Proof of Proposition 9.1

9.4 Proofs of the Main Estimates for the Eikonal Function Quatities

9.4.1 Proof of Proposition 9.1

With the help of the previous results, we are now ready to control the geometric quantities. Recall[u]₊:=

max{u,0}.

Proof ofυ≈er²in (9.17a). Recall equation (8.143a), we have:

L er⁻²υ

= tr

eg

/χ˜^(Small)−ΓΓΓL

er⁻²υ

. (9.124)

Integrating (9.124) along the integral curve ofL, we have:

er⁻²υ

(t,u,ω) = lim

τ↓[u]₊ er⁻²υ

(τ,u,ω) + Z t

[u]+

tr

ge/χ˜^(Small)−ΓΓΓ_L

er⁻²υ

dτ. (9.125)

Using Grownwall’s inequality, bootstrap assumptions for tr_g/χ˜^(Small)and∂∂∂~Ψ, and initial conditions forr⁻²υ,

we have:

er⁻²υ

(t,u,ω). lim

τ↓[u]+ er⁻²υ exp

tr

eg

/χ˜^(Small)−ΓΓΓL

L¹_tL^∞_ω(Cu)

.1. (9.126)

We get er⁻²υ

(t)&1 by applying similar argument to:

−er⁻²υ

(t,u,ω) = lim

τ↓[u]₊ −er⁻²υ +

Z _t [u]+

treg/χ˜^(Small)−ΓΓΓL

−er⁻²υ

dτ. (9.127)

Proof ofkb−1k_L∞(M).λ^−4ε0<¹₄in (9.17b). Recall equation (8.141). Taking the initial condition for the two separate casesu<0 andu≥0 into consideration, we have

b=1+ Z t

[u]+

b·f_(~L)·∂∂∂~Ψdτ, u≥0, (9.128a)

b=a+ Z _t

[u]+

b·f_(~L)·∂∂∂~Ψdτ, u<0, (9.128b)

whereais defined in (8.62). Using Grownwall’s inequality, bootstrap assumption for∂∂∂~Ψand initial condition (8.63a), we conclude the desired estimate.

Proof of f_(~L)

L_t^∞L^∞_uC^0,δω⁰(M).1in (9.10).

f_(~L)

_C^0,δ0

ω (St,u).1+

~Ψ _C^0,δ0

ω (St,u)+

~L _C^0,δ0

ω (St,u).1+

~L _C^0,δ0

ω (St,u). (9.129) Recalling equation (8.133a) and using (9.108), we have:

~L

Cω^0,δ0(St,u). ~L

C^0,δω⁰(S_[u]+,u)+ Z t

[u]+

f_(~L)·∂∂∂~Ψ

Cω^0,δ0(Sτ,u)dτ (9.130) .

~L

C^0,δω⁰(S_[u]+,u)+ Z t

[u]+

f_(~L)

C^0,δω⁰(Sτ,u)

∂∂∂~Ψ

C^0,δω⁰(Sτ,u)dτ.

By Grownwall’s inequality, we have:

f_(~L)

C^0,δω⁰(St,u).

1+

~L

Cω^0,δ0(S_[u]+,u)

exp

Z _t [u]+

∂∂∂~Ψ

C^0,δω⁰(Sτ,u)dτ

. (9.131)

By (7.7) and (9.104),

Z t [u]+

∂∂∂~Ψ

Cω^0,δ0(Sτ,u)dτ.λ^−7ε0. (9.132) Combining (9.101), (9.100c), (9.96) and (9.98), we have:

~L

Cω^0,δ0(S_[u]+,u).1+

3 i=1

∑

A=1,2max

Θⁱ_(A)(t,t,ω)

L^∞ω(S_[u]+,u).1+

∑

α

kL^αk

L^∞_uC^0,δω⁰(Σ₀).1. (9.133) Combining (9.131), (9.132) and (9.133) we conclude the desired estimate.

Proof ofkχkˆ _L2

tL^pω(Cu).λ^−1/2in (9.7a). Recalling equation (8.139b) and using the transport identity (9.76), we have:

er²χˆ

_g/(t,u,ω)≤ lim

τ↓[u]+

er²χˆ

_g/(τ,u,ω) (9.134)

+ Z t

[u]+

λ⁻¹

er²(C~,D) g/+

er²(∇/ ,/D_L)ξ _g/+

er²f_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ g/dτ.

Dividing (9.134) byer²(t,u)and taking the norm, we have:

kχkˆ _L2

tLω^p(Cu)≤ er⁻²

lim

τ↓[u]₊er²χˆ

L²_tLω^p(Cu)

(9.135) +

λ⁻¹er⁻² Z t

[u]+

er²(C~,D) g/dτ

L²_tL_ω^p(Cu)

+ er⁻²

Z t [u]+

er²(∇/ ,D/_L)ξ g/dτ

L_t²L_ω^p(Cu)

+ er⁻²

Z _t [u]₊

er²f_(~L)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ _g/dτ

L²_tLω^p(Cu)

.

We now consider the initial conditions. When u ≥0, by the initial condition on the cone-tip (8.64a), lim

τ↓[u]+

er²χˆ

_g/=0. Whenu<0, lim

τ↓[u]+

er²χˆ

_g/=w²χ(0,ˆ u,ω). Then, by (8.63b) and (9.82a), we have:

w^1/2χ(0,ˆ u,ω) _Lp

ω(Sw).

w^1/2θ(0,u,ω) _Lp

ω(Sw)+

w^1/2∂∂∂~Ψ(0,u,ω) _Lp

ω(Sw).λ^−1/2. (9.136) Then,

er⁻²w²χ(0,u,ˆ ω) _L2

tL_ω^p(Cu). _{Z T}

∗;(λ)

0

er⁻²w^3/2λ^−1/2 2

dτ 1/2

(9.137)

= (

λ⁻¹w³ − 1 (τ+w)³

T∗;(λ)

0

!)1/2

.λ^−1/2.

Now we estimate other terms in (9.135). By (9.89b),

λ⁻¹er⁻² Z t

[u]+

er²(C~,D) g/dτ

L²_tLω^p(Cu)

.λ^{−1/2−12ε0}. (9.138)

Using the estimate for Hardy-Littlewood maximal function (9.28) and estimate (9.82f), we have

er⁻²

Z _t [u]+

er²(∇/ ,/D_L)ξ g/dτ

L²_tL^p_ω(Cu)

.

1 t−[u]+

Z _t [u]+

|er(∇/ ,/D_L)ξ|_g/dτ L²_tL_ω^p(Cu)

(9.139) .ker(∇/ ,/D_L)ξk_L2

tLω^p(Cu).λ^−1/2. (9.140) Similarly, by the estimate for Hardy-Littlewood maximal function (9.28), bootstrap assumptions (6.13), (9.4), and (9.5a), we have:

er⁻²

Z t [u]+

er²f_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ g/dτ

L_t²Lω^p(Cu)

(9.141) .

erf_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ L²_tLω^p(Cu)

. er

tr

ge/χ˜^(Small),χ,ˆ er⁻¹ L^∞_tLω^p(Cu)

∂∂∂~Ψ

L_t²L^∞_ω(Cu)+λ^1/2−4ε0 er^1/2∂∂∂~Ψ

L^∞_tL^pω(Cu)

∂∂∂~Ψ

L²_tL^∞_ω(Cu)

.λ^{−1/2−4ε0}.

Combining (9.134)-(9.141) and we conclude the desired estimate.

Proof of er^1/2χˆ

L^∞_tL^pω(Cu).λ^−1/2in (9.7b). Dividing the equation (9.134) byer^3/2(t,u)and taking the norm, we have:

er^1/2χˆ

_L∞

tL_ω^p(Cu)≤ er^−3/2

lim

τ↓[u]₊er²χˆ

L_t^∞Lω^p(Cu)

(9.142) +

λ⁻¹er^−3/2 Z _t

[u]+

er²(C~,D) g/dτ

L^∞_tL^p_ω(Cu)

+ er^−3/2

Z _t [u]₊

er²(∇/ ,/D_L)ξ g/dτ

L_t^∞Lω^p(Cu)

+ er^−3/2

Z t [u]+

er²f_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ g/dτ

L^∞_tLω^p(Cu)

.

We control the initial condition as in the previous proof. In particular, using (9.136) foru<0, we have:

er^−3/2

lim

τ↓[u]₊er²χˆ

L^∞_tLω^p(Cu)

.λ^−1/2. (9.143)

By (9.89e),

λ⁻¹er^−3/2 Z t

[u]+

er²(C~,D) g/dτ

L^∞_tLω^p(Cu)

.λ^{−1/2−12ε0}. (9.144)

By Minkowski’s integral inequality and H¨older’s inequality, also using (9.82f), we have:

er^−3/2

Z _t [u]+

er²(∇/ ,/D_L)ξ g/dτ

L^∞_tL_ω^p(Cu)

≤sup

t

er^−1/2(t,u) Z _t

[u]+

|er(∇/ ,/D_L)ξ|_g/dτ L_ω^p(St,u)

(9.145) .sup

t

(t−[u]₊)^1/2

er^1/2(t,u) ker(∇/ ,/D_L)ξk_L2 tL_ω^p(Cu)

.λ^−1/2.

By the same argument as above and also using (9.141), we have

er^−3/2

Z t [u]+

er²f_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ g/dτ

L^∞_tL^pω(Cu)

(9.146) .sup

t

(t−[u]+)^1/2 er^1/2(t,u)

er

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ L²_tL^pω(Cu)

.λ^{−1/2−4ε0}.

Combining (9.142)-(9.146) and we conclude the desired estimate.

Proof ofkerD/_Lχkˆ _L2

tLω^p(Cu).λ^−1/2in (9.7a). Consider equation (8.139b):

erD/_Lχˆ= ertr

ge/χ˜^(Small)+1 ˆ

χ+er(∇/ ,/D_L)ξ+λ⁻¹erf_(~L)·(C~,D) (9.147) +erf_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹ ·∂∂∂~Ψ.

By (9.4) and (9.7b), we have:

ertr

ge

/χ˜^(Small)·χˆ

L_t²Lω^p(Cu).λ^1/2−4ε0 tr

eg /χ˜^(Small)

L²_tL^∞_ω(Cu)

er^1/2χˆ

L^∞_tLω^p(Cu).λ^{−1/2−2ε0}. (9.148) By (9.82e), (9.82f) and (9.141), we have:

ker(∇/ ,/D_L)ξk_L2

tLω^p(Cu),

λ⁻¹erf_(~L)·(C~,D)

L²_tLω^p(Cu), (9.149)

erf_(~L)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L_t²Lω^p(Cu).λ^−1/2.

Proof ofkζk_L2

tL^p_ω(Cu).λ^−1/2in (9.7a). Considering equation (8.140) and using (9.76), we have:

|erζ|_g/≤ lim

τ↓[u]+

|erζ|_g/(τ,u,ω) (9.150)

+ Z t

[u]+

λ⁻¹ er(C~,D)

g/+|er(∇/ ,/D_L)ξ|_g/ +

erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ _g/+

erf_(~L)·ζ·χˆ _g/dτ.

