5.15) + 2h`mh`m+ h2− 6hˆh + 15ˆh2− 12|hˆx|2 ψ 00 |x|2 − ψ0 |x|3 . Now, plugging our choice of the weight into (5.15) gives (recall (5.1)
S(x) = − 1 R2ˆh
2δ(|x| − R)
for r ≤ R and, after rearranging the terms, S(x) = 23ˆh − h ˆhR 2 |x|5 − 6(|hˆx| 2− ˆh2))R 2 |x|5 −2h`mh`m + h2− 6hˆh + 15ˆh2− 12|hˆx|2 1 − R |x| 2! 1 2|x|3
for |x| > R. As we can write hinv(x) = I + ε(x) (meaning εjk = hjk − δjk), we have
h`mh`m = δ`m`m+ 2δ`mε`m+ ε`mε`m = 3 + 2ε + ε`mε`m
as well as
ˆ
h = 1 + ˆε, a + h, |hˆx|2 = 1 + 2ˆε + |εˆx|2.
Notice also that by assumption (1.12) |ε(x)| = |hinv(x) − I| ≤ CIhxi−σ < 1 and
therefore
|ε| ≤ 3CIhxi−σ, |ˆε| ≤ CIhxi−σ, |εˆx| ≤ CIhxi−σ
so that 2h`mh`m + h2− 6hˆh + 15ˆh2− 12|hˆx|2 = 4ε − 12ˆε + 2ε`mε`m+ ε2 −6εˆε + 15ˆε2− 12|εˆx|2 ≥ 4ε − 12ˆε − 6εˆε − 12|εˆx|2 ≥ −46CIhxi−σ. (5.16) Also, as 1 − CI ≤ ˆh ≤ 1 + CI, we have −ˆh2 ≤ −(1 − CI)2,
and (5.17)
3ˆh − h ˆh ≤ 6CI(1 + CI) ≤ 12CI.
Therefore, under our assumptions and with our choice of the multiplier ψ, we obtain the estimates
S(x) ≤ −(1 − CI)2
1
R2δ(|x| − R) for |x| ≤ R,
and, with a bit more careful computations that essentially rely on (5.16) and (5.17)
S(x) ≤ 24CI R2 |x|5 + 1 |x|3hxiσ for |x| > R.
We now multiply times φ−1 and integrate in time (5.13) from 0 to T : this gives
− Z T 0 φ−1 Z Mh ∆2hψ|u|2 = − Z Mh ∆2hψkuk2L2 φ,T = I + II with I = − Z Mh S(x)kuk2L2 φ,T, II = − Z Mh R(x)kuk2L2 φ,T
and estimate the two terms separately. For the S(x) term we get, thanks to (2.12) and (2.13), I ≥ (1 − CI)2 1 R2 Z Mh∩SR kuk2 L2 φ,T − 72CI σ ν N 3/2 kuk2 XL2 φ,T.
The R(x) term can be instead estimated with II ≥ −36Ch(N + Ch) Z T 0 " Z Mh∩BR + Z Mh∩BRc # |u|2 φ(t)|x|2hxi1+σ where Bc
R is the complementary set of BR, that is the region where r > R. Thanks
to (2.13) we have (5.18) Z T 0 Z Mh∩BR |u|2 φ(t)|x|2hxi1+σ ≤ Z Mh∩BR kuk2 L2 φ,T |x|2hxi1+σ ≤ 2 σ ν N 3/2 kuk2 XL2 φ,T
and thanks to (2.5) and (2.16)
(5.19) Z T 0 Z Mh∩B1 |u|2 φ(t)|x|2hxi1+σ ≤ Z T 0 Z Mh∩B1 |u|2 φ(t)|x|2 ≤ 4k 5 uk 2 L2(M h∩B1)L2φ,T.
From (5.18) and (5.19) we thus obtain
II ≥ −324Ch(N + Ch) σ ν N 3/2 kuk2 XL2 φ,T + k 5 uk 2 L2(M h∩B1)L2φ,T .
Putting all together gives −1 2 Z T 0 φ−1 Z Mh ∆2hψ|u|2 ≥ (1 − CI) 2 2 1 R2 Z Mh∩SR kuk2 L2 φ,T −1 σ ν N 3/2h 36CIkuk2XL2 φ,T + 162Ch(N + Ch) kuk2 XL2 φ,T + k 5 uk 2 L2(M h∩B1)L2φ,T i
Recalling (2.5) eventually gives
−1 2 Z T 0 φ(t)−1 Z Mh ∆2hψ|u|2 ≥ (1 − CI) 2 2 1 R2 Z Mh∩SR kuk2 L2 φ,T (5.20) − CI ∆2kuk2XL2 φ,T − C II ∆2k 5 uk2Y L2 φ,T.
where the constants are explicitly given by (5.21) C∆I2 = ν N 3/236CI+ 162Ch(N + Ch) σ , C II ∆2 = ν N 3/2162Ch(N + Ch) σ
We now turn to the last term of (4.6) that is
(5.22)
Z
Mh
5hψ · 5hRh|u|2.
