4. The Schwarzschild exterior background
4.4. The `=0, 1 spherical harmonics and elliptic estimates
We collect in this final subsection some useful properties which require isolating the`=0,1 angular frequencies of a tensor. More specifically, after defining notation in §4.4.1, we shall recall in §4.4.2 the classical `=0,1 spherical harmonics and define what it means forSu,v2 tensors of various types to be supported on angular frequencies`>2. This will then allow us to infer in§4.4.3some useful elliptic estimates on spheres for such tensors.
4.4.1. Norms on spheres
Let (θ, φ) denote standard spherical coordinates as in§4.1.2where the spherical metric takes the form (89).
We define the following pointwise norm forSu,v2 -tensorsξA1...An of rankn:
|ξ|2:=/gA1B1... /gAnBnξA1...AnξB1...Bn. (104) We also define theL2(Su,v2 )-norm
kξk2S2 u,v:=
Z
Su,v2
r2(u, v) sinθ dθ dφ|ξ|2 (105) and note that(18)
kr−1·ξk2S2 u,v=
Z
Su,v2
sinθ dθ dφ|ξ|2. (106)
4.4.2. The `=0,1 spherical harmonics and tensors supported on `>2
Recall the well-known spherical harmonicsYm` (where`∈N0andm∈{−`, ..., `}admissible for fixed`) on the unit sphere. The`=0,1 spherical harmonics are given explicitly by
Ym=0`=0= 1
√4π (107)
and
Ym=0`=1 = r 3
4πcosθ, Ym=−1`=1 = r 3
4πsinθcosφ, Ym=1`=1 = r 3
4πsinθsinφ. (108) We have that the above family is orthogonal with respect to the standard inner product on the sphere, and any arbitrary function f∈L2(S2) can be expanded uniquely with respect to such a basis.
(18) We will often write quantities in this form, as it is easier to read off the decay.
Definition 4.1. We say that a functionfonMissupported on `>2 if the projections Z
sinθ dθ dφ f·Ym` = 0
vanish for (107) and (108). Any functionf can be uniquely decomposed orthogonally as
f=c(u, v)Ym=0`=0+
1
X
i=−1
ci(u, v)Ym=i`=1(θ, φ)+f`>2,
where f`>2 is supported on `>2. The functions c(u, v) and ci(u, v) inherit regularity fromf.
Recall that an arbitrary 1-formξonS2 has a unique representationξ=r /D?1(f, g) in terms of two unique functionsf and g on the unit sphere, both with vanishing mean.
We can use this to define an analogous decomposition forSu,v2 1-forms onM. We then have the following definition.
Definition 4.2. We say that a smooth Su,v2 1-formξ on M issupported on `>2 if the functionsf andgin the unique representation
ξ=r /D?1(f, g)
are supported on`>2. Any smooth Su,v2 1-form ξ on Mcan be uniquely decomposed orthogonally as
ξ=ξ`=1+ξ`>2, where the two scalar functions
(r /divξ`=1, r /curlξ`=1) are in the span of (108) andξ`>2is supported on `>2.
For symmetric tracelessSu,v2 2-tensors, we have the following result.
Proposition 4.4.1. Let ξ be a smooth symmetric traceless S2u,v 2-tensor. Then,ξ can be uniquely represented as
ξ=r2D/?2D/?1(f, g),
wheref and g are supported on `>2. In this sense, any symmetric traceless 2-tensor on S2is supported on`>2.
Proposition 4.4.1follows immediately by duality considerations from the following lemma concerning the angular operator
T =r2D/?2D/?1, (109)
which, for fixeduandv can be considered as an operator on the unit sphere(19)which maps a pair of functions (f1(θ, φ), f2(θ, φ)) to a symmetric traceless tensor onS2. Note that its adjoint,r2D/1D/2, has trivial kernel inL2.
For the computations in the following lemma we regardT as an operator defined on pairs of smooth functions, which are dense inL2(S2).
Lemma 4.4.1. The kernel of T is finite-dimensional. More precisely, if the pair of functions (f1, f2) is in the kernel,then which can be written
Z
Clearly, the constant solutions f1=c and f2=˜c satisfy this (and are obviously in the kernel). If we assume bothf1andf2to have mean value zero, we see using the Poincar´e
(19) More precisely, it acts on the round spheresSu,v2 which have been rescaled (this is the reason for the factorr2) to have unit radius.
inequality on the sphere that the only functions satisfying the above condition are the
`=1 modes. Finally, one checks directly that the`=1 modes are indeed in the kernel: In components, the equation forf1,
∇/A∇/BYm`=1+∇/B∇/AYm`=1=−2/g
ABYm`=1,
reads (using Γθφφ=−sinθcosθand Γφθφ=cosθ/sinθ in standard coordinates) (2∂2θ+2)Ym`=1= 0
∂θ∂φYm`=1−cosθ
sinθ∂φYm`=1= 0
∂2φYm`=1+sinθcosθ ∂θYm`=1+sin2θYm`=1= 0
and these identities are easily verified. The computation for f2 is similar or can be inferred by duality.