Dividing (9.150) byer(t,u)and taking theL_t²L_ω^p norm, we have:

kζk_L2

tL_ω^p(Cu)≤ er⁻¹

lim

τ↓[u]+erζ

L²_tL_ω^p(Cu)

(9.151) +

λ⁻¹er⁻¹ Z t

[u]+

er(C~,D)

g/dτ

L²_tLω^p(Cu)

+ er⁻¹

Z t [u]+

|er(∇/ ,/D_L)ξ|_g/dτ L²_tLω^p(Cu)

+ er⁻¹

Z t [u]+

erf_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),ζ,er⁻¹

·∂∂∂~Ψ g/dτ

L²_tL^pω(Cu)

+ er⁻¹

Z _t [u]+

|erζ·χ|ˆ _g/dτ L_t²Lω^p(Cu)

.

We now consider the initial condition. Whenu≥0, by the initial condition on the cone-tip (8.64a),er⁻¹

lim

τ↓[u]₊erζ

= 0. Whenu<0, we use (8.17), (8.63b), and bootstrap assumption for∂∂∂~Ψto deduce:

er⁻¹

lim

τ↓[u]+erζ

L²_tL^p_ω(Cu)

.

er⁻¹w∇/lna

L²_tL^pω(Cu)+

er⁻¹w∂∂∂~Ψ

L²_tLω^p(Cu) (9.152) .λ^−1/2

Z _T∗;(λ) 0

w

(τ+w)²dτ+λ^{−1/2−4ε0} .λ^−1/2.

By (9.89a),

λ⁻¹er⁻¹ Z t

[u]+

er(C~,D) g/dτ

L²_tL^pω(Cu)

.λ^{−1/2−12ε0}. (9.153)

Using the same method as in (9.139) and (9.141), we have :

er⁻¹

Z _t [u]+

|er(∇/ ,/D_L)ξ|_g/dτ L²_tLω^p(Cu)

.λ^−1/2, (9.154)

er⁻¹

Z t [u]+

erf_(~L)·

∂∂

∂~Ψ,tr

ge/χ˜^(Small),ζ,er⁻¹

·∂∂∂~Ψ g/dτ

L²_tLω^p(Cu)

.λ^{−1/2−4ε0}. (9.155)

Using the estimate for Hardy-Littlewood maximal function (9.28), bootstrap assumption (9.6), and the esti-

mate previously proven forer^1/2χˆin (9.7b), we have:

er⁻¹

Z t [u]+

|erζ·χ|ˆ _g/dτ L²_tLω^p(Cu)

.kerζ·χkˆ _L2

tL_ω^p(Cu) (9.156)

.λ^1/2−4ε0 er^1/2χˆ

L_t^∞Lω^p(Cu)kζk_L2

tL^∞_ω(Cu).λ^{−1/2−4ε0}.

Combining (9.150)-(9.156), we conclude the desired estimate.

Proof of er^1/2ζ

L^∞_t L^pω(Cu).λ^−1/2in (9.7b). Dividing the equation (9.150) byer^1/2(t,u)and taking theL^∞_t Lω^p

norm, we have:

er^1/2ζ

L^∞_tLω^p(Cu)≤ er^−1/2

lim

τ↓[u]₊erζ

L^∞_t L^pω(Cu)

(9.157) +

λ⁻¹er^−1/2 Z _t

[u]+

er(C~,D) _g/dτ

L^∞_tLω^p(Cu)

+ er^−1/2

Z _t [u]+

|er(∇/ ,/D_L)ξ|_g/dτ

L_t^∞Lω^p(Cu)

+ er^−1/2

Z t [u]+

erf_(~L)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ g/dτ

L_t^∞Lω^p(Cu)

+ er^−1/2

Z t [u]+

|erζ·χ|ˆ _g/dτ L^∞_tL_ω^p(Cu)

.

Let’s consider the initial condition. Whenu≥0, by the initial condition on the cone-tip (8.64a),er^−1/2

τ↓[u]lim₊erζ

= 0. Whenu<0, we use (8.17), (8.63b), and the estimate (9.82a) to deduce:

er^−1/2

lim

τ↓[u]₊erζ

L_t^∞Lω^p(Cu)

.

w^1/2/∇lna _L∞

t L_ω^p(Cu)+ er^1/2∂∂∂~Ψ

_L2

tL_ω^p(Cu).λ^−1/2. (9.158) By (9.89d), we have:

λ⁻¹er^−1/2 Z t

[u]+

er(C~,D)

g/dτ

L^∞_tLω^p(Cu)

.λ^{−1/2−12ε0}. (9.159)

Using the same method as in (9.145) and (9.146), we have :

er^−1/2

Z t [u]+

|er(∇/ ,/D_L)ξ|_g/dτ

L_t^∞L_ω^p(Cu)

.λ^−1/2, (9.160)

er^−1/2

Z _t [u]₊

erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),ζ,er⁻¹

·∂∂∂~Ψ _g/dτ

L_t^∞Lω^p(Cu)

.λ^{−1/2−4ε0}. (9.161)

With the help of (9.156), we have:

er^−1/2

Z t [u]+

|erζ·χ|ˆ _g/dτ

L^∞_tLω^p(Cu)

.sup

t er^−1/2 Z t

[u]+

kerζ·χkˆ _L^p

ω(Sτ,u)dτ (9.162) .sup

t

(t−[u]₊)^1/2

(t−u)^1/2 kerζ·χkˆ _L2 tL^pω(Cu)

.λ^{−1/2−4ε0}.

Combining (9.157)-(9.162) and we conclude the desired estimate.

Proof ofkerD/_Lζk_L2

tL_ω^p(Cu).λ^−1/2in (9.7a). Consider equation (8.140):

erD/_Lζ=ζ+er(∇/ ,/D_L)ξ+λ⁻¹erf_(~L)·(C~,D) +erf_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ (9.163) +erf_(~L)·ζ·

tr

ge/χ˜^(Small),χ,∂ˆ ∂∂~Ψ

.

By (6.13), (9.4) and (9.7b), we have:

er

tr

eg

/χ˜^(Small),χ,ˆ ∂∂∂~Ψ ·ζ

L_t²Lω^p(Cu).λ^1/2−4ε0 tr

eg

/χ˜^(Small),χ,ˆ ∂∂∂~Ψ L²_tL^∞_ω(Cu)

er^1/2ζ

L_t^∞Lω^p(Cu) (9.164) .λ^{−1/2−2ε0}.

By (9.82e), (9.82f) and (9.141), we have:

ker(∇/ ,/D_L)ξk_L2

tL_ω^p(Cu),

λ⁻¹erf_(~L)·(C~,D)

L²_tLω^p(Cu), (9.165)

erf_(~L)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L_t²Lω^p(Cu).λ^−1/2.

Combining (9.163)-(9.165) with (9.7a) forkζk_L2

tLω^p(Cu), we conclude the desired estimates.

Proof of

er^1/2(χ,ζ)ˆ

L²ω^pL^∞_t(Cu).λ^−1/2whenCu⊂M^(Int)in (9.14). By the Sobolev inequality (9.32), the previously proven estimate (9.7a), and the bootstrap assumptions (9.6), we have:

er¹²(ˆχ,ζ)

2

L^2pωL_t^∞(Cu).

kerD/_L(ˆχ,ζ)k_L^p

ωL²_t(Cu)+k(χ,ˆ ζ)k_L^p

ωL²_t(Cu)

k(χ,ˆ ζ)k_L∞

ωL²_t(Cu).λ⁻¹. (9.166) In deriving (9.166), we also used Minkowski’s inequality for integrals to switch the order ofL_t,L_ω norms.

Proof ofertr

eg

/χ˜≈1in (9.8a). It is sufficient to show ertr

eg /χ˜^(Small)

g/.λ^−4ε0. We plug (8.138a) into (9.77),

whereG=tr

ge/χ˜^(Small)+f_(~L)·∂∂∂~Ψ. Then dividing the estimate (9.79) byer(t,u), we have:

ertr

ge/χ˜^(Small)

g/.er⁻¹ lim

τ↓[u]+

er²tr

ge/χ˜^(Small)

g/ (9.167)

+er⁻¹ Z t

[u]+

er²λ⁻¹f_(~L)·(C~,D) +er²f_(~L)· ˆ

χ,er⁻¹,∂∂∂~Ψ

·∂∂∂~Ψ+er²|χ|ˆ ²_g/ g/dτ.

We now consider the initial conditions. When u ≥0, by the initial condition on the cone-tip (8.64a), er⁻¹ lim

τ↓[u]₊

er²tr

eg /χ˜^(Small)

g/≤ lim

τ↓[u]₊

ertr

ge /χ˜^(Small)

g/=0. Whenu<0, by (8.61) and (8.63a), we have:

er⁻¹ lim

τ↓[u]₊

er²tr

eg /χ˜^(Small)

_g/=1

er

2w(1−a)

a ≤2(1−a)

a .λ^−4ε0. (9.168)

By (9.89f), we have:

er⁻¹ Z t

[u]+

er²λ⁻¹f_(~L)·(C~,D)dτ.λ^−16ε0. (9.169) By bootstrap assumptions (6.13) and (9.4), we have:

er⁻¹ Z _t

[u]+

er²f_(~L)·(χ,∂ˆ ∂∂~Ψ)·(χ,∂ˆ ∂∂~Ψ)

g/dτ.λ^1−8ε0 χ,∂ˆ ∂∂~Ψ

L²_tL^∞ω(Cu)

χ,∂ˆ ∂∂~Ψ

L²_tL^∞ω(Cu).λ^−4ε0. (9.170)

er⁻¹ Z _t

[u]+

erf_(~L)·∂∂∂~Ψ _g/dτ.

∂∂∂~Ψ

_L2

tL^∞_ω(Cu)λ^1/2−4ε0 .λ^−8ε0. (9.171) Combining (9.167)-(9.171) and we conclude the desired estimate.

ertr

ge

/χ˜^(Small)

L_t^∞Lω^p(Cu).λ^−4ε0 follows from the proof ofertr

ge/χ˜≈1.

Proof of er^1/2tr

ge/χ˜^(Small)

L^∞(M).λ^−1/2in (9.8b). Dividing (9.167) byer^1/2, we have

er^1/2tr

ge/χ˜^(Small)

_g/.er^−3/2 lim

τ↓[u]₊

er²tr

eg /χ˜^(Small)

_g/ (9.172)

+er^−3/2 Z _t

[u]₊

er²λ⁻¹f_(~L)·(C~,D) +er²f_(~L)· ˆ

χ,er⁻¹,∂∂∂~Ψ

·∂∂∂~Ψ+er²|χ|ˆ ²_{g/ g/}dτ.

