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3.1. Higher Order Estimates [01Y4]

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3.1. Higher Order Estimates

A central analytic estimate from [ChCo1] in the proof of Lemma 1.7 states that if RicMn≥−(n−1)​δ{\rm Ric}_{M^{n}}\geq-(n-1)\delta and if u:B2​(p)→ℝu:B_{2}(p)\to\mathds{R} is a harmonic function, then on B1​(p)B_{1}(p) we have the L2L^{2} estimates of the form

⨏Br​(x)|∇2u|2≤C⁡(n)​infc⨏B2​r​(x)||∇u|2−c|+δ​⨏B2​r​(x)|∇u|2.\fint_{B_{r}(x)}|\nabla^{2}u|^{2}\leq C(n)\inf_{c}\fint_{B_{2r}(x)}\Big|\,|\nabla u|^{2}-c\Big|+\delta\fint_{B_{2r}(x)}|\nabla u|^{2}\,. (3.5)

Since the technique of proof will occur repeatedly in the sequel, we will recall it here.

According to [ChCo1] if RicMn≥−δ{\rm Ric}_{M^{n}}\geq-\delta, for any Br​(x)⊂MnB_{r}(x)\subset M^{n} there exists a cutoff function, with 0≤φ≤10\leq\varphi\leq 1, such that

φ⁡(x)\displaystyle\varphi(x) ≡1​ if ​x∈B9​r/5​(x),\displaystyle\equiv 1\text{ if }x\in B_{9r/5}(x)\,, (3.6)
supp​φ\displaystyle{\rm supp}\,\varphi ⊂B2​r​(x),\displaystyle\subset B_{2r}(x)\,,

and such that

r​|∇φ|\displaystyle r|\nabla\varphi| ≤C⁡(n),\displaystyle\leq C(n)\,, (3.7)
r2​|Δ​φ|\displaystyle r^{2}|\Delta\varphi| ≤C⁡(n).\displaystyle\leq C(n)\,.

Now using Bochner’s formula, we get for φ\varphi as above and any constant cc,

C⁡(n)​⨏B2​r​(x)||∇u|2−c|\displaystyle C(n)\fint_{B_{2r}(x)}|\,|\nabla u|^{2}-c| ≥⨏B2​r​(x)|Δφ|⋅|∇u|2−c|\displaystyle\geq\fint_{B_{2r}(x)}|\Delta\varphi|\cdot|\nabla u|^{2}-c| (3.8)
≥|⨏B2​r​(x)φ​Δ​(|∇u|2−c)|\displaystyle\geq\left|\fint_{B_{2r}(x)}\varphi\Delta(|\nabla u|^{2}-c)\right|
=|⨏B2​(x)φ⁡(|∇2u|2+Ric⁡(∇u,∇u))|\displaystyle=\left|\fint_{B_{2}(x)}\varphi(|\nabla^{2}u|^{2}+{\rm Ric}(\nabla u,\nabla u))\right|
≥⨏Br​(x)|∇2u|2−δ​|∇u|2,\displaystyle\geq\fint_{B_{r}(x)}|\nabla^{2}u|^{2}-\delta|\nabla u|^{2}\,,

which implies (3.5).

Part (1) of the following Theorem 1.9, whose statement is recalled below, is actually a sharpening of (3.5).

For every ϵ>0\epsilon>0 there exists δ⁡(n,ϵ)>0\delta(n,\epsilon)>0 such that if RicMn≥−δ{\rm Ric}_{M^{n}}\geq-\delta with u:B2​(p)→ℝku:B_{2}(p)\to\mathds{R}^{k} a δ\delta-splitting map, then the following hold:

  1. (1)

    There exists α⁡(n)>0\alpha(n)>0 such that for each 1≤a≤k1\leq a\leq k,

    ⨏B3/2​(p)|∇2ua|2|∇ua|1+α<ϵ.\displaystyle\fint_{B_{3/2}(p)}\frac{|\nabla^{2}u^{a}|^{2}}{|\nabla u^{a}|^{1+\alpha}}<\epsilon\,. (3.9)
  2. (2)

    Let ωℓ≡d​u1∧⋯∧d​uℓ\omega^{\ell}\equiv du^{1}\wedge\cdots\wedge du^{\ell}, 1≤ℓ≤k1\leq\ell\leq k. Then

    ⨏B3/2​(p)|Δ​|ωℓ||<ϵ.\displaystyle\fint_{B_{3/2}(p)}\big|\Delta|\omega^{\ell}|\big|<\epsilon\,. (3.10)
Proof of Theorem 1.9.

