High-frequency analysis

We now analyzeP_~ near fiber infinity ∂^bT^∗M. By Lemma 8.5, we only need to study the propagation of regularity within∂Σ=G⁻¹(0)∩^bS^∗M along the flow of the rescaled Hamilton vector field

HG= ˆ%HG∈ Vb(^bS^∗M),

see (3.26); here ˆ%is a boundary defining function of fiber infinity ^bS^∗M⊂^bT^∗M. Note thatH_G is equal to the Hamilton vector field of the real part ofσ_b,~(P_~).

Let`=σ<sub>b,~</sub>(L<sub>1,~</sub>), so the eigenvalues of±ˆ%`are positive near∂Σ^±. In order to prove the propagation of regularity (i.e. of estimates) in the future direction, as explained after the proof of Lemma8.5, we need to choose a positive definite inner product on π^∗bT^∗M (with π:∂Σ!M being the projection), so that ±¹₂%(`+`ˆ ^∗) is positive at (and hence near)∂Σ^±in the sense of self-adjoint endomorphisms; see also [72, Proposition 3.12]. We phrase this in a more direct way in Lemma8.8below. For real-principal-type propagation

estimates, we only need to arrange this locally in∂Σ, but we can in fact arrange it globally, which will make the estimates at radial points and at the trapped set straightforward.

Lemma8.8. Fix the positive definite inner product

g_R=dt²_∗+dr²+/g (8.11)

on ^bT^∗M, used to define adjoints below. Then, there exists a pseudodifferential operator Q∈Ψ⁰_b,~(M;^bT^∗M)which is elliptic near ∂Σ,with microlocal inverse Q⁻, so that

±%σˆ b,~

1

2i(QPe,~Q⁻−(QPe,~Q⁻)^∗)

>0 (8.12)

holds near ∂Σ^± for alle close to 1.

Proof. Since^CPg,~has a real scalar principal symbol, it does not contribute to (8.12);

thus, we need to arrange (8.12) for−iLe,~ in place ofPe,~. Moreover, if (8.12) holds for e=1 for some choice ofQ, then one can take the sameQforeclose to 1 by compactness considerations. Explicitly, then, we may diagonalize the symbol ofL1,~, given in (8.10), near∂Σ^± by conjugating it by an endomorphism-valued order-zero symbolq; quantizing q gives an operator Q with the desired properties. Concretely, near ∂Σ, we may take

%=|cˆ ²σ+νξ|⁻¹, and then

q=







1 ∓ˆ%(νσ−µξ) ±ˆ%r⁻²iη

0 1 0

0 0 1





.

The proof is complete.

This immediately gives the propagation of regularity/estimates.

Proposition 8.9. Let B₁, B₂, S∈Ψ⁰_b,~(M;^bT^∗M)be operators with wave front set disjoint from ∂Σ⁻ (resp.∂Σ⁺),as well as from the zero section o⊂^bT^∗M. Suppose that all backward (resp. forward)null-bicharacteristics of HG from WF⁰_b,~(B2)∩^bS^∗M reach Ellb,~(B1)in finite time while remaining in Ellb,~(S),and with S elliptic onWF⁰_b,~(B2).

Then for all s, %∈Rand N∈R, there exists ~0>0 such that one has kB2uk_H^s,%

b,~.kB1uk_H^s,%

b,~+~⁻¹kSP_~uk_Hs−1,%

b,~ +~^Nkuk_H−N,%

b,~ , 0<~<~0, (8.13) in the strong sense that if the norms on the right are finite, then so is the norm on the left, and the estimate holds. The same holds if one replaces P_~ by its adjoint P^∗

~

(with respect to any non-degenerate fiber inner product)and interchanges ‘backward’ and

‘forward’. See Figure 8.3.

Ellb,~(B1)

Ellb,~(S)

WF⁰_b,~(B2)

∂Σ⁺

HG

Figure 8.3. Propagation of regularity in the future direction within∂Σ⁺: we can propagate microlocal control from Ellb,~(B1) to Ellb,~(B2). In∂Σ⁻, the arrows are reversed, correspond-ing to propagation in the backwards direction along HG(which is still the future direction in∂Σ).