We consider the initial conditions. Whenu≥0, by the initial condition on the cone-tip (8.64a), we have:

er^−3/2 lim

τ↓[u]+

er²tr

ge/χ˜^(Small)

g/≤er^−1/2 lim

τ↓[u]+

ertr

eg /χ˜^(Small)

g/. lim

τ↓[u]+er^1/2=0. (9.173)

Whenu<0, by (8.61) and (8.63b), we have:

er^−3/2 lim

τ↓[u]+

er²tr

ge/χ˜^(Small) g/= 1

er^3/2

2w(1−a)

a ≤w^−1/2(1−a).λ^−1/2. (9.174) By (9.89g), we have:

λ⁻¹ er⁻³²

Z _t [u]+

er²(C~,D) g/dτ

L^∞(M).λ^{−1/2−12ε0}. (9.175) By bootstrap assumptions (6.13) and (9.4), we have:

er^−3/2 Z t

[u]+

er²f_(~L)·(ˆχ,∂∂∂~Ψ)·(χ,∂ˆ ∂∂~Ψ)

g/dτ.λ^1/2−4ε0 χ,∂ˆ ∂∂~Ψ

L²_tL^∞_ω(Cu)

χ,∂ˆ ∂∂~Ψ

L_t²L^∞_ω(Cu) (9.176) .λ^{−1/2−4ε0}.

er^−3/2 Z t

[u]+

erf_(~L)·∂∂∂~Ψ

g/dτ.er^−1/2 Z t

[u]+

f_(~L)·∂∂∂~Ψ

L^∞_x(Σ_t)dτ.(t−[u]₊)^1/2 (t−u)^1/2

∂∂∂~Ψ

L_t²L^∞_x(M) (9.177) .λ^{−1/2−4ε0}.

Combining (9.172)-(9.177) and we conclude the desired estimate.

er^1/2tr

ge/χ˜^(Small)

L^∞_tLω^p(Cu) .λ^−1/2 and er^1/2tr

ge/χ˜^(Small)

L^2pωL_t^∞(Cu).λ^−1/2 follows from the previously proven estimate

er^1/2tr

ge/χ˜^(Small)

L^∞(M).λ^−1/2. Proof of

treg/χ˜^(Small)

L²_tC^0,δω⁰(Cu).λ^−1/2in (9.8e). Using equation (8.138a) and dividingL(er²tr

ge

/χ˜^(Small))by er²(t,u), we have:

tr

eg /χ˜^(Small)

_g/≤er⁻² lim

τ↓[u]₊

er²tr

ge /χ˜^(Small)

_g/+er⁻²

Z _t [u]+

er²λ⁻¹f_(~L)·(C~,D)dτ

_g/ (9.178)

+er⁻² Z _t

[u]₊

er²f_(~L)·

tr

ge/χ˜^(Small),χ,ˆ er⁻¹,∂∂∂~Ψ ·∂∂∂~Ψ

+er²|χ|ˆ ²_g/+er²tr

ge/χ˜^(Small)·tr

ge /χ˜^(Small)

g/dτ.

Using (9.108), we have:

tr

eg

/χ˜^(Small) _C^0,δ0

ω (St,u). er⁻² lim

τ↓[u]₊

er²tr

ge/χ˜^(Small) Cω^0,δ0(Cu)

+λ⁻¹

C~,D _L1

tC_ω^0,δ0(Cu) (9.179) +

tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ

2

L²_tC^0,δ_ω⁰(Cu)+M ∂∂∂~Ψ

_C^0,δ0

ω (St,u)

.

We now consider the initial condition. Whenu≥0, by (8.64a), lim

τ↓[u]+ er²tr

eg /χ˜^(Small)

=0, we have:

er²tr

ge/χ˜^(Small)≤ Z _t

[u]+

er²λ⁻¹f_(~L)·(C~,D)

_g/dτ (9.180)

+ Z t

[u]+

er²f_(~L)· tr

eg

/χ˜^(Small),χ,ˆ er⁻¹,∂∂∂~Ψ ·∂∂∂~Ψ

+er²|χ|ˆ ²_g/+er²tr

ge/χ˜^(Small)·tr

ge /χ˜^(Small)

_g/dτ.

Whenu<0, by (8.61), we have:

er⁻² lim

τ↓[u]+

er²tr

ge/χ˜^(Small)

2

L²_tC^0,δ_ω⁰(Cu)

. Z T_∗;(λ)

0

w^3/2 er² λ^−1/2

!2

dτ≤λ⁻¹. (9.181)

Using the estimate for Hardy-Littlewood maximal function (9.28) and estimate (9.104), we have:

M

∂∂∂~Ψ

Cω^0,δ0(St,u)

L²_t

. ∂∂∂~Ψ

L²_tCω^0,δ0(Cu).λ^{−1/2−3ε0}. (9.182) By bootstrap assumptions (6.13), (9.4), and estimate (9.104), we have:

λ⁻¹

C~,D _L1

tC_ω^0,δ0(Cu).λ^−1−7ε0, (9.183)

tr

eg

/χ˜^(Small),∂∂∂~Ψ,χˆ

2

L_t²C_ω^0,δ0(Cu).λ^−1+4ε0. (9.184) Taking theL²_t norm of (9.179), combining (9.180)-(9.184), we conclude the desired result.

tr

eg

/χ˜^(Small)

L²_tLω^p(Cu).λ^−1/2follows directly from tr

ge/χ˜^(Small)

L²_tC^0,δω⁰(Cu).λ^−1/2. Proof of

erD/_Ltr

ge/χ˜^(Small)

L²_tL^pω(Cu).λ^−1/2in (9.7a). Using equation (8.138a), we have:

erD/_Ltr

eg

/χ˜^(Small)=tr

ge

/χ˜^(Small)+erλ⁻¹f_(~L)·(C~,D) +erf_(~L)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ (9.185) +er|χ|ˆ²_g/+ertr

ge

/χ˜^(Small)·tr

eg /χ˜^(Small). By (9.93a), we have:

λ⁻¹ er(C~,D)

L²_tL^pω(Cu).^{−1/2−8ε0}. (9.186)

Using the same method as in (9.141), we have:

erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L²_tLω^p(Cu).λ^{−1/2−4ε0}. (9.187) By bootstrap assumtions (9.4) and the previously proven results (9.7b), we have:

er

trge/χ˜^(Small),χ,ˆ ζ

·

treg/χ˜^(Small),χ,ˆ ζ

L²_tL^pω(Cu) (9.188)

.λ^1/2−4ε0 tr

eg

/χ˜^(Small),χ,ˆ ζ

L²_tC^0,δω⁰(Cu)

er^1/2

ˆ χ,tr

ge/χ˜^(Small),ζ L^∞_tLω^p(Cu)

.λ^{−1/2−2ε0}.

Combining (9.185)-(9.188) and we conclude the desired estimate.

Remark 9.14. From (9.7a), (9.141), (9.161), (9.188) and (9.164), it follows:

erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ζ,ˆ er⁻¹

·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ ζ

L²_tLω^p(Cu).λ^−1/2. (9.189) Proof of

er /∇tr

eg

/χ˜^(Small),/∇χˆ

L²_tL^pω(Cu).λ^−1/2in (9.8d). First, we bound er∇/tr

ge/χ˜^(Small)

L²_tLω^p(Cu). Plugging equation (8.138b) into (9.77), whereG=f_(~L)·

∂

∂

∂~Ψ,tr

eg

/χ˜^(Small),χˆ

, then dividing byer²(t,u), we have:

er∇/tr

ge/χ˜^(Small)

g/.er⁻² lim

τ↓[u]+

er³/∇tr

eg /χ˜^(Small)

g/+λ⁻¹er⁻² Z t

[u]+

er³f_(~L)·/∇(C~,D)

g/dτ (9.190)

+λ⁻¹er⁻² Z t

[u]+

er³f_(~L)·(C~,D)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹ g/dτ +er⁻²

Z _t [u]₊

er³f_(~L)·/∂∇∂∂~Ψ·

∂∂

∂~Ψ,tr

ge/χ˜^(Small),er⁻¹

+er³f_(~L)·/∇χˆ·χˆ _g/dτ +er⁻²

Z t [u]+

er³f_(~L)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),er⁻¹

·∂∂∂~Ψ g/dτ.

Let’s consider the initial conditions. Whenu≥0, we use the estimate for Hardy-Littlewood maximal function (9.28) and (8.64a) to deduce

er⁻² lim

τ↓[u]₊(er³/∇tr

eg /χ˜^(Small))

L²_tLω^p(Cu)

=0. Whenu<0, we use the initial condition (8.63g) to deduce:

er⁻² lim

τ↓[u]+

(er³/∇tr

ge /χ˜^(Small))

L²_tL^p_ω(Cu)

.λ^−1/2 er⁻²w^3/2

L_t²Lω^p(Cu).λ^−1/2. (9.191)

By (9.90b), we have:

λ⁻¹ er⁻²

Z t [u]+

er³f_(~L)·/∇(C~,D) g/dτ

L_t²Lω^p(Cu)

.λ^{−1/2−8ε0}. (9.192)

By (9.91b), we have:

λ⁻¹ er⁻²

Z _t [u]+

er³f_(~L)·(C~,D)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

g/dτ

L²_tL^p_ω(Cu)

.λ^{−1/2−12ε0}. (9.193)

By the estimate for Hardy-Littlewood maximal function (9.28), bootstrap assumptions (6.13), (9.4), and estimate (9.82f), we have:

er⁻²

Z t [u]+

er³f_(~L)·/∇∂∂∂~Ψ·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),er⁻¹ dτ

L²_tLω^p(Cu)

(9.194) .

Z t [u]+

er∇/∂∂∂~Ψ

Lω^p(S_τ,u)

∂∂∂~Ψ,tr

ge

/χ˜^(Small)

L^∞_ω(S_τ,u)dτ L²_t

+ er⁻¹

Z t [u]+

er∇/∂∂∂~Ψ

Lω^p(S_τ,u)dτ L²_t

.λ^1/2−4ε0 er∇/∂∂∂~Ψ

L_t²Lω^p(Cu)

∂∂∂~Ψ,tr

eg

/χ˜^(Small)

L_t²L^∞_ω(Cu)+ er∇/∂∂∂~Ψ

L_t²Lω^p(Cu)

.λ^−1/2.

By the estimate for Hardy-Littlewood maximal function (9.28), bootstrap assumptions (6.13) , (9.4), (9.5a), and estimate (9.82e), we have:

er⁻²

Z _t

[u]+er³f_(~L)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),er⁻¹

·∂∂∂~Ψdτ L_t²Lω^p(Cu)

(9.195) .

Z t [u]+

er

tr

ge/χ˜^(Small),χ,ˆ er⁻¹ Lω^p(Sτ,u),

er∂∂∂~Ψ

Lω^p(Sτ,u)

∂∂∂~Ψ,tr

eg /χ˜^(Small)

L^∞_ω(Sτ,u)

∂∂∂~Ψ

L^∞_ω(Sτ,u)dτ _L2

t

+ er⁻¹

Z _t [u]+

∂∂∂~Ψ

_L∞

ω(Sτ,u)dτ L_t²

.λ^1/2−4ε0 ∂∂∂~Ψ,tr

eg

/χ˜^(Small) L²_tL^∞_ω(Cu)

∂∂∂~Ψ

L²_tL^∞_ω(Cu)+ ∂∂∂~Ψ

L²_tL^∞_ω(Cu)

.λ^{−1/2−4ε0}.

By bootstrap assumption (9.4), we have:

er⁻²

Z t [u]+

er³f_(~L)·/∇χˆ·χdτˆ L²L^pω(Cu)

. ker∇/χkˆ _L2

tL^p_ω(Cu)kχkˆ _L2 tL^∞_ω(Cu)

L_t² (9.196)

Combining (9.190)-(9.196) and we have:

er∇/tr

eg /χ˜^(Small)

L²_tL^pω(Cu).λ^−1/2+λ^−2ε0ker∇/χkˆ _L2

tL^p_ω(Cu). (9.197) Now we considerker∇/χkˆ _L2

tLω^p(Cu). Plugging equation (8.139a) into the Hodge estimate (9.111), we have:

ker∇/χkˆ _L2

tL^pω(Cu). er∇/tr

eg

/χ˜^(Small)

L²_tLω^p(Cu) (9.198)

+ er∇/∂∂∂~Ψ

L²_tLω^p(Cu)+ erf_(~L)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L²_tLω^p(Cu).