We begin with the proof of (3.9).

The Bochner formula for |∇u|1−α|\nabla u|^{1-\alpha} is given by

Δ​|∇u|1−α=(1−α)​(|∇2u|2−(1+α)​|∇|∇u||2+Ric⁡(∇u,∇u))|∇u|1+α.\displaystyle\Delta|\nabla u|^{1-\alpha}=(1-\alpha)\frac{\Big(|\nabla^{2}u|^{2}-(1+\alpha)|\nabla|\nabla u||^{2}+{\rm Ric}(\nabla u,\nabla u)\Big)}{|\nabla u|^{1+\alpha}}\,. (3.11)

To estimate the left hand side, we observe that since trace⁡(∇2u)=Δ​u=0{\rm trace}(\nabla^{2}u)=\Delta u=0, if follows if λ1,…,λn\lambda_{1},\ldots,\lambda_{n} are the eigenvalues of ∇2u\nabla^{2}u then ∑λi=0\sum\lambda_{i}=0. In particular, if λn\lambda_{n} is the largest eigenvalue then by the Schwarz inequality,

λ12+⋯+λn2≥1n−1​(λ1+⋯+λn−1)2+λn2≥nn−1​λn2.\displaystyle\lambda_{1}^{2}+\cdots+\lambda^{2}_{n}\geq\frac{1}{n-1}(\lambda_{1}+\cdots+\lambda_{n-1})^{2}+\lambda_{n}^{2}\geq\frac{n}{n-1}\lambda_{n}^{2}\,. (3.12)

This leads to the improved Kato inequality

|∇2u​(v)|2≤(1−1n)​|∇2u|2,\displaystyle|\nabla^{2}u({\rm v})|^{2}\leq\big(1-\frac{1}{n}\big)|\nabla^{2}u|^{2}\,, (3.13)

where v{\rm v} is any vector with |v|=1|{\rm v}|=1.

If rewrite

|∇|∇u||=|∇2u​(∇u∇u)|,\displaystyle|\nabla|\nabla u||=\big|\nabla^{2}u\Big(\frac{\nabla u}{\nabla u}\Big)\big|\,, (3.14)

and apply the improved Kato inequality, then we get the Bochner formula

Δ​|∇u|1−α≥1−(n−1)​αn​|∇2u|2|∇u|1+α−(1−α)​δ​|∇u|1−α.\displaystyle\Delta|\nabla u|^{1-\alpha}\geq\frac{1-(n-1)\alpha}{n}\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}}-(1-\alpha)\delta|\nabla u|^{1-\alpha}\,. (3.15)

which gives nontrivial information for any α<1n−1\alpha<\frac{1}{n-1}. Namely,

|∇2u|2|∇u|1+α≤C⁡(n,α)​(Δ​|∇u|1−α+δ​|∇u|1−α).\displaystyle\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}}\leq C(n,\alpha)\Big(\Delta|\nabla u|^{1-\alpha}+\delta|\nabla u|^{1-\alpha}\Big)\,. (3.16)

As in the proof of (3.5), let φ≥0\varphi\geq 0 be a smooth function as in [ChCo1], with supp​φ⊂B2​(p){\rm supp}\,\varphi\subset B_{2}(p), |∇ϕ|,|Δ​ϕ|≤C⁡(n)|\nabla\phi|,|\Delta\phi|\leq C(n).

