Boundedness of the pure gauge solution G ∧

Y-part in theD[

(1)

Y,

(1)

Z]-norm, from Proposition13.5.6 applied withn=2+εfor the

(1)

Z-part in the D[

(1)

Y,

(1)

Z]-norm, and finally from Proposition13.5.9for the remaining part in (244).

For Corollary10.2, one uses the classical Sobolev embedding onS_u,v² in conjunction with the estimates of Corollary13.3forbχ, Proposition13.3.4for

(1)

χ, Propositionb 13.5.3for

(1)η and⁽¹⁾η (recall that the`=0,1 modes vanish for

S∨ ⁰), Proposition13.5.12for all metric quantities, Proposition13.5.5for

(1)

(Ω trχ), Corollary13.10for

(1)

(Ω trχ), Proposition13.5.2 for ⁽¹⁾%, ⁽¹⁾σ,

(1)

β and

(1)

β, and Corollary 12.3 for ⁽¹⁾α and ⁽¹⁾α. The pointwise bounds for S_∨ itself follow from the identityS_∨ =S_∨ ⁰+Km,s_i, together with the fact that, as is checked by direct computation, reference Kerr solutions Km,s_i indeed satisfy the boundedness property of the corollary with right-hand side controlled by a constant depending only on the parametersmandsi.

14. Proof of Theorem4

In this final section of the paper, we turn to the proof of Theorem 4. The reader can again refer to the overview in§2.4.4.

In§14.1, we shall show that the pure gauge solution G^∧, and thus also S^∧, satisfies a uniform boundedness statement and an asymptotic flatness statement. This gives statement (1) of Theorem4. In§14.2, we obtain statement (2) of the theorem concerning integrated local energy decay. Finally, we prove the final statement (3) of the theorem concerning polynomial decay in§14.3.

14.1. Boundedness of the pure gauge solutionG^∧

LetG^∧denote the pure gauge solution in the statement of Theorem 4. The goal of this section is to prove the boundedness ofG^∧, from which a similar statement will follow for S∧=

S∨ +G^∧, in view of Theorem3applied to S∨.

We will begin in §14.1.1 below with certain preliminary estimates for

S∧ on the horizon. We shall then use these in§14.1.2to infer estimates for the functionfdefining

G∧. The precise boundedness statements that follow will be given in§14.1.3.

To distinguish between the quantities (129) associated with

S∨ or S^∧, we agree on the followingconvention: The geometric quantities of the solution

S∨ will, from now on,

be denoted with an additional [

S∨ ] next to them, while those of S^∧will appear without any additional notation, unless there is potential confusion, in which case we add [

S∧].

The general rationale is to always write an estimate for a quantity of

S∧on the left in terms of initial quantities of

S∨ on the right.

For the geometric quantities associated with the pure gauge solution G^∧, we shall always add [G^∧].

14.1.1. Decay bounds on the ingoing shear

(1)

χb at the horizon

Recall from§13.4(part of the proof of Theorem3) that the “first” obstruction to proving decay forS_∨ arose from the quantitity

(1)

χ[bS_∨ ]. We will show in this section that our choice of

G∧ensures that

(1)

χ=b

(1)

χ[b

S∧] does indeed decay along the event horizonH⁺. The estimates obtained will then allow us in the next section to infer bounds for the gauge functionf defining

G∧.

First, some preliminary remarks: We note that the pure gauge solution

G∧has van-ishing linearised shear

(1)

χ[b

G∧]=0. Therefore, in addition to the estimates on the gauge invariant quantities, also the estimates on

(1)

χb proven in §13.3 remain valid as stated for

(1)

χ=b

(1)

χ[b

S∧]. We also recall from Proposition9.3.1that Ω⁻²

(1) Finally, again by Proposition9.3.1and Lemma6.1.1, we have

(1)η[G^∧] = 0,⁽¹⁾%[G^∧] = 0 on the event horizonH⁺. (406) Hence, in particular, the gauge condition (194) holds for bothS_∨ and

S∧. Since (406) and (212) hold on the horizonH⁺, we conclude

D/^?₂⁽¹⁾η[

S∧] =−D/^?₂⁽¹⁾η[

S∧] =−D/^?₂⁽¹⁾η[S_∨ ]. (407) We now deduce the following flux bounds on the horizon.