By (9.82f) and (9.189), we have:

ker∇/χkˆ _L2

tL^pω(Cu).λ^−1/2+ er∇/tr

ge/χ˜^(Small) _L2

tL_ω^p(Cu). (9.199) Combining (9.197) and (9.199), we have the desired estimates

er /∇tr

eg

/χ˜^(Small),/∇χˆ

L²_tLω^p(Cu).λ^−1/2. Proof of

er^3/2/∇tr

eg /χ˜^(Small)

L_t^∞L^∞_uLω^p(M).λ^−1/2in (9.8c). Plugging equation (8.138b) into (9.77), whereG= f_(~L)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χˆ

, then dividing byer^3/2(t,u), we have:

er∇/tr

ge/χ˜^(Small)

g/ (9.200)

.er^−3/2 lim

τ↓[u]₊

er³/∇tr

eg /χ˜^(Small)

g/+λ⁻¹er^−3/2 Z t

[u]+

er³f_(~L)·/∇(C~,D) g/dτ +λ⁻¹er^−3/2

Z t [u]+

er³f_(~L)·(C~,D)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹ g/dτ +er^−3/2

Z t [u]+

er³f_(~L)·/∇∂∂∂~Ψ·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),er⁻¹

+er³f_(~L)·/∇χ·ˆ χˆ g/dτ +er^−3/2

Z _t [u]+

er³f_(~L)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹ ·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),er⁻¹ ·∂∂∂~Ψ

g/dτ.

We now consider the initial conditions. Whenu≥0, we use (8.64a) to deduceer^−3/2 lim

τ↓[u]₊ er³/∇tr

eg /χ˜^(Small)

=0.

Whenu<0, we use the initial condition (8.63g) to deduce

er^−3/2 lim

τ↓[u]₊

er³/∇tr

eg /χ˜^(Small)

g/

L^∞_uLω^p(Σ0)

.λ^−1/2. By (9.90a), we have:

λ⁻¹er^−3/2 Z _t

[u]₊

er³f_(~L)·/∇(C~,D) _g/dτ

L^∞_tL^∞_uL^pω(M)

.λ^{−1/2−8ε0}. (9.201)

By (9.91a), we have:

λ⁻¹er^−3/2 Z t

[u]+

er³f_(~L)·(C~,D)·

∂∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹ g/dτ

L_t^∞L^∞_uLω^p(M)

.λ^{−1/2−12ε0}. (9.202)

By bootstrap assumptions (6.13) and (9.4), estimate (9.82f), we have:

er^−3/2

Z _t [u]+

er³f_(~L)·/∇∂∂∂~Ψ·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),er⁻¹

g/dτ

L^p_ω(St,u)

(9.203) .λ^1/2−4ε0

er∇/∂∂∂~Ψ

L²_tLω^p(S_t,u)

∂∂∂~Ψ,tr

eg /χ˜^(Small)

L²_tL^∞_ω(S_t,u)+(t−[u]₊)^1/2 (t−u)^1/2

er∇/∂∂∂~Ψ

L_t²Lω^p(Cu)

.λ^−1/2.

By bootstrap assumptions (6.13), (9.4) and (9.82a), we have:

er^−3/2

Z t [u]+

er³f_(~L)·

∂∂

∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),er⁻¹

·∂∂∂~Ψ g/dτ

Lω^p(S_t,u)

(9.204) .λ^1/2−4ε0

Z _t [u]+

er

tr

ge

/χ˜^(Small),χ,ˆ er⁻¹ _Lp

ω(Sτ,u)

∂∂∂~Ψ,tr

eg

/χ˜^(Small) _L∞

ω(Sτ,u)

∂∂∂~Ψ

_L∞

ω(Sτ,u)dτ +λ^1−8ε0

Z t [u]+

er^1/2∂∂∂~Ψ

Lω^p(S_τ,u)

∂∂∂~Ψ,tr

ge /χ˜^(Small)

L^∞_ω(S_τ,u)

∂∂∂~Ψ

L^∞_ω(S_τ,u)dτ +er^−1/2

Z t [u]+

∂∂∂~Ψ

L^∞_ω(Sτ,u)dτ .λ^1/2−4ε0

∂∂∂~Ψ,tr

ge

/χ˜^(Small) _L2

tL^∞_ω(Cu)

∂∂∂~Ψ

_L2

tL^∞_ω(Cu)+(t−[u]₊)^1/2 (t−u)^1/2

∂∂∂~Ψ

_L2

tL^∞_ω(Cu)

.λ^{−1/2−4ε0}.

By bootstrap assumptions (6.13) and the previously proven result (9.8d), we have:

er^−3/2

Z t [u]+

er³f_(~L)·/∇χˆ·χˆ g/dτ

L^p_ω(Sτ,u)

.λ^1/2−4ε0ker∇/χkˆ _L2

tL^pω(Cu)kχkˆ _L2

tL^∞_ω(Cu).λ^{−1/2−2ε0}. (9.205)

Proof of tr

eg

/χ˜^(Small),tr_g/χ−²

er,χˆ

L²_tL^∞_uC_ω^0,δ0(M^(Int)).λ^{−1/2−3ε0} in (9.15). We first bound tr

eg

/χ˜^(Small). Using equa-

tion (8.138a) and initial condition (8.64a), and dividingL(er²tr

ge/χ˜^(Small))byer²(t,u), we have

tr

eg /χ˜^(Small)

g/.er⁻²

Z t [u]+

er²λ⁻¹f_(~L)·(C~,D)

g/dτ (9.206)

+er⁻² Z _t

[u]+

er²f_(~L)·

tr

ge

/χ˜^(Small),χ,ˆ er⁻¹,∂∂∂~Ψ ·∂∂∂~Ψ

+er²|χ|ˆ ²_g/+er²tr

eg

/χ˜^(Small)·tr

ge/χ˜^(Small) g/dτ.

Using (9.108), we have:

tr

eg

/χ˜^(Small) _L∞

uC^0,δ_ω⁰(Σ_t^(Int)).λ⁻¹

C~,D _L1

tL^∞_uC^0,δ_ω⁰(M^(Int)) (9.207)

+ tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ

2

L²_tL^∞_uCω^0,δ0(M^(Int))+M ∂∂∂~Ψ

L^∞_uC^0,δω⁰(Σ^(Int)_t )

.

Using the estimate for Hardy-Littlewood maximal function (9.28) and estimate (9.104), we have:

M

∂∂∂~Ψ

L^∞_uCω^0,δ0(Σ^(Int)_t )

_L2

t

. ∂∂∂~Ψ

L²_tL^∞_uCω^0,δ0(Cu).λ^{−1/2−3ε0}. (9.208) By bootstrap assumption (6.13), (9.6) and (9.104), we have:

λ⁻¹

C~,D _L1

tL^∞_uC_ω^0,δ0(Cu).λ^−1−7ε0, (9.209)

tr

ge

/χ˜^(Small),∂∂∂~Ψ,χˆ

2

L²_tL^∞_uCω^0,δ0(Cu).λ⁻¹. (9.210) Taking theL²_t norm of (9.206), combining (9.206)-(9.210), we conclude the desired result for tr

ge /χ˜^(Small). Now, since tr_g/χ−²

er =tr

eg

/χ˜^(Small)−ΓΓΓL, we have:

tr_g/χ−2 er

L²_tL^∞_uCω^0,δ0(M^(Int))

= tr

ge/χ˜^(Small)

L²_tL^∞_uC^0,δω⁰(M^(Int))+

f_(~L)·∂∂∂~Ψ

L²_tL^∞_uCω^0,δ0(M^(Int)). (9.211) By the previously proven result (9.10) and estimate (9.104), we have:

f_(~L)·∂∂∂~Ψ

L²_tL^∞_uCω^0,δ0(M^(Int)). f_(~L)

L^∞_t L^∞_uCω^0,δ0(M^(Int))

∂∂∂~Ψ

L_t²L^∞_uC^0,δω⁰(M^(Int)).λ^{−1/2−3ε0}. (9.212) Combining (9.211)-(9.212) with the result for tr

ge/χ˜^(Small), we conclude the proof for tr_g/χ−²

er. We now prove kχkˆ

L²_tL^∞_uC_ω^0,δ0(M^(Int)).λ^{−1/2−3ε0}. Plugging equation (8.139a) into the Hodge estimate

(9.114) withQ:=_1−δ²

0, where we recallδ0<1−²_p,Q<p, we have:

kχkˆ

L²_tL^∞_uC^0,δω⁰(M^(Int)). tr

ge /χ˜^(Small)

L_t²L^∞_uC^0,δω⁰(M^(Int)) (9.213)

+ ∂∂∂~Ψ

L_t²L^∞_uC^0,δω⁰(M^(Int))

+ erf_(~L)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L²_tL^∞_uLω^p(M^(Int)).

By bootstrap assumptions (6.13), (9.5a) and (9.82a), we have:

erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L²_tL^∞_uL^pω(M^(Int)) (9.214)

.

λ^1/2−4ε0 er^1/2∂∂∂~Ψ

Lω^p(St,u), er

trge/χ˜^(Small),χ,ˆ er⁻¹

Lω^p(S_t,u)

∂∂∂~Ψ

L^∞ω(St,u)

L_t²L^∞_u

. ∂∂∂~Ψ

L²_tL^∞_uL^∞_ω(M).λ^{−1/2−4ε0}.

Combining (9.213)-(9.214) with previously proven result for tr

eg

/χ˜^(Small) and estimate (9.104), we conclude the desired estimate.

Proof of

treg/χ˜^(Small),χ,ˆ tr_g/χ−²

er

L

q

t2L^∞_uCω^0,δ0(M).λ

2 q−1−4ε₀

4 q−1

in (9.11). We first bound tr

ge/χ˜^(Small). Using equation (8.138a) and dividing byer²(t,u), we have:

tr

ge /χ˜^(Small)

_g/.er⁻² lim

τ↓[u]₊

er²tr

ge/χ˜^(Small) _g/+er⁻²

Z _t [u]+

er²λ⁻¹f_(~L)·(C~,D)

_g/dτ (9.215) +er⁻²

Z _t [u]+

er²f_(~L)· tr

eg

/χ˜^(Small),χ,ˆ er⁻¹,∂∂∂~Ψ ·∂∂∂~Ψ

+er²|χ|ˆ ²_g/+er²tr

ge

/χ˜^(Small)·tr

eg /χ˜^(Small)

g/dτ.

Using (9.108), by bootstrap assumptions (6.13), (9.4) and (9.104), we have:

tr

ge/χ˜^(Small) _L∞

uC^0,δ_ω⁰(Σt). er⁻² lim

τ↓[u]₊

er²tr

ge/χ˜^(Small)

L^∞_uC^0,δω⁰(Σ_t)

+λ⁻¹

C~,D _L1

tL^∞_uC^0,δ_ω⁰(M) (9.216) +

tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ

2

L²_tL^∞_uC_ω^0,δ0(M)+M ∂∂∂~Ψ

_L∞

uC^0,δ_ω⁰(Σt)

.