By multiplying both sides of (3.16) by φ\varphi and integrating we obtain

∫φ​|∇2u|2|∇u|1+α\displaystyle\int\varphi\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}} ≤C⁡(n,α)​∫(φ​Δ​|∇u|1−α+φ​δ​|∇u|1−α),\displaystyle\leq C(n,\alpha)\int\Big(\varphi\Delta|\nabla u|^{1-\alpha}+\varphi\delta|\nabla u|^{1-\alpha}\Big)\,,
≤C⁡(n,α)​∫Δ​φ​(|∇u|1−α−⨏B2​(p)|∇u|1−α)+C⁡(n,α)​δ​∫B2​(p)|∇u|1−α,\displaystyle\leq C(n,\alpha)\int\Delta\varphi\Big(|\nabla u|^{1-\alpha}-\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\Big)+C(n,\alpha)\delta\int_{B_{2}(p)}|\nabla u|^{1-\alpha}\,,
≤C⁡(n,α)​∫B2​(p)||∇u|1−α−⨏B2​(p)|∇u|1−α|+C⁡(n,α)​δ​∫B2​(p)|∇u|1−α,\displaystyle\leq C(n,\alpha)\int_{B_{2}(p)}\Big||\nabla u|^{1-\alpha}-\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\Big|+C(n,\alpha)\delta\int_{B_{2}(p)}|\nabla u|^{1-\alpha}\,, (3.17)

Now we use that if uu is a harmonic δ\delta-splitting map then |∇u|1−α|\nabla u|^{1-\alpha} is bounded and ⨏B2​(p)||∇u|1−α−⨏B2​(p)|∇u|1−α|\fint_{B_{2}(p)}\Big||\nabla u|^{1-\alpha}-\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\Big| is small. In particular, for δ\delta sufficiently small, we have

⨏B3/2​(p)|∇2u|2|∇u|1+α\displaystyle\fint_{B_{3/2}(p)}\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}} ≤C⁡(n)​⨏B2​(p)φ​|∇2u|2|∇u|1+α\displaystyle\leq C(n)\fint_{B_{2}(p)}\varphi\frac{|\nabla^{2}u|^{2}}{|\nabla u|^{1+\alpha}}
≤C⁡(n,α)​⨏B2​(p)||∇u|1−α−⨏B2​(p)|∇u|1−α|+C⁡(n,α)​δ​⨏B2​(p)|∇u|1−α≤ϵ,\displaystyle\leq C(n,\alpha)\fint_{B_{2}(p)}\Big||\nabla u|^{1-\alpha}-\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\Big|+C(n,\alpha)\delta\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\leq\epsilon\,, (3.18)

which proves (3.9).

Now we proceed with the proof of (3.10).

We begin with some computations. Given ωℓ=d​u1∧⋯∧d​uℓ\omega^{\ell}=du^{1}\wedge\cdots\wedge du^{\ell} we have

Δωℓ=∑bdu1∧⋯Ric(dua)∧⋯∧duℓ+∑a,bdu1∧∇j(dua)∧⋯∧∇j(dub)∧⋯∧duℓ,\displaystyle\Delta\omega^{\ell}=\sum_{b}du^{1}\wedge\cdots{\rm Ric}(du^{a})\wedge\cdots\wedge du^{\ell}+\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell}\,,
Δ|ωℓ|=|∇ωℓ|2−|∇|ωℓ||2|ωℓ|+⟨∑bdu1∧⋯Ric(dua)∧⋯∧duℓ,ωℓ|ωℓ|⟩\displaystyle\Delta|\omega^{\ell}|=\frac{|\nabla\omega^{\ell}|^{2}-|\nabla|\omega^{\ell}||^{2}}{|\omega^{\ell}|}+\langle\sum_{b}du^{1}\wedge\cdots{\rm Ric}(du^{a})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle
+⟨∑a,bd​u1∧∇j(d​ua)∧⋯∧∇j(d​ub)∧⋯∧d​uℓ,ωℓ|ωℓ|⟩.\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,+\langle\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle\,. (3.19)

In particular, if uu is a δ\delta-splitting map then

Δ​|ωℓ|−⟨∑a,bd​u1∧∇j(d​ua)∧⋯∧∇j(d​ub)∧⋯∧d​uℓ,ωℓ|ωℓ|⟩+C⁡(n)​δ|ωℓ|≥0,\displaystyle\Delta|\omega^{\ell}|-\langle\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle+C(n)\delta|\omega^{\ell}|\geq 0\,, (3.20)

As in the proof of (3.5), let φ≥0\varphi\geq 0 be a smooth function as in [ChCo1], with supp​φ⊂B2​(p){\rm supp}\,\varphi\subset B_{2}(p), |∇φ|,|Δ​ϕ|≤C⁡(n)|\nabla\varphi|,|\Delta\phi|\leq C(n). Then we have