Proposition 14.1.1. On the horizon H⁺, the geometric quantities of S^∧ in Theo-rem 4 satisfy,for i>3 and any v>v₀,

and

Proof. Restricting the angular commuted (141) to the horizon, we have, onH⁺,

Ω∇/₄ Hence, contracting with A^[i]

(1)

χ/Ω and applying the Cauchy–Schwarz inequality on theb right, in particular Taking into account (405) on the sphereS_∞,v² ₀, integration yields

We now use (407), recall

(1) holds onH⁺ from (145)) to obtain the first estimate.

For the second, we proceed similarly. Commuting (135) and restricting to the horizon H⁺yields H⁺(for bothS_∨ andS^∧). Integrating the identity (410) as in the previous case yields the second estimate after applying Proposition 13.2.3 to control the flux on the right-hand side (recall (406)).

Corollary 14.1. On the horizonH⁺,the geometric quantities of S^∧in Theorem 4

Proof. Follows directly from (408), recalling (407) and using the flux bounds of Propositions14.1.1and13.2.3.

Corollary 14.2. On the horizonH⁺,the geometric quantities of S^∧in Theorem4 satisfy in addition the L^∞_v L²(S²_∞,v)-bound, for i>2,

Proof. Revisit (408) and use theL^∞_u,v-bound of Proposition 14.1.1, the L^∞_u,v-bound onA^[i]

If we use the polynomial decay estimates of Propositions 11.5.1and12.3.4, we also have, using Proposition14.1.1 in conjunction with a pigeonhole principle, the following corollary.

Corollary 14.3. On the horizonH⁺,the geometric quantities of S^∧in Theorem 4 satisfy the decay estimate

Proof. For the bound on

(1)

χ, we combine Propositionb 14.1.1 with a simple dyadic argument. In particular, we use the fact that the fluxes appearing on the horizon on the right-hand side of (409) (after integration) satisfy the polynomial decay estimates of Proposition 13.2.4. To derive the bound for∇/₄

(1)

χ, we revisit the identity (408) withb i=2 and show that all other terms have the desired decay. The bound for two angular derivatives of

(1)

χbhas just been obtained. From Proposition12.3.5and the identity (332) restricted to the horizon, we see we have the decay bound for two angular derivatives of

(1)

χbonS_∞,v² . Finally, to estimate the term involving three derivatives of⁽¹⁾η=−⁽¹⁾η=−⁽¹⁾η[ S∨ ] in (408), we use the identity (333) and the fact that an estimate for kΨk_S2

∞,v follows directly from Proposition11.5.1and 1-dimensional Sobolev embedding.

14.1.2. Controlling the gauge function

With Proposition14.1.1controlling

(1)

χbon the horizon (from data in

S∨ ) and Corollary13.3 controlling

(1)

χ[b

S∨ ] on the horizon (also from data in

S∨ ), we can infer boundedness of the gauge function.

Proposition14.1.2. The gauge functionf=f(v, θ, φ)associated with the pure gauge solution G^∧in Theorem 4 via Proposition 9.3.1satisfies,for i=5,

Z

S²

sinθ dθ dφ|A^[i]r²D/^?₂∇f/ |².kA^[i]Ω⁻¹

(1)

χ[b S∨ ]k²_S2

∞,v0

+E0, (411) Z

S²

sinθ dθ dφ|A^[i−1]r²D/^?₂∇∂/ vf|².kA^[i−1]Ω⁻¹

(1)