We now consider the initial condition. Whenu≥0, by (8.64a), lim

τ↓[u]+ er²tr

eg /χ˜^(Small)

=0, we have:

er²tr

ge/χ˜^(Small) _g/.

Z _t [u]+

er²λ⁻¹f_(~L)·(C~,D)

_g/dτ (9.217)

+ Z t

[u]+

er² f_(~L)·

tr

eg

/χ˜^(Small),χ,ˆ er⁻¹,∂∂∂~Ψ ·∂∂∂~Ψ

+er²|χ|ˆ²_g/+er²tr

ge/χ˜^(Small)·tr

eg /χ˜^(Small)

_g/dτ.

Whenu<0, by (8.61), we have:

er⁻² lim

τ↓[u]+

er²tr

ge/χ˜^(Small) L

q

t2L^∞_uC_ω^0,δ0(M). _{Z T}

∗;(λ)

0

er^−1/2λ^{−1/2 q}₂

dτ ²_q

(9.218) .

λ^{(1−8ε0)(1−}

q 4)

λ⁻

q 4

²_q

=λ

2 q−1−4ε₀

4 q−1

.

Using the estimate for Hardy-Littlewood maximal function (9.28), (9.104), and H¨older’s inequality int, we have:

M

∂∂∂~Ψ

_L∞

uC^0,δ_ω⁰(Σt)

L

q t2

. ∂∂∂~Ψ

L

q

t2L^∞_uC_ω^0,δ0(M)

(9.219) .

∂∂∂~Ψ

L²_tL^∞_uC^0,δω⁰(M)λ^(1−8ε0)(^1−q₄)²_q .λ

2 q−1−4ε₀

4

q−1

−3ε₀

.

By bootstrap assumptions (6.13) and (9.4), and (9.104), we have:

λ⁻¹

C~,D

L¹_tL^∞_uCω^0,δ0(M)

L

q t2

.λ^(1−8ε0)

2 q−1−7ε₀

=λ

2 q−1−4ε₀

4

q−1

−11ε₀

, (9.220)

tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ

2

L²_tL^∞_uC_ω^0,δ0(M)

L

q t2

.λ^(1−8ε0)

2 q−1+4ε₀

=λ

2 q−1−4ε₀

4

q−1

. (9.221)

Combining (9.215)-(9.221) and we conclude the desired estimate for tr

eg /χ˜^(Small). Now since tr_g/χ−²

er =tr

eg

/χ˜^(Small)−ΓΓΓL, we have:

tr_g/χ−2 er L

q

t2L^∞_uC^0,δω⁰(M)

= tr

ge/χ˜^(Small) L

q

t2L^∞_uC^0,δ_ω⁰(M)+

f_(~L)·∂∂∂~Ψ L

q

t2L^∞_uC^0,δ_ω⁰(M). (9.222) By previously proven result (9.10) and previous estimate (9.219), we have:

f_(~L)·∂∂∂~Ψ

L

q

t2L^∞_uC^0,δω⁰(M). f_(~L)

L^∞_tL^∞_uC^0,δω⁰(M)

∂∂∂~Ψ

L

q

t2L^∞_uCω^0,δ0(M).λ

2 q−1−4ε₀

4

q−1

−3ε₀

. (9.223)

Combining the above with the result for tr

eg

/χ˜^(Small), we conclude the proof for tr_g/χ−²

er. Now we provekχkˆ

L

q

t2L^∞_uC_ω^0,δ0(M).λ

2 q−1−4ε₀

4 q−1

. Plugging equation (8.139a) into the Hodge estimate (9.114) withQ:=_1−δ²

0, and recallingδ₀<1−²_p,Q<p, we have:

kχkˆ

L

q

t2L^∞_uC^0,δω⁰(M). tr

eg /χ˜^(Small)

L

q

t2L^∞_uC^0,δω⁰(M)

(9.224) +

∂∂∂~Ψ

L

q

t2L^∞_uC^0,δ_ω⁰(M)

+ erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ L

q

t2L^∞_uL_ω^p(M)

.

Using previous estimate (9.214), we have:

erf_(~L)·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ L

q

t2L^∞_uLω^p(M) (9.225)

.λ^(1−8ε0)(1−^q₄)²_q erf_(~L)·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L²_tL^∞_uLω^p(M)

.λ^(1−8ε0)(^1−q₄)²_q

λ^{−1/2−4ε0}

=λ

2

q−1−4ε₀(⁴_q−1)−4ε₀

.

Combining (9.224)-(9.225) with previously proven result for tr

ge

/χ˜^(Small)and (9.223), we conclude the desired estimates.

Proof ofkχkˆ

L²_tCω^0,δ0(Cu).λ^−1/2in (9.8e). Using the Sobolev inequality (9.33b) withQ:=p, and the previ- ously proven estimates (9.8d) and (9.7a), we have:

kχkˆ

L_t²C^0,δω⁰(Cu).ker∇/χkˆ _L2

tLω^p(Cu)+kχkˆ _L2

tL²_ω(Cu).λ^−1/2. (9.226)

Proof of (9.16a) and (9.16b). We first prove (9.16a). Integrating equation (8.142) along the integral curves ofL, we find that:

er⁻²g/

∂

∂ω^A, ∂

∂ω^B

−/e ∂

∂ω^A, ∂

∂ω^B

(9.227)

=lim

t↓[u]+

er⁻²g/

∂

∂ω^A, ∂

∂ω^B

−/e ∂

∂ω^A, ∂

∂ω^B

+ Z t

[u]+

treg/χ˜^(Small)−ΓΓΓ_L

er⁻²g/ ∂

∂ω^A, ∂

∂ω^B

−e/ ∂

∂ω^A, ∂

∂ω^B

dτ +

Z t [u]+

treg/χ˜^(Small)−ΓΓΓ_L

e/ ∂

∂ω^A, ∂

∂ω^B

+ 2 er²χˆ

∂

∂ω^A, ∂

∂ω^B

dτ.

Using the initial conditions (8.63c) and (8.64c), noticing that ˆχ

∂

∂ω^A, ^∂

∂ω^B

=er²χˆ_AB, and using Grownwall’s inequality, we have:

er⁻²g/

∂

∂ω^A, ∂

∂ω^B

−/e ∂

∂ω^A, ∂

∂ω^B

g/

(9.228) .

λ^−4ε0+ tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ L¹_tL^∞_ω(Cu)

exp

tr

eg

/χ˜^(Small),∂∂∂~Ψ L¹_tL_ω^∞(Cu)

.

By bootstrap assumptions (6.13), and the previously proven result (9.11) withq:=4, we have:

er⁻²g/

∂

∂ω^A, ∂

∂ω^B

−e/ ∂

∂ω^A, ∂

∂ω^B

_L∞(M).λ^−4ε0. (9.229) We now prove (9.16b), we first apply ^∂

∂ω^C to equation (8.142). Note that tr

eg

/χ˜^(Small)−ΓΓΓ_L=tr_g/χ−²

er. SinceL and ^∂

∂ω^C commute, we have:

d dt

∂

∂ω^C

er⁻²g/ ∂

∂ω^A, ∂

∂ω^B

−e/ ∂

∂ω^A, ∂

∂ω^B

= ∂

∂ω^Ctr_g/χer⁻²g/ ∂

∂ω^A, ∂

∂ω^B

(9.230) +

tr_g/χ−2

er ∂

∂ω^C

er⁻²g/ ∂

∂ω^A, ∂

∂ω^B

−/e ∂

∂ω^A, ∂

∂ω^B

+

tr_g/χ−2 er

∂

∂ω^C/e ∂

∂ω^A, ∂

∂ω^B

+2er⁻² ∂

∂ω^Cχˆ ∂

∂ω^A, ∂

∂ω^B

.

Integrating equation along the integral curves ofL, taking theL_ω^p norm, and using the initial condition (8.63d), (8.64d), and then applying Grownwall’s inequality, we have:

∂

∂ω^C

er⁻²g/ ∂

∂ω^A, ∂

∂ω^B

−e/ ∂

∂ω^A, ∂

∂ω^B

L^pω(St,u)

(9.231)

. λ^−4ε0+ er∇/tr_g/χ

L_t¹Lω^p(Cu)+

tr_g/χ−2 er

∂

∂ω^Ce/ _L1

tL_ω^p(Cu)

+ker∇/χkˆ _L1

tLω^p(Cu)+kΓΓΓ·χkˆ _L1 tLω^p(Cu)

·exp

tr_g/χ−2 er L¹_tL^∞_ω(Cu)

! .

By bootstrap assumptions (6.13), and the previously proven results (9.7a), (9.8d) and (9.11) withq:=4, we have:

∂

∂ω^C

er⁻²g/ ∂

∂ω^A, ∂

∂ω^B

−/e ∂

∂ω^A, ∂

∂ω^B

_Lp

ω(St,u)

.λ^−4ε0. (9.232)

Proof of

b⁻¹−1 er

_L2

tL^∞_x(M).λ^−1/2in (9.9). We first bound

b⁻¹−1 er

_L2

tL^∞_x(M^(Int)). Integrating equation (8.141) along along the integral curves ofLemanating from the cone-tip, and using the initial condition (8.64a), we have:

b⁻¹−1=− Z t

u

b⁻¹−1

f_(~L)·∂∂∂~Ψdτ− Z t

u

f_(~L)·∂∂∂~Ψdτ. (9.233)

Using Grownwall’s inequality and the bootstrap assumption (6.13), we have:

b⁻¹−1 er

g/

.M ∂∂∂~Ψ

L^∞ω(St,u)

. (9.234)

Hence,

b⁻¹−1 er

L²_tL^∞_x(M^(Int))

. ∂∂∂~Ψ

L_t²L^∞_x(M).λ^{−1/2−4ε0}. (9.235) Now we consider the case whenu<0. Integrating equation (8.141) along along the integral curves of L emanating fromΣ₀, we have:

b⁻¹−1 er

g/

≤

b⁻¹−a⁻¹ er

g/

+

a⁻¹−1 er

g/

.M

∂∂∂~Ψ

L^∞_ω(S_t,u)

+

a⁻¹−1 er

g/

. (9.236)

By initial condition (8.63a), we have:

a⁻¹−1 er

L²_tL^∞_x(M)

.

a⁻¹−1 w^1/2

L^∞_x(M)

Z _T∗;(λ) 0

w (τ+w)²dτ

1/2

.λ^−1/2. (9.237)

Combining (9.236)-(9.237), we have:

b⁻¹−1 er

_L2

tL^∞_x(M^(Ext))

.λ^−1/2. (9.238)

Combining (9.235) and (9.238), we conclude the desired estimate.

