Global estimates

We now piece together the estimates obtained in the previous sections to establish an a-priori estimate for usolving P_~u=f on decaying b-Sobolev spaces on the domain Ω defined in (3.33).

To do so, fix a weight α as in Proposition 8.21. Suppose that u∈C_c^∞(Ω;^bT_Ω^∗M)

vanishes to infinite order at Σ0, and letf=P_~u. Forδ∈Rsmall, let Ω_δ= [0,1]_τ×[r−−ε_M−δ, r++ε_M+δ]_r×S²⊂M

be a small modification of Ω, so Ω0=Ω, and Ωδ1⊆Ωδ2 ifδ16δ2. LetΩ:=Ωe ε_M/2. Denote by ˜f∈C_c^∞(eΩ;^bT^∗

ΩeM) an extension off, vanishing fort_∗60, and with kf˜k_H0,α

b (eΩ)^,−62kfk_H^0,α

b (Ω)^,−. (8.46)

We can then uniquely solve the forward problem P_~u˜= ˜f

in Ω. Ife u⁰ is any smooth compactly supported extension ofu to Ω, thene P_~(˜u−u⁰) is supported in {r>r++εM}∪{r6r−−εM}, hence by the support properties of forward solutions ofP_~, we find that ˜u−u⁰ has compact support, and moreover ˜u≡uin Ω.

Now, by the energy estimate near Σ₀, Proposition8.26, we have kuk_H1

~(Ωδ∩t⁻¹∗ ([0,1]))6C~⁻¹kfk˜ _L2(t⁻¹∗ ([0,2]))

for fixedδ∈ 0,¹₂εM

. But then we can use this information to propagateH_b,^1,α_~ regularity ofu: at fiber infinity, this uses Propositions8.9,8.11and8.12, while we can use Propo-sition8.21to obtain an estimate onunear the critical set r=r_c of ∇t∗, and propagate this control outwards in the direction of increasingrforr>r_cand decreasingrforr<r_c, by means of Proposition8.19. Away from the semiclassical characteristic set ofP_~, we simply use elliptic regularity. We obtain an estimate

kuk_H1,α

b,~(Ω−δ)+kuk_H1

~(Ωδ∩t⁻¹∗ ([0,1]))6C(~⁻¹kf˜k_H0,α

b (eΩ)+~kuk_H1,α

b,~(Ω)) (8.47) for small ~>0. The error term in u here is measured on a larger set than the con-clusion on the left-hand side, so we now use the energy estimate beyond the horizons, Proposition8.28, in order to bound

kuk_H^1,α

b,~(Ω)6C(~⁻¹kf˜k_H0,α

b (eΩ)+kuk_H^1,α

b,~(Ω_−δ)+kuk_H1

~(Ω_δ∩t⁻¹∗ ([0,1]))).

Plugging (8.47) into this estimate and choosing 0<~<~0 small, we can thus absorb the term~kuk_H1,α

b,~(Ω) from (8.47) into the left-hand side and, in view of (8.46), we obtain the desired a-priori estimate

kuk_H1,α

b,~(Ω)^,−6C~⁻¹kP_~uk_H0,α

b (Ω)^,−, 0<~<~0, (8.48)

which, by a simple approximation argument, continues to hold for allu∈H_b^1,α(Ω)^,− for whichP_~u∈H_b^0,α(Ω)^,−.

Let us fix 0<~<~0and drop the subscript ‘~’. There are a number of ways in which (8.48) can be used to rule out resonances ofP in Imσ>−α. One way is to notice that the a-priori estimate (8.48) for P yields the solvability of the adjoint P^∗ (the adjoint taken with respect to the fiber inner product b) on growing function spaces by a stan-dard application of the Hahn–Banach theorem (see, e.g. [82, Proof of Theorem 26.1.7]);

concretely, there is a bounded inverse

(P^∗)⁻¹:H_b^−1,−α(Ω;^bT_Ω^∗M)^−,−!H_b^0,−α(Ω;^bT_Ω^∗M)^−, (8.49) for the backwards problem. Now, ifσwith Imσ >−αwere a resonance ofP, then there would exist a dual resonant stateψ∈D⁰(Y;^bT_Y^∗M)withP^∗(τ^i¯σψ)=0, and in fact by the radial point arithmetic (see the proof of Proposition8.11), we haveψ∈L²(Y;^bT_Y^∗M)if our fixed~>0 is small enough. Lettingχ(x) denote a smooth cutoff,χ≡0 forx60 and χ≡1 for x>1, we put

vj:=χ(j−t∗)τ^i¯σψ and gj:=P^∗vj= [P^∗, χ(j− ·)]τ^i¯σψ.