⨏B2​(p)φ|\displaystyle\fint_{B_{2}(p)}\varphi\Big| Δ|ωℓ​|−⟨∑a,bd​ua1∧∇j(d​uaa)∧⋯∧∇j(d​uab)∧⋯∧d​uak,ωℓ|ωℓ|⟩+C⁡(n)​δ​|ωℓ||\displaystyle\Delta|\omega^{\ell}|-\langle\sum_{a,b}du_{a_{1}}\wedge\nabla^{j}(du_{a_{a}})\wedge\cdots\wedge\nabla_{j}(du_{a_{b}})\wedge\cdots\wedge du_{a_{k}},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle+C(n)\delta|\omega^{\ell}|\Big|
=⨏B2​(p)φ⁡(Δ​|ωℓ|−⟨∑a,bd​ua1∧∇j(d​uaa)∧⋯∧∇j(d​uab)∧⋯∧d​uak,ωℓ|ωℓ|⟩+C⁡(n)​δ​|ωℓ|)\displaystyle=\fint_{B_{2}(p)}\varphi\Big(\Delta|\omega^{\ell}|-\langle\sum_{a,b}du_{a_{1}}\wedge\nabla^{j}(du_{a_{a}})\wedge\cdots\wedge\nabla_{j}(du_{a_{b}})\wedge\cdots\wedge du_{a_{k}},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle+C(n)\delta|\omega^{\ell}|\Big)
≤⨏B2​(p)Δ​φ​(|ωℓ|−1)+∑⨏B2​(p)|∇2uaj|2+C​δ​⨏B2​(p)|ωℓ|,\displaystyle\leq\fint_{B_{2}(p)}\Delta\varphi\Big(|\omega^{\ell}|-1\Big)+\sum\fint_{B_{2}(p)}|\nabla^{2}u_{a_{j}}|^{2}+C\delta\fint_{B_{2}(p)}|\omega^{\ell}|\,,
≤C⁡(n)​⨏||ωℓ|−1|+C​δ<ϵ3,\displaystyle\leq C(n)\fint\big||\omega^{\ell}|-1\big|+C\delta<\frac{\epsilon}{3}\,, (3.21)

if δ⁡(n,ϵ)\delta(n,\epsilon) is sufficiently small. Thus, if δ=δ⁡(n,ϵ)\delta=\delta(n,\epsilon) is sufficiently small,

⨏B2​(p)\displaystyle\fint_{B_{2}(p)} φ|Δ​|ωℓ||≤⨏B2​(p)ϕ​|Δ​|ωℓ​|−⟨∑a,bd​u1∧∇j(d​ua)∧⋯∧∇j(d​ub)∧⋯∧d​uℓ,ωℓ|ωℓ|⟩+C⁡(n)​δ|​ωℓ||\displaystyle\varphi\Big|\Delta|\omega^{\ell}|\Big|\leq\fint_{B_{2}(p)}\phi\Big|\Delta|\omega^{\ell}|-\langle\sum_{a,b}du^{1}\wedge\nabla^{j}(du^{a})\wedge\cdots\wedge\nabla_{j}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle+C(n)\delta|\omega^{\ell}|\Big|
+⨏B2​(p)φ|⟨∑a,bdu1∧∇c(dua)∧⋯∧∇c(dub)∧⋯∧duℓ,ωℓ|ωℓ|⟩|+C(n)δ⨏B2​(p)φ|ωℓ|,\displaystyle+\fint_{B_{2}(p)}\varphi\Big|\langle\sum_{a,b}du^{1}\wedge\nabla^{c}(du^{a})\wedge\cdots\wedge\nabla_{c}(du^{b})\wedge\cdots\wedge du^{\ell},\frac{\omega^{\ell}}{|\omega^{\ell}|}\rangle\Big|+C(n)\delta\fint_{B_{2}(p)}\varphi|\omega^{\ell}|\,,
≤ϵ3+C​∑⨏B2​(p)|∇2ua|2+C​δ​⨏B2​(p)|ωℓ|.\displaystyle\leq\frac{\epsilon}{3}+C\sum\fint_{B_{2}(p)}|\nabla^{2}u^{a}|^{2}+C\delta\fint_{B_{2}(p)}|\omega^{\ell}|\,.
≤ϵ,\displaystyle\leq\epsilon\,, (3.22)

This completes the proof of (3.10). ∎

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