Remaining estimates

We follow the procedure outlined in the introduction. Define the energy quantities Q0(u, u) =δ²

According to Corollary 14.2, for all possible choices of the vector fieldsX, Y and Z in the set{e₄, e₃}, we are led to the identity with ⁽⁴⁾πand ⁽³⁾πbeing the deformation tensors ofe₄ ande₃, respectively. Every term appearing in the above integrands is linear in⁽⁴⁾πor ⁽³⁾πand quadratic with respect to R. Also, all the components of⁽⁴⁾πcan be expressed in terms of our Ricci coefficientsχ, ω,η,ω andη. In fact one can easily check that

Similar formulas hold for ⁽³⁾π, with χ replaced by χ. Observe, in particular, that the term trχ can only occur in connection with ⁽³⁾π. Thus, all terms appearing in the E

integrand are of the formφΨ₁Ψ₂, whereφis one of the Ricci coefficients and Ψ₁ and Ψ₂ are null curvature components. Consider first the contribution toQ0 of the anomalous terms

As all the terms of the form φΨ₁Ψ₂ have the same overall signature 4. Thus, it is easy to derive the scale-invariant norm estimates The gain ofδ^1/2 is a reflection of the product estimates of type (2.19). Now, the only null curvature component which is anomalous with respect to the scale-invariant norms L²_(sc)(Hu^(0,u)) is α. On the other hand, the only Ricci coefficient which is anomalous in L^∞_(sc) is trχ. Indeed we have to decompose trχ=trχ+ trχ₀, where trχ₀is the flat value of trχ₀ and therefore independent ofδ. This leads to a loss ofδ^1/2 in the corresponding estimates. Now, since trχcannot appear among the components of ⁽⁴⁾π, we can lose at most a power ofδon the right-hand side of (14.6), which occurs only when Ψ₁=Ψ₂=α.

Fortunately the terms on the left of our integral inequality are also anomalous with respect to the same power of δ. Therefore, since φ_L^∞_(sc)C, with C=C(I⁰,R,R) we derive

R²0[α]+R²0[β](I⁽⁰⁾)²+δ^1/2CR²0. Hence, for smallδ >0, we derive the bound

R0[α]+R0[β]I⁽⁰⁾+δ^1/4C(R,R), (14.7)

withCbeing a universal constant depending only on the curvature normsRandR. We would like to show that all other error terms can be estimated in the same fashion, i.e.

we would like to prove an estimate of the form

R0+R0I⁽⁰⁾+δ^1/4C(R,R). (14.8) Assuming that a similar estimate holds for R1+R1, we would thus conclude, for suffi-ciently smallδ >0, that

R+RI0.

To prove (14.8), we observe that all remaining terms in (14.5) are scale invariant (i.e. they have the correct powers ofδ). In estimating the corresponding error terms, appearing on the right-hand side, we only have to be mindful of those which contain trχ and α. All other terms can be estimated by δ^1/2p(R,R) exactly as above. It is easy to check that all terms involving trχ can only appear through ⁽³⁾πˆ₃₄. Thus, it is easy to see that all such terms are of the form

Q₃₄₄₄⁽³⁾πˆ³⁴≈ −|β|²trχ, Q₃₄₃₄⁽³⁾πˆ³⁴≈ −(²+σ²) trχ, Q₃₄₃₃⁽³⁾πˆ³⁴=−|β|²trχ.

Hence, since trχ=trχ+ trχ₀, we easily deduce that all error terms containing trχ can be estimated by

δ⁻¹ u

0 Q0(u, u)du+δ^1/2C(R,R).

It is easy to check that the integral term can be absorbed on the left by a Gr¨onwall-type inequality. It thus remains to consider only the terms linear(²⁴) inα_L

(sc)(H_u^(0,u)), which we have already estimated above. These lead to error terms with no excess powers ofδ, which could be potentially dangerous. In fact we have to be a little more careful, because we would get an estimate of the form