Then,vj is the unique backwards solution ofP^∗v=gj, sovj=(P^∗)⁻¹gj; however,gj!0 inH_b^−1,−α(Ω;^bT^∗Ω)^−,asj!∞, whilekvjk_H^0,−α

b

converges to a non-zero number. This contradicts the boundedness of (8.49), and establishes Theorem8.1after reducingα>0 by an arbitrarily small positive amount.

Another, somewhat more direct way of proving Theorem 8.1 proceeds as follows:

using the same arguments as above forP_~, but in reverse, one can prove an estimate for P_~^∗ of the form

kuk_H^1,−α

b,~ (Ω)^−,6C~⁻¹kP_~^∗uk_H^0,−α

b,~ (Ω)^−,, 0<~<~0. (8.50) This relies on versions of the propagation estimates proved in§8.2and§8.3in which the direction of propagation is reversed; since the effect of passing from P_~=^CPg,~−iL_~ to P^∗

~=(^CPg,~)^∗+iL^∗

~is a change of sign in the skew-adjoint part, the adjoint version of the crucial Proposition8.21now requires that the weight satisfy%>−α. The estimate (8.50) then gives the solvability of the forward problem forP_~ on the dual spaces, which are spaces of decaying functions. Concretely, we obtain a forward solution operator

P⁻¹

~ :H_b^−1,α(Ω;^bT_Ω^∗M)^,−−!H_b^0,α(Ω;^bT_Ω^∗M)^,−, 0<~<~0. (Using elliptic regularity and propagation estimates, one also has P⁻¹

~ :H_b^s−1,α!H_b^s,α for s>0, or in fact, for any fixed s∈R provided ~>0 is sufficiently small.) Since the

forward problem for P_~ is uniquely solvable, the operator P_~ cannot have resonances with Imσ >−α, since otherwise solutions of P_~u=f for suitable (generic)f∈C_c^∞ would have an asymptotic expansion with such a resonant state appearing with a non-zero coefficient, contradicting the fact that the unique solutionulies in the spaceH_b^0,α.

This concludes the proof of Theorem 8.1.

Remark 8.29. The argument we presented above in some sense does more than what is strictly necessary; after all we only want to rule out resonances of the normal operator ofe^CPg in the closed upper half-plane, not study the solvability properties ofe^CPg , though the two are closely related. Thus, the ‘right’ framework would be to work fully on the Mellin transform side, where one would have two large parameters,σ andγ=γ1, which can be thought of as a joint parameter (σ, γ) lying in a region of C×R. (This is also related to the notion of a ‘suspended algebra’ in the sense of Mazzeo and Melrose [105].) In this case, one could use complex absorption around r=r−−εM and r=r++εM, as was done in [140], without the need for the initial or final hypersurfaces for Cauchy problems. Energy estimates still would play a minor role, as in [140,§3.3], to ensure that in a neighborhood of the black hole exterior, the resonant states are independent of the particular ‘capping’ used (complex absorption vs. Cauchy hypersurfaces). We remark that if the final Cauchy surface is still used (rather than complex absorption), the proof of the absence of resonances in the closed upper half-plane for the Mellin transformed normal operatorNb(e^CPg ) can be obtained by takingu=e^−iσt∗vin the above arguments, wherev is a function onY=Ω∩X, and only integrating inX (not inM) in the various pairings, dropping any cutoffs or weights int∗(Imσplays the role of weights, contributing to the skew-adjoint part ofN(b e^CPg ) in the arguments), with the b-Sobolev spaces thus being replaced by large-parameter versions of standard Sobolev spaces. In any case, we hope that not introducing further microlocal analysis machinery, but rather working on Ω directly, makes this section more accessible.

9. Spectral gap for the linearized gauged Einstein equation (ESG)