This propagates H_b,^s,%

~-control ofuin the forward (resp. backwards) direction along theH_G-flow in∂Σ⁺(resp.∂Σ⁻).

Proof. First of all, with Qas in Lemma 8.8, we can write QP_~u=QP_~Q⁻(Qu)+Ru

with WF⁰_b,~(R)∩WF⁰_b,~(S)=∅; from this, one easily sees that it suffices to prove the proposition for

P_~⁰:=QP_~Q⁻ (8.14)

in place ofP_~. This then follows from a standard positive commutator argument, con-sidering that

i~⁻¹(hAu, AP_~⁰ui−hAP_~⁰u, Aui) =hi~⁻¹[P_~⁰, A²]u, ui−h(i~)⁻¹(P_~⁰−(P_~⁰)^∗)A²u, ui (8.15) for a suitable principally scalar commutant A=A^∗∈τ^−2rΨ^s−1/2_b,~ (M;^bT^∗M) with wave front set contained in a small neighborhood of∂Σ^±, quantizing a non-negative symbola whoseHG-derivative has a sign on Ellb,~(B2), with an error term with the opposite sign on Ellb,~(B1). For example, in the case of ∂Σ⁺, the term coming from the commutator can be written σb,~(i~⁻¹[P_~⁰, A²])=HGa²=−b²₂+b²₁, while by the previous lemma, the skew-adjoint part gives a contribution (in fact of order~⁻¹, rather than 1, due to the strict positivity in (8.12)) with the same sign as the term −b²₂; thus, this contribution can be dropped from the estimate.

More concretely, quantizingb₁andb₂, one can evaluate the right-hand side of (8.15);

the a-priori control termkB1uk_H^s,%

b,~ (resp. conclusion termkB2uk_H^s,%

b,~) in the estimate (8.13) arises from the quantization ofb²₁ (resp.b²₂). The left-hand of (8.15) is estimated bykAuk²+~⁻²kAP⁰

~uk, and the first term can be absorbed into thekB₂uk_H^s,%

b,~ if one chooses a carefully. We refer the reader to [80, §2] and [140, §2.5] for a more detailed discussion and further references.

S

B1

B2

∂R⁺±

bS_X^∗M

Figure 8.4. Propagation of singularities near a component∂R⁺_±of the radial set within∂Σ⁺: we can propagate microlocal control from Ellb,~(B1) to Ellb,~(B2), in particular into the boundary, along the forward direction of theHG-flow. In ∂Σ⁻, we can propagate into the boundary as well by propagating backwards along theHG-flow.

Remark 8.10. Using the strict positivity of the contribution of the skew-adjoint part in (8.15), the estimate (8.13) can be improved, to wit

kB₂uk_H^s,%

b,~.~^1/2kB₁uk_H^s,%

b,~+kSP_~uk_Hs−1,%

b,~ +~^Nkuk_H−N,%

b,~

, 0<~<~0. (8.16) Indeed, the right-hand side of (8.15) controls ~⁻¹kAuk² (rather than merely kB2uk²), due to the presence of the second term, provided one has a-priori control ofkB1uk²; the left-hand side on the other hand can be estimated byε~⁻¹kAuk²+Cε~⁻¹kAP_~⁰uk², the first term of which can be absorbed, giving (8.16). The same improvement can be made in the statements of Propositions8.11and8.12below.

Likewise, we have microlocal estimates for the propagation near the radial set∂R.

Proposition 8.11. Suppose the wave front sets of B1, B2, S∈Ψ⁰_b,~(M;^bT^∗M) are contained in a small neighborhood of ∂Σ⁺ (resp. ∂Σ⁻), with B2 elliptic at ∂R⁺±, resp.