R0+R0I⁽⁰⁾+C(R,R),

which is useless for large curvature norms RandR. To avoid this problem we need to refine our use of the^(S)O0,∞norms. We observe that, among all the termsφΨ₁Ψ₂ linear in α, we can get better estimates for all, except those which contain a Ricci-coefficient

(²⁴) By signature considerations there can be no terms quadratic inα.

component φ which is anomalous in L⁴_(sc)(S). All other terms gain a power of δ^1/4. Indeed the corresponding error terms inE1can be estimated by(²⁵)

δ^1/2φ_L⁴_(sc)(u,u)Ψ_L2

(sc)(H_u^(0,u))∇α^1/2

L(sc)(H_u^(0,u))α^1/2

L(sc)(H_u^(0,u))

δ^{1/4 (S)}O0,4R0R0[α]^1/2R1[α]^1/2. Denoting byEg all such error terms, we thus have

|Eg|δ^1/4C(R,R).

It remains to check the terms linear inαfor which the Ricci-coefficient is anomalous in theL⁴_(sc) norm, i.e. terms for whichφis eitherχorχ. It is easy to check that there are no terms linear in α which containχ, and thus we only have to consider terms of the formχαΨ, which we denote by Eb. Asχ_L⁴_(sc)(u,u)loses a power ofδ^1/4, we now have

δ^1/2χ_L⁴_(sc)(u,u)Ψ_L2

(sc)(H_u^(0,u))∇α^1/2

L(sc)(H_u^(0,u))α^1/2

L(sc)(H_u^(0,u))

^(S)O0,4[χ]R0R0[α]^1/2R1[α]^1/2.

Since we are left with no positive power ofδ, we must now be mindful of the fact that the estimates for^(S)O0,4 depend at least linearly on the curvature norms Rand R, in which caseEb is super-quadratic inR andR. We can however trace back theδ^1/4 loss ofχ_L⁴_(sc)(u,u) to initial data, i.e. upon a careful inspection we find (see estimate (2.12) of Theorem A)

χ_L⁴_(sc)(u,u)δ^−1/4I⁽⁰⁾+C(R,R).

Thus,

EbI⁽⁰⁾R0R0[α]^1/2R1[α]^1/2+δ^1/4C(R,R).

The above considerations lead us to conclude, back to (14.5), that

R0+R0I⁽⁰⁾+cR0[α]^1/2R1[α]^1/2+δ^1/8C(R,R), (14.9) with a constantc=c(I⁽⁰⁾) depending only on the initial data.

Remark. In the analysis above we have not considered the possibility that, among the terms in the integrands ofE0, we can have terms of the formφΨ₁Ψ₂with at least one of the curvature terms being the null componentα, which cannot be estimated alongH_u. Among these terms, only those containing trχlead to terms which areO(1) inδ. These can be treated usingH, which leads to estimates of the form

Q0(u, u)+Q₀(u, u)I0²+ u

0 Q0(u, u)du+δ⁻¹ u

0 Q₀(u, u)du

+Cδ^1/2,

(²⁵) This follows from the Gagliardo–Nirenberg inequalityα²_L4(u,u)∇α_L2(u,u)α_L2(u,u).

withC=C(I⁽⁰⁾,R,R). The final estimate will follow from the lemma below (which can be easily proved by the method of continuity).

Lemma 14.5. Let f(x, y) and g(x, y)be positive functions defined in the rectangle 0xx₀ and 0yy₀ which satisfy the inequality

f(x, y)+g(x, y)J+a x

0

f(x, y)dx+b y

0

g(x, y)dy

for some non-negative constants a, b and J. Then, for all 0xx₀ and 0yy₀, f(x, y), g(x, y)J e^ax+by.

We summarize the results of this section in the following proposition.

Proposition 14.6. The following estimates hold with constants C=C(I⁽⁰⁾,R,R) and c=c(I⁽⁰⁾),and a sufficiently small δ:

R0[α]+R0[β]I⁽⁰⁾+Cδ^3/4,

R0+R0I⁽⁰⁾+cR^1/2+δ^1/8C.