Notice that it involves only terms with derivatives on h (and indeed vanishes in the flat case). Therefore, using repeatedly assumptions (1.13), it is not difficult to show that | 5 Rh(x)| ≤ 3Ch(1 + 18N2) hxi3+3σ + 3N C2 h+ 9N Ch2+ 9N3Ch2 hxi2+2σ|x| + 3N2C3 h hxi1+σ|x|2 ≤ CR hxi1+σ|x|2. (5.23)
where the constant CR is the sum of the three numerators above, that is
CR = 3Ch(1 + 18N2) + 3N Ch2(4 + 3N
2) + 3C3 hN
2.
We now multiply as usual (5.22) times φ−1 and integrate in time between 0 and T : following calculations and relying on (2.13)-(2.16) yield the estimate
Z T 0 φ(t)−1 Z Mh 5hψ · 5hRh|u|2 ≥ −CR Z T 0 " Z Mh∩BR + Z Mh∩BRc # |u|2 φ(t)|x|2hxi1+σ ≥ −CR 2 σ ν N 3/2 kuk2 XL2φ,T + 2 ν N 3/2 k 5 ukY L2 φ,T = −4CRIkuk2 XL2 φ,T − C II Rk 5 ukY L2 φ,T (5.24)
with (5.25) CRI = CR 2σ ν N 3/2 , CRII = CR 2 ν N 3/2 .
5.4. Conclusion of the proof. We multiply times φ−1 and integrate in time iden-tity (4.6) from 0 to T , exchange integrals and use (5.11), (5.20), (5.24) for the left hand side and (5.6) for the right hand side to obtain
(5.26) (1 − CI) 2 2 1 R2 Z Mh∩SR kuk2 L2 φ,T + 2ν2 3R Z Mh∩BR k 5 vk2 L2 φ,T −(CI ∆2 + CRI)kuk2XL2 φ,T − (CD 2 + C∆II2 + CRII)k 5 ukY L2 φ,T ≤ Cν,N,σkHu(T )k2L2 Mh + kHu(0)k 2 L2 Mh
where the constants are explicit and given by (5.12), (5.21) and (5.25). We also stress that the constant C = Cν,N,σ does not depend on R. We now take the sup
over R > 1 on the left hand side of (5.26) (notice that only the first two terms of inequality above depend on R). We use (2.14) to estimate, for 0 < θ < 1,
(5.27) (1 − CI) 2 2 supR>1 1 R2 Z Mh∩SR kuk2 L2 φ,T ≥ (1 − θ) (1 − CI)2 2 supR>1 1 R2 Z Mh∩SR kuk2 L2 φ,T +θ(1 − CI)2 1 4 ν N 3/2 kuk2 XL2 φ,T − 13 4 k 5 ukY L2φ,T . Thanks to our assumption (1.12), we can take ν = 1 − CI, such that
sup R>1 2ν2 3R Z Mh∩BR k 5 uk2 L2φ,T ≥ 2 3(1 − CI) 2k 5 uk2 Y L2φ,T.
Choosing θ in (5.27) such that 13θ 4 ≤
2
3 (e.g. θ = 1/5) and using the simple property
sup R (F1(R) + F2(R)) ≥ 1 2 sup R F1(R) + sup R F2(R) therefore yields (1 − CI)2 2 supR>1 1 R2 Z Mh∩SR kuk2 L2 φ,T + supR>1 2ν2 3R Z Mh∩BR k 5 uk2 L2 φ,T ≥ (1 − CI)2 1 40 ν N 3/2 kuk2 XL2 φ,T + 1 120k 5 uk 2 Y L2 φ,T
which plugged into (5.26) finally gives M1kuk2XL2 φ,T + M2k 5 uk 2 Y L2φ,T ≤ Cν,N,σkHu(T )k 2 L2Mh + kHu(0)k 2 L2Mh
with M1 = (1 − CI)2 40 − C I ∆2 − CRI and M2 = (1 − CI)2 120 − CD2 − C II ∆2 − CRII.
The proof is concluded provided the constants M1 and M2 are positive, i.e. if the
constants CI and Ch are small enough, by letting T to infinity and using the
conser-vation of the L2-norm of Hu, which is standard.
6. warped products
We dedicate this section to prove Theorem 1.2. First of all, we notice that if h is in the form (1.17) the following result holds.
Proposition 6.1. Let h be a warped product. We have
(6.1) Rh = −2 d00 d + 1 2 d0 d 2 + 1 dRκ and
(6.2) Λ1,ij = 0 if i = 1 or j = 1 and Λ1,ij = −1 2
d0 d2κ
ij otherwise.
Proof. The proof is straightforward computation.
The strategy to prove Theorem 1.2 is the same we have seen in details in the pre-vious section to deal with the asymptotically flat case, and thus consists in applying the virial identity (4.6) to an appropriate function ψ, and then estimate the different terms. We will deal with the three different cases separately.