∂R⁻± (see §3.4), and so that all backward (resp. forward) null-bicharacteristics from WF⁰_b,~(B2)∩^bS^∗M either tend to ∂R⁺± (resp. ∂R⁻±), or enter Ellb,~(B1) in finite time, while remaining in Ellb,~(S);assume further that S is elliptic on WF⁰_b,~(B2). Then,for all s, %∈Rand N∈R,there exists ~0>0such that the estimate(8.13)holds for0<~<~0. See Figure 8.4.

The same estimate holds if one replacesP_~by its adjoint and interchanges ‘backward’

and ‘forward’.

Proof. This follows again from a positive commutator argument; see [75, Propo-sition 2.1]. We may replace P_~ by P⁰

~ defined in (8.14). Using the non-negativity of (i~)⁻¹(P⁰

~−(P⁰

~)^∗) near∂Σ⁺, we get an estimate under a threshold condition on the reg-ularity s relative to the weight % of the form s−β%−¹₂>0, with β=β±; but the strict positivity of (i~)⁻¹(P⁰

~−(P⁰

~)^∗) gives an extra contribution of size~⁻¹, so the main term

B1

B2

B1

B1

∂Γ⁺

∂Γ⁺_fw

∂Γ⁺_fw

∂Γ⁺_bw

bS_X^∗M

Figure 8.5. Propagation of singularities near the trapped set∂Γ⁺: we can propagate microlo-cal control from Ellb,~(B1) to Ellb,~(B2), and hence from the forward trapped set∂Γ⁺_fwinto the trapped set∂Γ⁺(and from there into the backward trapped set∂Γ⁺_bw), along the forward direction of the HG-flow. For the corresponding propagation result in∂Γ⁻, the subscripts

‘bw’ and ‘fw’ are interchanged.

in the positive commutator argument is (up to an overall sign) >s−β%−¹₂+δ~⁻¹ for some (small)δ >0, which is>¹2δfor small~. This explains why the radial point estimate doesnot require any above-threshold assumptions which appear in the estimate given in the reference.

Finally, at the trapping, we have the following result.

Proposition 8.12. There exist operators B1, B2, S∈Ψ⁰_b,~(M;^bT^∗M),with B2 and Selliptic near∂Γ⁺(resp.∂Γ⁻)and WF⁰_b,~(B1)∩∂Γ⁺_bw=∅(resp. WF⁰_b,~(B1)∩∂Γ⁻_fw=∅), where Γ⁺_bw⊂^bT_X^∗M is the backward trapped set (with respect to the HG flow)in Σ⁺ and Γ⁻_fw⊂^bT_X^∗M is the forward trapped set in Σ⁻, so that, for all s, %∈R and N∈R, there exists ~0>0 such that the estimate (8.13)holds for 0<~<~0. See Figure 8.5.

The same estimate holds if one replaces P_~ by its adjoint, now with WF⁰_b,~(B₁)∩

∂Γ⁺_fw=∅(resp. WF⁰_b,~(B₁)∩∂Γ⁻_bw=∅), where Γ⁺_fw⊂^bT^∗M is the forward trapped set in Σ⁺ and Γ⁻_bw∈^bT^∗M the backward trapped set in Σ⁻.

Proof. This follows from a straightforward adaption of the proof of the last part of [74, Theorem 3.2]; again, the strict positivity of (i~)⁻¹(P_~⁰−(P_~⁰)^∗), and the fact that it has size ~⁻¹ rather than O(1) (which is the size of the main term of the commutator argument, where the main term of the commutant isτ^−2r, whose H_G-derivative has a sign near Γ^±), shifts the threshold weight (r=0 in the reference) toCfor any fixedC >0, provided~>0 is sufficiently small. This is the reason for the trapping estimate holding forall weighted spaces, unlike in the reference.

Combining these propagation results with elliptic regularity in^bS^∗M\∂Σ and using the global structure of the null-geodesic flow (see [140,§6] or [77,§2]), we can thus control umicrolocally near fiber infinity ^bS_U^∗M over any open set UbΩ, provided we have a-priori control ofuat∂Σ near the Cauchy surface Σ0and the two spacelike hypersurfaces [0,1]τ×∂Y. (That is,U is disjoint from Σ0and [0,1]τ×∂Y.)