In particular, we have the following corollary.
Corollary4.1. Let ξbe a smooth symmetric traceless Su,v2 2-tensor on M. Then, Z
S2u,v
sinθ dθ dφ /D1D/2ξ·(c+ciYm=i`=1,c˜+˜ciYm=i`=1) = 0.
for any choice of constants c, ci, ˜cand ˜ci.
Note that this, in particular, means that, ifξis a symmetric tracelessSu,v2 2-tensor, then the scalarsdiv/ div/ ξandcurl/ div/ ξare supported on`>2.
4.4.3. Elliptic estimates and positivity for angular operators onS2u,v-tensors We end with a discussion of elliptic estimates giving positivity for various angular oper-ators acting onSu,v2 tensors supported on`>2.
We first give an estimate associated with the operatorT from (109) acting on pairs of scalar functions supported on`>2.
Proposition 4.4.2. Let (f1, f2) be a pair of functions on Su,v2 supported on `>2.
Then, we have the elliptic estimate
2
X
i=0
Z
S2u,v
sinθ dθ dφ(|ri∇/if1|2+|ri∇/if2|2). Z
Su,v2
sinθ dθ dφ|r2D/?2D/?1(f1, f2)|2. Proof. This follows immediately revisiting the computation of Lemma4.4.1.
We next give identities (in the formulas below, recall thatK=1/r2) associated with operators acting on symmetric traceless 2-tensors and 1-forms.
Proposition 4.4.3. Let ξ be a smooth symmetric traceless S2u,v 2-tensor on M.
Then, Z
Su,v2
sinθ dθ dφ(|∇ξ|/ 2+2K|ξ|2) = 2 Z
Su,v2
sinθ dθ dφ|div/ ξ|2, (110) Z
S2u,v
sinθ dθ dφ|D/?2div/ ξ|2= Z
Su,v2
sinθ dθ dφ 14|∆ξ|/ 2+K|∇ξ|/ 2+K2|ξ|2
. (111)
Now,let η be a smooth Su,v2 1-form on M. Then,we have kD/?2D/?1D/1ηk2S2
u,v=k2D/?2div/ D/?2ηk2S2
u,v+k2K /D?2ηk2S2
u,v+8Kkdiv/ D/?2ηk2S2
u,v. (112) Proof. See [14] for the first and note D/?2div/ ξ= −12∆+K/
ξ for the second. For (112) observe that D/?2D/?1D/1η=D/?2(−∆+K)η/ =2D/?2div/ D/?2η+2K /D?2η and integrate the cross-term by parts.
Remark 4.2. The identities (110) and (111) can be paraphrased as saying that the operatorA[n] defined in (103) acting on symmetric tracelessSu,v2 2-tensors is uniformly elliptic and positive definite.
The identity (112), on the other hand, when combined with Proposition4.4.2 and the identity (111) leads to the following corollary, which can be thought of as an elliptic estimate associated with the operatorD/?2 acting onSu,v2 1-formsη supported on`>2.
Corollary 4.2. Let η be a smooth S2u,v 1-form supported on `>2. Then, we have
3
X
i=0
Z
S2
sinθ dθ dφ|ri∇/iη|2. Z
S2
sinθ dθ dφ|A[2]D/?2η|2.
The statement remains true replacing 3 by 1 in the sum on the left and removing A[2]
on the right-hand side.
We also remark, at this point already, the following result.
Proposition4.4.4. Let ξbe a smooth symmetric traceless Su,v2 2-tensor. Then,we have the estimate
− Z
S2
sinθ dθ dφ
∆−/ 4 r2
ξ
AB
ξAB> 6 r2
Z
S2
sinθ dθ dφ|ξ|2. (113)
Proof. We only outline the proof. The desired estimate follows from
− Z
S2
sinθ dθ dφ /∆ξ·ξ= Z
S2
sinθ dθ dφ|∇ξ|/ 2> 2 r2
Z
S2
sinθ dθ dφ|ξ|2, (114) which holds for any symmetric tracelessS2u,v2-tensorξ. The latter can in turn be shown by representing the tensorξasξ=D/?2D/?1(f, g) for unique functionsf andgsupported on
`>2 as in Proposition4.4.1, so in particular
− Z
S2
sinθ dθ dφ /∆f·f>6r−2 Z
S2
sinθ dθ dφ|f|2,
and the same estimate forg) and diligently integrating by parts using the properties of spherical harmonics (in particular, their orthogonality).