Let us be more precise. The multiplier we choose is very similar to the flat case or, indeed, the asymptotically flat case. First, we take ψ to be radial, which means here that it depends only on the priviledged variable x1 = r. We divide ψ into two sectors : one below a chosen R (r ≤ R) and one above R.
Below R, we choose the map ψ0 to be affine. This is important because of the integral
Z
Mh
D2(ψ)ij∂iu∂ju;
as ψ is radial, this term is equal to Z Mh h11ψ00|∂ru|2− Z Mh ψ0Λ1,ij∂iu∂ju.
Taking ψ affine (and choosig it not constant), the first part controls the L2 norm of ∂ru. In the subflat and flat cases, we choose ψ0 linear because Λ1,ij is proportional
to −1rhij thus we need r to compensate this loss. In the hyperbolic case, we have
Λ1,ij proportional to −hij thus we need a constant term.
Above R, we take ψ0(r) = A − Bd(r)−1, choosing A and B such that ψ is C2. This
has many advantages : since d is increasing, ψ0 is increasing and positive; taking the Laplace-Beltrami of ψ yields 4hψ = A4hr = Ad
0
d; this choice makes ψ
00differentiable
but not C1, which induces a Dirac delta in 42 hψ.
6.1. Hyperbolic-type metrics. We start with the choice d(r) = er/2 that, as one may re-scale, includes some hyperbolic manifolds. In this case we have
Rh = − 3 8+ e −r/2R κ, ∂1Rh = − 1 2e −r/2R κ, Λ1,ij = − 1 4h ij .
We recall that under the hypothesis of Theorem1.2for the hyperbolic type metrics, the curvature of κ is positive, we recall our notation : Rκ > 0.
We make the following choice for ψR, it is a function depending only on x1 = r
and, with the notation ψR0 = ∂1ψR, we take, for some M ≤ Rκ,
(6.3) ψR0 (r) = 1 + Mre
−R/2
if r ≤ R 1 + M Re−R/2(2 + R) − 2M e−r/2 if r > R. With this choice, we have that ψ is C2 and the following identities hold
ψ00R = M e−R/21r≤R+ M e−r/21r>R 4hψR = 1 2 + M e −R/2 (1 + r 2) 1r≤R+ 1r>R 1 2 + M e −R/2 (2 + R) 42ψ R = − M 2 e −R/2 δ(r − R) + 1r≤R M 4 e −R/2
where δ is the Dirac delta.
We start by computing the terms involving |u|2 in the virial identity.
Lemma 6.2. We have −1 2 Z Mh 42 hψ + 1 25hψ · 5hRh |u|2 ≥ Z SR dκ|u|2 where SR is the domain r = R.
Proof. We use the fact that Rκ is positive, that ψ0 ≥ 0 and that below R, ψ0(r) ≥ 1,
to get −1r≤R M 4 e −R/2− 1 25hψ · 5hRh ≥ 1r≤R e−r/21 4Rκ− e −R/2M 4 . Since M ≤ min Rκ, we get
−1r≤R
M 4 e
−R/2− 1
Therefore, we have −1 2 Z Mh 42 hψ + 1 2 5hψ · 5hRh |u|2 ≥ Z Mh M 2 e −R/2 δ(r − R)|u|2. What is more, Z Mh δ(r − R)|u|2 = Z R 0 drer/2 Z Sr |u|2dκ = eR/2 Z SR |u|2dκ
which yields the result.
We now deal with the terms involving the gradient of u. Lemma 6.3. Assuming M ≤ 14, we have
Z Mh D2(ψ)ij∂iu∂ju ≥ M e−R/2 Z BR | 5hu|2.
Proof. If i = j = 0, we have D2(ψ)ij = ψ00(r). And if none of them is 0, we have
D2(ψ)ij = 1 4h
ijψ0. We use that above R, ψ0 and ψ00 are non negative to get
Z
r>R
D2(ψ)ij∂iu∂ju ≥ 0.
Below R, we use that ψ00≥ M e−R/2 and ψ0 ≥ 1 ≥ M e−R/2, to get the result.
Lemma 6.4. We have, for any η > 0
Z ∞ 0 hri−(1+η)dr sup R Z SR Z dtφ−1(t)|u2|dκ ≥ Z Mh hri−(1+η)e−r/2 Z dtφ−1(t)|u|2. Proof. Indeed, Z Mh hri−(1+η)e−r/2 Z dtφ−1(t)|u|2 = Z ∞ 0 hri−(1+η) Z Sr Z dtφ−1(t)|u|2dκ. Lemma 6.5. For any η2 > 0, there exists Cη2 such that
Cη2sup R e−R/2 Z BR Z dtφ−1(t)| 5hu|2 ≥ Z Mh hri−(1+η2)e−r/2 Z dtφ−1(t)| 5hu|2.
Proof. Indeed, let χ = hri−(1+η2)e−r/2, we have
Z Mh χ(r) Z dtφ−1(t)| 5hu|2 = − Z Mh Z ∞ r χ0(y)dy Z dtφ−1(t)| 5hu|2(r).