Proof of

er(D/_L,/∇)

b⁻¹−1 er

L²_tLω^p(Cu).λ^−1/2in (9.9). We first prove erD/_L

b⁻¹−1 er

L²_tLω^p(Cu).λ^−1/2. By (8.141) and the fact thatL(er) =1, we have:

erD/_L

b⁻¹−1 er

=−b⁻¹f_(~L)·∂∂∂~Ψ−b⁻¹−1

er . (9.239)

the last proof, we have:

erD/_L

b⁻¹−1 er

_L2

tL_ω^p(Cu)

. ∂∂∂~Ψ

L²_tLω^p(Cu)+

b⁻¹−1 er

_L2

tL_ω^p(Cu)

.λ^−1/2. (9.240)

Now we bounder∇/

b⁻¹−1 er

. Recall (8.17),ζ=/∇lnb+f_(~L)·∂∂∂~Ψ. Therefore,

er∇/

b⁻¹−1 er

=−b⁻¹/∇lnb=b⁻¹

−ζ+f_(~L)·∂∂∂~Ψ

. (9.241)

By the bootstrap assumption (6.13), and the proven results (9.7a) and (9.17b), we have:

er∇/

b⁻¹−1 er

_L2

tL^p_ω(Cu)

.λ^−1/2. (9.242)

Proof of

b⁻¹−1 er^1/2

_L∞

t L^∞_uL^2p_ω(M).λ^−1/2in (9.9). By the Sobolev inequality (9.32), we have:

b⁻¹−1 er^1/2

2

L^∞_tL^∞_uL^2pω(M)

(9.243)

. erD/_L

b⁻¹−1 er

_L2

tL^∞_uL_ω^p(M)

+

b⁻¹−1 er

_L2

tL^∞_uL^p_ω(M)

!

b⁻¹−1 er

_L2

tL^∞_uL^∞_ω(M)

.

Using the proven first and third estimates of (9.9), we conclude the desired result.

Proof of (9.18). We first prove

∇/ln er⁻²v

L²_tLω^p(Cu) .λ^−1/2. Plugging equation (8.143b) into estimate (9.76), we have:

er∇/ln er⁻²v g/=lim

t↓τ

er∇/ln er⁻²v g/+

Z _t [u]+

er

1 2

tr

ge

/χ˜^(Small)−ΓΓΓL

+f_(~L)·χˆ

/∇ln er⁻²v g/

dτ (9.244) +

Z t [u]+

er∇/

trg/eχ˜^(Small)−ΓΓΓ_L

g/dτ.

Applying the Grownwall’s inequality, and using the proven results (9.11) withq:=4, we have:

er∇/ln er⁻²v g/.

limt↓τ

er∇/ln er⁻²v g/+

Z _t [u]+

er∇/

tr

ge

/χ˜^(Small)−ΓΓΓL

_g/dτ

(9.245)

·exp

tr_g/χ−2 er,χˆ

_L1

tL^∞_ω(Cu)

!

.lim

t↓τ

er∇/ln er⁻²v g/+

Z _t [u]+

er∇/

trg/eχ˜^(Small)−ΓΓΓ_L

g/dτ.

We now consider the initial conditions. When u≥0, by initial condition (8.64c) and (8.64d), we have limt↓τ

er∇/ln er⁻²v

g/=0. Whenu<0, by (8.63e),

limt↓τer^1/2/∇ln er⁻²v _Lp

ω(S_w)

.λ^−1/2. Now dividing both sides of (9.245) byer(t,u), then take the norms and use the estimate for Hardy-Littlewood maximal function (9.28) to deduce:

∇/ln er⁻²v

L²_tL^pω(Cu).λ^−1/2 _{Z T}

∗;(λ)

0

w (τ+w)²dτ

1/2

+ er

/∇tr

eg

/χ˜^(Small),/∂∇∂∂~Ψ

L²_tLω^p(Cu). (9.246) By estimate (9.82f) and proven result (9.8d), we conclude the desired estimate.

Now we bounder^1/2/∇ln er⁻²v

. Dividing both sides of (9.245) byer^1/2, taking theL^p_ωnorm, by estimate (9.82f) and proven result (9.8d), we have:

er^1/2/∇ln er⁻²v

Lω^p(St,u).λ^−1/2+(t−[u]₊)^1/2 (t−u)^1/2

er

/∇tr

eg

/χ˜^(Small),/∂∇∂∂~Ψ

L²_tL^pω(Cu).λ^−1/2. (9.247) Now we prove forerL∇/ln er⁻²v

. Using equation (8.143b), we have:

erL∇/ln er⁻²v

g/ (9.248)

. er∇/

tr

ge

/χ˜^(Small)−ΓΓΓL

+er

tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ

·/∇ln er⁻²v

+∇/ln er⁻²v g/.

By the bootstrap assumptions (6.13), the estimate (9.82f), and the proven results (9.11) withq:=4, (9.8d), we have:

erL∇/ln er⁻²v

L_t²Lω^p(Cu). er

/∇tr

eg

/χ˜^(Small),/∂∇∂∂~Ψ

_L2

tL_ω^p(Cu) (9.249)

+λ^1/2−4ε0 tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ L²_tL^∞_ω(Cu)

er^1/2/∇ln er⁻²v

L^∞_tLω^p(Cu)

+

∇/ln er⁻²v L²_tLω^p(Cu)

.λ^−1/2.

Proof ofker∇/ζk_L2

tLω^p(Cu),kζk

L²_tC^0,δω⁰(Cu).λ^−1/2in (9.19) and (9.8e). Plugging equation (8.148a) and (8.148b)

into the Hodge estimate (9.110) withQ:=p, we have:

ker∇/ζk_L2

tL_ω^p(Cu).

λ⁻¹erf_(~L)·(C~,D)

L²_tLω^p(Cu)+ er∇/∂∂∂~Ψ

L²_tLω^p(Cu) (9.250) +

erf_(~L)·χ·ˆ χˆ

L_t²Lω^p(Cu)+

erf_(~L)·ζ·ζ L²_tLω^p(Cu)

+

erf_(~L)·/∇ln er⁻²v

·(∂∂∂~Ψ,ζ) L²_tLω^p(Cu)

+

erf_(~L)·(∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹)·∂∂∂~Ψ

L_t²Lω^p(Cu).

By estimates (9.82d), (9.82f), the already proven results (9.7a), (9.7c), (9.11), (9.18), (9.189), we have:

ker∇/ζk_L2

tLω^p(Cu).λ^−1/2+λ^−4ε0kζk_L2

tL^∞_ω(Cu). (9.251)

Using the Sobolev inequality (9.33b) withQ:=p, and the already proven result (9.7a), we have:

kζkL²_tCω^0,δ0(Cu).ker∇/ζk_L2

tL^pω(Cu)+kζk_L2

tL²_ω(Cu).λ^−1/2+ker∇/ζk_L2

tL^pω(Cu). (9.252) Combining (9.251) and (9.252), we conclude the desired estimates.

Proof ofkerµk_L2

tL_ω^p(Cu).λ^−1/2in (9.19). Using equation (8.144), by the estimates (9.82d), (9.82f), and al- ready proven results (9.7a), (9.7c), (9.8e), (9.18), (9.189), we have:

kerµk_L2

tLω^p(Cu).

λ⁻¹erf_(~L)·(C~,D)

L²_tLω^p(Cu)+ er∇/∂∂∂~Ψ

L²_tLω^p(Cu)+

erf_(~L)·χˆ·χˆ

L²_tL^pω(Cu) (9.253) +

erf_(~L)·/∇ln er⁻²v

·(∂∂∂~Ψ,ζ) L²_tLω^p(Cu)

+

erf_(~L)·(∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹)·∂∂∂~Ψ L_t²Lω^p(Cu)

.λ^−1/2.

Proof ofkζk_L2

tL^∞_x(M^(Int)).λ^{−1/2−3ε0} andkζk

L

q

t2L^∞_x(M).λ

2 q−1−4ε₀

4 q−1

in (9.12). Plugging equations (8.148a)

and (8.148b) into the Hodge estimate (9.116) withQ:=p,c=2,F=f_(~L)·∂∂∂~Ψ, we have:

kζk_L2

tL^∞_x(M^(Int)). ν^δ0P_ν

f_(~L)·∂∂∂~Ψ

L²_tL^∞_ul²_νL^∞_ω(M^(Int)) (9.254)

+ ∂∂∂~Ψ

L²_tL^∞_x(M^(Int))+

λ⁻¹er(C~,D)

L²_tL^∞_uLω^p(M^(Int))

+ker(χˆ·χ,ζˆ ·ζ)k_L2

tL^∞_uLω^p(M^(Int))

+ er(∂∂∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹)·∂∂∂~Ψ _L2

tL^∞_uL_ω^p(M^(Int))

+

er∇/ln er⁻²v

·(∂∂∂~Ψ,ζ)

L²_tL^∞_uLω^p(M^(Int)).

Using the estimates (9.93a), (9.104), the already proven results (9.7c), (9.15) for ˆχ, (9.18), and (9.141), we have:

kζk_L2

tL^∞_x(M^(Int)).λ^{−1/2−3ε0}+λ^−4ε0kζk_L2

tL^∞_x(M^(Int)). (9.255)

By absorbing the second term on RHS of (9.255) into the left, we conclude the desired result forL²_tL^∞_x norm.

To bound theL

q

t2L^∞_x norm, we apply the H¨older’s inequality to (9.254) and use the same argument given above to deduce:

kζk

L

q

t2L^∞_x(M).λ

2 q−1−4ε₀

4 q−¹₄

+λ^−4ε0kζk

L

q t2L^∞_x(M)

. (9.256)

By absorbing the second term on RHS of (9.256) into the left, we conclude the desired result.

Proof of (9.20a)-(9.20c). Using equation (8.37a) and initial condition forσ(8.37b), we have:

σ=1 2

Z t

uΓΓΓLdτ. (9.257)

Using the bootstrap assumptions (6.13), we find that:

kσk_L∞(M(Int)).λ^−8ε0, (9.258)

er^−1/2σ

_L∞(M(Int)).sup

t,u

(t−u)^1/2

er^1/2 λ^{−1/2−4ε0} .λ^{−1/2−4ε0}. (9.259) To provek∇/σk_L2

tL^p_ω(Cu).λ^−1/2, we plug equation (8.145) into (9.76), and use the initial condition (8.64a) to

deduce:

|er∇/σ|_g/. Z t

u er

/∂∇∂∂~Ψ

g/+

tr

ge/χ˜^(Small),∂∂∂~Ψ,χˆ

·/∇σ g/

dτ. (9.260)

Using the Grownwall’s inequality, the bootstrap assumption (6.13) and the already proven results (9.8e), we have:

|er∇/σ|_g/. Z _t

u er /∇∂∂∂~Ψ

_g/dτexp

tr

ge

/χ˜^(Small),∂∂∂~Ψ,χˆ _L1

tL^∞_ω(Cu)

.