χ[bS_∨ ]k²_S2

∞,v0

+E0, (412)

where we have introduced the shorthand notation D/^?₂∇f/ =D/^?₂D/^?₁(−f,0). We also have the flux bound

Z ∞

v₀

d¯v Z

S²

sinθ dθ dφ|A^[i−1]r²D/^?₂∇∂/ _vf|².kA^[i−1]Ω⁻¹

(1)

χ[b S∨ ]k²_S2

∞,v0

+E0 (413)

and the decay bound v²

Z

S²

sinθ dθ dφ|A^[2]r²D/^?₂∇∂/ vf|².kA^[3]Ω⁻¹

(1)

χ[b S∨ ]k²_S2

∞,v0

+E0. (414) Proof. We have, from Lemma6.1.1,

r·Ω⁻¹r

(1)

χ[bS^∧]−r·Ω⁻¹r

(1)

χ[bS_∨ ] =r·Ω⁻¹r

(1)

χ[bG^∧] =−2r²D/^?₂∇f,/ (415) which, when restricted to the horizon u=∞ (where r=2M), leads to (411) after us-ing Proposition 14.1.1 and Corollary 13.3. For the second estimate, we commute the defining equation (214) with A^[i−1]r²D/^?₂∇, and estimate/ f by (411) and the quan-tity Aⁱ⁻¹r /D^?₂(⁽¹⁾η+⁽¹⁾η) from Proposition 13.5.11and (the twice angular commutedS_∞,v² -estimate of) Proposition13.2.2. For the third estimate, we use again Lemma 6.1.1 to conclude that, on the horizonH⁺,

Ω∇/₄(r

(1)

χ[bS^∧]Ω⁻¹)−Ω∇/₄(r

(1)

χ[b

S∨ ]Ω⁻¹) =− 2

2Mr²D/^?₂∇∂/ vf.

The flux estimates of Corollary14.1(withi=4) and Corollary13.8produce (413). Com-bining Proposition13.4.3with Corollary14.3yields the bound (414).

14.1.3. Boundedness of the pure gauge and horizon-renormalised solution Combining the estimates of Proposition14.1.2with Lemma6.1.1, we can easily deduce the uniform boundedness of

G∧in Theorem4, as well as deduce a uniform boundedness statement for

S∧from the estimate onS_∨ and G∧.

Proposition 14.1.3. The curvature components ⁽¹⁾%,⁽¹⁾σ,

(1)

β and

(1)

β of the pure gauge solution G^∧in Theorem 4 satisfy the same boundedness estimates as these quantities for S∨ in Proposition 13.5.2,provided the term

A^[5]Ω⁻¹

is added on all right-hand sides of that proposition. Furthermore, the Ricci and metric coefficients of the solution G^∧satisfy, for all uand v,

rkr⁻¹·A^[4]r /D^?₂⁽¹⁾ηkS²_u,v+r²kr⁻¹·A^[2]r /D^?₂⁽¹⁾ηkS²_u,v+rkr⁻¹·A^[4]r /D^?₂⁽¹⁾ηkS_u,v²

Proof. Use Lemma6.1.1in conjunction with Corollary14.1.2.

We leave stating the estimate for five angular derivatives of

(1)

(Ω trχ) arising from Corollary13.10and more refined estimates for the metric coefficients to the reader. Note that the estimate for

(1)

χb is unchanged, as G^∧has

(1)

χ=0. We finally remark that, forb

(1)

b, stronger estimates hold, but Corollary14.4would not be true.

In view of S^∧=

S∨ +G^∧, we immediately conclude the following.

Corollary 14.4. The estimates of Proposition 14.1.3hold also for the solution S∧. Proof. Compare the estimates with

• Propositions 13.5.3and13.5.10for⁽¹⁾η;

• Propositions 13.5.3and13.5.11for⁽¹⁾η;

• Corollary13.3for

(1)

χ;b

• Proposition 13.5.5for

(1)

(Ω trχ);

• Corollary13.10for

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