Z _t uer

/∂∇∂∂~Ψ

_g/dτ. (9.261) Dividing both sides of (9.261) byer(t,u), taking theL²_tL_ω^p norms, using the estimate for Hardy-Littlewood maximal function (9.28) and the estimate (9.83a), we have:

k∇/σk_L2

tL_ω^p(Cu). er∇/∂∂∂~Ψ

L²_tL^pω(Cu).λ^−1/2, (9.262) as desired. To show

er^1/2/∇σ _Lp

ωL^∞_t(Cu).λ^−1/2, dividing estimate (9.261) byer^1/2(t,u), then taking theL_ω^p norms, we have:

er^1/2/∇σ

_Lp

ω(St,u).(t−u)^1/2 er^1/2

er∇/∂∂∂~Ψ

_L2

tL_ω^p(Cu).λ^−1/2. (9.263) We now prove

er^1/2Lσ

L_t^∞L^2p_ω(Cu) .λ^{−1/2−2ε0}. By the Sobolev equality (9.32), the estimate (9.82d) and (9.83a), we have:

er^1/2Lσ

2

L^∞_tL^2pω(Cu).

erD/_L∂∂∂~Ψ

Lω^pL_t²(Cu)+ ∂∂∂~Ψ

Lω^pL²_t(Cu)

∂∂∂~Ψ

L^∞_ωL²_t(Cu).λ^−1−4ε0. (9.264)

Proof of

er /∇µ,ˇ /∇ζ˜

L²_uL²_tL^pω(M^(Int)).λ^−4ε0 in (9.21a). In this paragraph, we are assumingCu⊂M^(Int). We will silently use the fact that 0≤u≤t≤T∗;(λ).λ^1−8ε0silently. We start by deriving a preliminary estimate for

er∇/ζ˜

L²_uL²_tLω^p(M^(Int)). Plugging equations (8.149a) and (8.149b) into the Hodge estimate (9.110), we have:

er∇/ζ˜

L²_uL_t²Lω^p(M^(Int)). er∇/∂∂∂~Ψ

L²_uL²_tLω^p(M^(Int))+λ⁻¹

er(C~,D)

L²_uL²_tL^pω(M^(Int)) (9.265) +ker(ζ·ζ,χ·ˆ χ)kˆ _L2

uL²_tL^pω(M^(Int))

+ er

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er

·∂∂∂~Ψ

L²_uL²_tLω^p(M^(Int))+kerµkˇ _L2

uL²_tL_ω^p(M^(Int)).

By estimates (9.82f), (9.93c), (9.189), we have:

er∇/ζ˜

_L2

uL_t²L_ω^p(M^(Int)).λ^−4ε0+kerµkˇ _L2

uL_t²Lω^p(M^(Int)). (9.266)

We now boundkerµkˇ _L2

uL_t²L_ω^p(M^(Int)). Plugging (8.146) into (9.76), and using the initial condition on cone- tip (8.64a), we have:

er²µˇ

g/. Z _t

u er²

J₍₁₎ g/+

J₍₂₎ g/

dτ+

Z _t u er²

tr

eg

/χ˜^(Small),∂∂∂~Ψ

_g/µdτ.ˇ (9.267) Using the Grownwall’s inequality, the bootstrap assumption (6.13) and the already proven estimate (9.15), we have:

er²µˇ

_g/. Z t

u er² J₍₁₎

g/+ J₍₂₎

g/

dτexp

tr

ge

/χ˜^(Small),∂∂∂~Ψ L_t¹L^∞_ω(Cu)

(9.268) .

Z t u er²

J₍₁₎ g/+

J₍₂₎ g/

dτ.

We divide estimate (9.268) byer(t,u)and then take theL²_uL²_tL_ω^p norm. We now estimate terms inJ₍₁₎andJ₍₂₎ defined in (8.147). First, by the estimate for Hardy-Littlewood maximal function (9.28), estimates (9.82f), (9.82a), we have:

er⁻¹

Z _t

u er∇/∂∂∂~Ψdτ

L²_uL²_tLω^p(M^(Int))

. er∇/∂∂∂~Ψ

_L2

uL_t²L_ω^p(M^(Int)).λ^−4ε0, (9.269)

er⁻¹

Z t u

∂

∂∂~Ψdτ _L2

uL²_tLω^p(M^(Int))

. ∂∂∂~Ψ

L²_uL²_tL^pω(M^(Int)).λ^−4ε0. (9.270) Next, by the estimates (9.92a), (9.91c), we have:

λ⁻¹ er⁻¹

Z _t u

er²∂∂∂(C~,D) _g/dτ

L²_uL²_tLω^p(M^(Int))

.λ^−12ε0, (9.271) λ⁻¹

er⁻¹

Z t [u]+

er²(∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S)·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹ g/dτ

L²_uL²_tL_ω^p(M^(Int))

.λ^−16ε0. (9.272)

By the already proven estimate (9.15), we have:

er⁻¹

Z _t uer²

∇/ζ˜·χˆ g/dτ

L²_uL²_tLω^p(M^(Int))

. er∇/ζ˜

L²_tLω^p(Cu)kχkˆ _L2 tL^∞_ω(Cu)

_L2

uL²_t (9.273) .λ^−7ε0

er∇/ζ˜ _L2

uL²_tL_ω^p(M^(Int)).

By the bootstrap assumptions (9.6), and the estimates (9.82f), (9.8d) and (9.189), we have:

er⁻¹

Z t u

er²/∇σ·

/∇∂∂∂~Ψ,/∇tr

ge/χ˜^(Small),

∂∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

g/dτ

_L2

uL²_tL^p_ω(M^(Int))

(9.274) .

k∇/σk_L2 tL^∞_ω(Cu)

er

/∂∇∂∂~Ψ,/∇tr

ge/χ˜^(Small),

∂∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

_L2

tL^p_ω(Cu)

L²_uL²_t

.λ^−4ε0.

By the bootstrap assumptions (6.13), the already proven estimate (9.8d) and (9.12), we have:

er⁻¹

Z _t uer²

/∇tr

eg

/χ˜^(Small)·(∂∂∂~Ψ,ζ) _g/dτ

L²_uL²_tLω^p(M^(Int))

(9.275) 1.

er∇/tr

eg /χ˜^(Small)

L²_tLω^p(Cu)

∂∂∂~Ψ,ζ

L_t²L^∞_ω(Cu)

_L2

uL²_t

.λ^−11ε0.

By the bootstrap assumptions (6.13), already proven result (9.15), and the estimate (9.189), we have:

er⁻¹

Z t u er²

∂∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ ζ,er⁻¹

·

∂∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ ζ

·

∂∂

∂~Ψ,χˆ g/dτ

_L2

uL²_tL^p_ω(M^(Int))

(9.276) .

er

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹

·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ _L2

tL^p_ω(Cu)

∂∂∂~Ψ,χˆ

_L2

tL^∞_ω(Cu)

L²_uL_t²

.λ^−11ε0.

By the bootstrap assumptions (6.13), the estimates (9.82b), (9.12), (9.15), and the Fubini’s theorem, we have:

er⁻¹

Z _t u er²

(∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ζ)ˆ ·∂∂∂²~Ψ _g/dτ

L²_uL²_tLω^p(M^(Int))

(9.277) .

er∂∂∂²~Ψ

L²_tLω^p(Cu)

∂∂∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ ζ L_t²L^∞_ω(Cu)

_L2

uL²_t

.λ^−7ε0 er∂∂∂²~Ψ

_L2

tL²_uL_ω^p(M^(Int))

.λ^−11ε0.

J₍₁₎is bounded by (9.269)-(9.270), andJ₍₂₎is bounded by (9.271)-(9.277). Therefore, we conclude

kerµkˇ _L2

uL²_tL^pω(M^(Int)).λ^−4ε0+λ^−7ε0 er∇/ζ˜

L²_uL²_tLω^p(M^(Int)). (9.278) Combining this estimate with the estimate (9.266) for

er∇/ζ˜ _L2

uL_t²L_ω^p(M^(Int)), and soaking the second term on

the RHS of (9.266) to the LHS of (9.278), we conclude the desired results.

Proof of er^3/2µˇ

_L2

uL^∞_t L^p_ω(M^(Int)).λ^−4ε0in (9.21b). We divide (9.268) byer^1/2(t,u)and then take theL²_uL^∞_t L_ω^p norms. We first estimate the terms inJ₍₁₎andJ₍₂₎defined in (8.147).

By the estimates (9.82f), (9.82a), we have:

er^−1/2

Z _t

u er∇/∂∂∂~Ψdτ

L²_uL^∞_t L^pω(M^(Int))

. er∇/∂∂∂~Ψ

L²_uL²_tL^pω(M^(Int)).λ^−4ε0, (9.279)

er^−1/2

Z t u

∂∂∂~Ψdτ _L2

uL^∞_t L^p_ω(M^(Int))

. ∂∂∂~Ψ

L²_uL²_tLω^p(M^(Int)).λ^−4ε0. (9.280) By the estimates (9.92b), (9.91d), we have:

λ⁻¹ er^−1/2

Z _t u

er²∂∂∂(C~,D) _g/dτ

L²_uL_t^∞Lω^p(M^(Int))

.λ^−12ε0, (9.281) λ⁻¹

er^−1/2

Z t [u]+

er²(∂∂∂~Ψ,∂∂∂~ω,∂∂∂~S)·

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ ζ,er⁻¹ g/dτ

L²_uL_t^∞Lω^p(M^(Int))

.λ^−16ε0. (9.282)

By the proven estimates (9.15) and (9.21a), we have:

er^−1/2

Z _t uer²

∇/ζ˜·χˆ g/dτ

L²_uL^∞_tLω^p(M^(Int))

.λ^1/2−4ε0 er∇/ζ˜

L²_tLω^p(Cu)kχkˆ _L2 tL^∞_ω(Cu)

_L2

uL^∞_t (9.283) .λ^−11ε0.

By the bootstrap assumption (9.6), the estimates (9.82f), (9.8d) and (9.189), we have:

er^−1/2

Z t u

er²/∇σ·

/∂∇∂∂~Ψ,/∇tr

eg

/χ˜^(Small),

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

g/dτ

L²_uL^∞_tLω^p(M^(Int))

(9.284) .λ^1/2−4ε0

k∇/σk_L2

tL^∞_ω(Cu)

· er

/∂∇∂∂~Ψ,/∇tr

ge/χ˜^(Small),

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹ ·∂∂∂~Ψ

L²_tL^pω(Cu)

L²_uL^∞_t

.λ^−4ε0.

By the bootstrap assumptions (6.13), proven estimates (9.8d) and (9.12), we have:

er^−1/2

Z t uer²

/∇tr

eg

/χ˜^(Small)·(∂∂∂~Ψ,ζ) g/dτ

_L2

uL_t^∞L_ω^p(M^(Int))

(9.285) .λ^1/2−4ε0

er∇/tr

eg /χ˜^(Small)

L_t²Lω^p(Cu)

∂∂∂~Ψ,ζ

L_t²L^∞_ω(Cu)

L²L^∞

.λ^−11ε0.

By bootstrap assumptions (6.13), proven result (9.15), estimate (9.189), we have:

er^−1/2

Z t uer²

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹

·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ

·

∂∂∂~Ψ,χˆ g/dτ

_L2

uL^∞_tL_ω^p(M^(Int))

(9.286) .λ^1/2−4ε0

er

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹ ∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ζˆ _L2

tL^p_ω(Cu)

· ∂∂∂~Ψ,χˆ

L²_tL^∞_ω(Cu)

L²_uL^∞_t

.λ^−11ε0.

By bootstrap assumptions (6.13), estimates (9.82b), (9.12), (9.15), using Fubini’s theorem, we have:

er^−1/2

Z _t u er²

(∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ζ)ˆ ·∂∂∂²~Ψ _g/dτ

L_u²L^∞_tLω^p(M^(Int))

(9.287) .λ^1/2−4ε0

er∂∂∂²~Ψ

L²_tL^pω(Cu)

∂∂∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ ζ L²_tL^∞_ω(Cu)

L²_uL^∞_t

.λ^−7ε0 er∂∂∂²~Ψ

L²_tL²_uLω^p(M^(Int))

.λ^−11ε0.

J₍₁₎is bounded by (9.279)-(9.280), andJ₍₂₎is bounded by (9.281)-(9.287). Combining all the estimates for terms inJ₍₁₎andJ₍₂₎, we conclude the desired result.

Proof ofk∇/σk

L²_uL²_tCω^0,δ0(M^(Int)).λ^−4ε0 in (9.21a). Plugging (8.56) into the Sobolev equality (9.33b) withQ:=

p, we have:

k∇/σk

L²_uL²_tCω^0,δ0(M^(Int)).

er(∇/ζ,˜ /∇ζ)

L²_uL²_tL^pω(M^(Int))+k∇/σk_L2

uL_t²L²_ω(M^(Int)). (9.288) By the proven estimates (9.19), (9.20a), (9.21a) for

er∇/ζ˜

L²_uL_t²Lω^p(M^(Int)), we conclude the desired estimate.

Proof of (9.22). Plugging (8.55) into the Hodge estimate (9.110), using (9.123) and the proven estimate (9.21a), we have:

ker∇/µ/ ,µ/k_L2

tL²_uLω^p(M^(Int)).kerµkˇ _L2

tL²_uL^pω(M^(Int))+ erµ¯ˇ

_L2

tL²_uL_ω^p(M^(Int)).λ^−4ε0. (9.289)

By the Sobolev inequality (9.33b) withQ:=p, we have:

kµ/k

L²_tL²_uCω^0,δ0(M^(Int)).ker∇/µ/k_L2

tL²_uL^pω(M^(Int))+kµ/k_L2

tL²_uL²_ω(M^(Int)).λ^−4ε0. (9.290)

Proof of ζ−˜ µ/

L_t²L^∞_uL^∞_ω(M^(Int)).λ^{−1/2−4ε0} in (9.25a). Plugging equations (8.150a) and (8.150b) into the Hodge estimate (9.116) withQ:=p,c:=2 andδ⁰≤δ0, we have:

ζ˜−µ/

L²_tL^∞_uL^∞_ω(M^(Int)).

ν^δ0P_ν∂∂∂~Ψ _L2

tL_u^∞l²_νL^∞_ω(M^(Int))+ ∂∂∂~Ψ

_L2

tL^∞_uL^∞_ω(M^(Int)) (9.291) +λ⁻¹

er(C~,D)

L_t²L^∞_uLω^p(M^(Int))

+ker(ζ·ζ,χˆ·χ)kˆ _L2

tL^∞_uLω^p(M^(Int))

+ er

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er

·∂∂∂~Ψ

L²_tL^∞_uLω^p(M^(Int)).

By the bootstrap assumptions (6.13), the estimates (9.93a), (9.141), and the proven results (9.7c), (9.12) and (9.15), we conclude the desired bound:

ζ˜−µ/

L²_tL^∞_uL^∞_ω(M^(Int)).λ^{−1/2−4ε0}. (9.292)

Proof of (9.24). By (8.154), it suffices to show lim

t↓uerµ/ =O(er). Plugging definition (8.55) into the Hodge estimate (9.116) withF:=0 andQ:=p, and using the initial condition (8.64a), we have:

lim

t↓uerµ/ L^∞_ω(St,u)

.

lim

t↓uer²µˇ Lω^p(St,u)

=O(er). (9.293)

Proof of µ/₍₁₎

L²_tL^∞_uL^∞_ω(M^(Int)).λ^{−1/2−4ε0} in (9.25a). LetH:=/D_Lµ/₍₁₎+¹₂tr_g/χµ/₍₁₎. By (9.76) and the initial condition forerµ/₍₁₎, we have:

erµ/₍₁₎

g/=

Z t u

erH+er

tr

ge/χ˜^(Small)+∂∂∂~Ψ

µ/₍₁₎

g/dτ. (9.294)

Using the Grownwall’s inequality, by the bootstrap assumption (6.13) and the proven estimate (9.15), we

have:

erµ/₍₁₎

g/.

Z t u

|erH|_g/dτexp

tr

eg

/χ˜^(Small),∂∂∂~Ψ L¹_tL^∞_ω(Cu)

.

Z t u

|erH|_g/dτ. (9.295)

Dividing both sides of (9.295) byer, taking the norm, and then substituting equations (8.152a) and (8.152b) into the Hodge estimate (9.116) withF:=er⁻¹div/ξ,Q:=p,c:=2 andδ⁰≤δ₀, we have:

µ/₍₁₎

_L2

tL^∞_uL_ω^∞(M^(Int)). er⁻¹

Z _t u er

er⁻¹

ν^δ0Pν∂∂∂~Ψ _l2

νL^∞_ω(Sτ,u)+ er⁻¹∂∂∂~Ψ

_L∞

ω(Sτ,u) (9.296) +

er⁻¹∂∂∂~Ψ

_Lp

ω(Sτ,u)

dτ

L²_tL^∞_u

.

By the estimate for Hardy-Littlewood maximal function (9.28) and bootstrap assumptions (6.13), we conclude the desired estimate.

Proof of µ/₍₂₎

L²_uL_t^∞L^∞_ω(M^(Int)).λ^{−1/2−3ε0} in (9.25b). Using the same argument as in the proof of (9.25a), we have

erµ/₍₂₎

g/.

Z _t u

|erH|_g/dτ, (9.297)

whereH:=D/_Lµ/₍₂₎+¹₂tr_g/χµ/₍₂₎. Now divide both sides of (9.297) byerand take the norms. Notice that:

µ/₍₂₎

L²_uL^∞_tL^∞_ω(M^(Int)). er⁻¹

Z t u

|erH|_g/dτ

L²_uL^∞_t L^∞_ω(M^(Int))

.kHk_L2

uL¹_tL^∞_ω(M^(Int)). (9.298) Applying equations (8.153a) and (8.153b) into Hodge estimate (9.116) withF:=0 andQ:=p, we have:

kHk_L2

uL¹_tL^∞_ω(M^(Int)). erJ₍₂₎

L²_uL¹_tL^pω(M^(Int))+kerχ·ˆ /∇µ/k_L2

uL¹_tL_ω^p(M^(Int)) (9.299)

+ er

/∇∂∂∂~Ψ,/∇tr

ge/χ˜^(Small)

·µ/

L²_uL_t¹Lω^p(M^(Int))

+ er

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ·µ/

L²_uL¹_tLω^p(M^(Int))

+

er tr_g/χ−tr_g/χ ˇ µ

L²_uL¹_tL_ω^p(M^(Int)).

By (9.93d), we have:

λ⁻¹

er∂∂∂(C~,D)

L²_uL¹_tL^pω(M^(Int)).λ^{−1/2−8ε0}. (9.300)

By (9.93e), we have:

λ⁻¹ er

∂∂

∂~Ψ,∂∂∂~ω,∂∂∂~S

·

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er⁻¹

L²_uL_t¹Lω^p(M^(Int)).^{−12−10ε0}. (9.301) By the proven estimates (9.21a) and (9.15), we have:

er∇/ζ˜·χˆ

L_u²L¹_tLω^p(M^(Int)). er∇/ζ˜

L²_uL²_tL^pω(M^(Int))

er∇/ζ·˜ χˆ

L^∞_uL²_tL^∞_ω(M^(Int)).λ^{−1/2−7ε0}. (9.302) By (9.82f), (9.8d), (9.189) and (9.21a), we have:

er∇/σ·

/∇∂∂∂~Ψ,∇/tr

eg

/χ˜^(Small),

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

L²_uL¹_tLω^p(M^(Int)) (9.303)

.k∇/σk_L2

uL²_tL^∞_ω(M^(Int))

er

/∂∇∂∂~Ψ,/∇tr

eg

/χ˜^(Small),

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ er⁻¹

·∂∂∂~Ψ

_L∞

uL²_tL^p_ω(M^(Int))

.λ^{−1/2−4ε0}.

By (9.8d), (6.13) and (9.12), we have:

er∇/tr

eg

/χ˜^(Small)·(∂∂∂~Ψ,ζ)

L²_uL¹_tL^pω(M^(Int)). er∇/tr

ge /χ˜^(Small)

L²_uL²_tL^pω(M^(Int))

∂∂∂~Ψ,ζ

L^∞_uL²_tL^∞_ω(M^(Int)) (9.304) .λ^{−1/2−3ε0}.

By (9.189), (6.13) and (9.15), we have:

er

∂∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ζ,ˆ er

·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ζˆ

·(∂∂∂~Ψ,χ)ˆ

L²_uL¹_tL^pω(M^(Int)) (9.305) .

er

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ,er

·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χ,ˆ ζ

L²_uL_t²Lω^p(M^(Int))

∂∂∂~Ψ,χˆ

L^∞_uL²_tL^∞_ω(M^(Int))

.λ^{−1/2−7ε0}.

By (9.82b), (6.13), (9.15) and (9.12), we have:

er

∂

∂

∂~Ψ,tr

ge/χ˜^(Small),ζ

·∂∂∂²~Ψ _L2

uL¹_tL_ω^p(M^(Int)). er∂∂∂²~Ψ

_L2

uL_t^∞L_ω^p(M^(Int))· (9.306)

∂∂∂~Ψ,tr

ge/χ˜^(Small),ζ

L^∞_uL_t¹L^∞ω(M^(Int))

.λ^{−1/2−7ε0}.

By (9.22) and (9.15), we have:

kerχˆ·/∇µ/k_L2

uL_t¹Lω^p(M^(Int)).ker∇/µ/k_L2

uL²_tLω^p(M^(Int))kχkˆ _L∞

uL_t²L^∞_ω(M^(Int)).λ^{−1/2−7ε0}. (9.307) By (9.82f), (9.8d), (9.189) and (9.22), we have:

erµ/·

/∂∇∂∂~Ψ,/∇tr

ge/χ˜^(Small),

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χ,ˆ er⁻¹ ·

∂

∂∂~Ψ,tr

eg

/χ˜^(Small),χˆ

L²_uL¹_tLω^p(M^(Int)) (9.308) .kµ/k_L2

uL²_tL^∞_ω(M^(Int))·

er

/∇∂∂∂~Ψ,/∇tr

eg

/χ˜^(Small),

∂∂

∂~Ψ,tr

ge/χ˜^(Small),χ,ˆ er⁻¹

·

∂∂∂~Ψ,tr

ge

/χ˜^(Small),χˆ

L^∞_uL²_tL^pω(M^(Int))

.λ^{−1/2−4ε0}.

Note that ¹

er =¹

er (see Definition 8.7 for the definition of ¹

er). By (6.13), (9.15), and (9.21a), we have:

er tr_g/χ−tr_g/χ ˇ µ

L²_uL¹_tL^pω(M^(Int)). ∂∂∂~Ψ,tr

eg /χ˜^(Small)

L^∞_uL_t²L^∞_ω(M^(Int))kerµkˇ _L2

uL²_tL^p_ω(M^(Int)) (9.309) .λ^{−1/2−7ε0}.

Combining (9.298)-(9.309), we conclude the desired estimate.

CHAPTER 10

Conformal Energy Estimates

In this section, with the control of the acoustic geometry that we derived in Proposition 9.1, we prove the boundness theorem for the conformal energy in Theorem 10.2.