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2.3 L1L^{1}-stability estimate

In complex pluripotential theory, an L1L^{1}-stability estimate is an assertion about the C0C^{0}-closeness of two Kähler potentials given that their volume densities are close in L1L^{1}. We now adapt an argument of Kolodziej [30] to prove a uniform version which allows the complex structure to be highly degenerate and the volume to collapse. Our formulation also brings out the asymmetrical role of the two Kähler potentials; in fact one of them is set to zero. We do not pursue optimality.

009J

Lemma 2.4. (Comparison principle)[30, Thm. 2.1] If uu and vv are ω\omega-psh on YY, then on Ω={u<v}\Omega=\{u<v\}, we have

∫Ωωun≥∫Ωωvn.\int_{\Omega}\omega_{u}^{n}\geq\int_{\Omega}\omega_{v}^{n}.
009K

Lemma 2.5. (Concavity of d​e​t1/ndet^{1/n}) On an open domain, suppose u,vu,v are continuous ω\omega-psh functions, with

ωun=f​ωn,ωvn=g​ωn\omega_{u}^{n}=f\omega^{n},\quad\omega_{v}^{n}=g\omega^{n}

for f,g∈L∞f,g\in L^{\infty}. Then for 0<s<10<s<1, we have ωs​u+(1−s)​vn≥(s​f1/n+(1−s)​g1/n)n​ωn\omega_{su+(1-s)v}^{n}\geq(sf^{1/n}+(1-s)g^{1/n})^{n}\omega^{n}.

009L

Proof. In the smooth case this is a pointwise inequality expressing the concavity of A↦det1/nAA\mapsto\det^{1/n}A on the set of Hermitian matrices. In general one shows this by an approximation argument [30, Lemma 1.2]. ∎

009M

Theorem 2.6. (Uniform L1L^{1}-stability) Let (Y,ω)(Y,\omega) be a compact Kähler manifold, and ϕ∈P​S​H​(Y,ω)∩C0\phi\in PSH(Y,\omega)\cap C^{0}, satisfying the complex MA equations

ωnVol​(Y)=d​μ,ωϕnVol​(Y)=d​ν\frac{\omega^{n}}{\text{Vol}(Y)}=d\mu,\quad\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}=d\nu

for probability measures d​μd\mu and d​νd\nu. Assume

  • •

    There is a Skoda estimate

    ∫Ye−α​u​𝑑μ≤A,∀u∈P​S​H​(Y,ω)​ with ​supYu=0.\int_{Y}e^{-\alpha u}d\mu\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.
  • •

    The complement of E0={ϕ>0}E_{0}=\{\phi>0\} has a mass lower bound

    ∫E0c𝑑μ≥λ>0.\int_{E_{0}^{c}}d\mu\geq\lambda>0.
  • •

    (L1L^{1}-stability assumption) The total variation ∫Y|𝑑μ−𝑑ν|≤s2​n+3<1\int_{Y}|d\mu-d\nu|\leq s^{2n+3}<1.

  • •

    ϕ\phi is smooth away from a (possibly empty) closed subset SS with d​μd\mu-measure zero. Globally ‖ϕ‖C0≤A′\left\lVert\phi\right\rVert_{C^{0}}\leq A^{\prime}.

Then for 0<s<s0​(λ,n,α,A,A′)≪10<s<s_{0}(\lambda,n,\alpha,A,A^{\prime})\ll 1, there is a uniform estimate

supYϕ≤C⁡(λ,n,α,A,A′)​s.\sup_{Y}\phi\leq C(\lambda,n,\alpha,A,A^{\prime})s.
009N

Proof. We can reduce to the case with d​μ​(E0)≥12d\mu(E_{0})\geq\frac{1}{2} by shifting ϕ\phi by a constant. We construct an auxiliary continuous ω\omega-psh function ρ\rho by solving the complex MA equation with L∞L^{\infty}-density [15]

ωρnVol​(Y)=1d​μ​(E0)​d​μ  E0,supYρ=0.\frac{\omega_{\rho}^{n}}{\text{Vol}(Y)}=\frac{1}{d\mu(E_{0})}d\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{0},\quad\sup_{Y}\rho=0.

where d​μ  E0​(F)=d​μ​(E∩F)d\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{0}(F)=d\mu(E\cap F) is the restricted measure. By Theorem 2.2 we have ‖ρ‖C0≤C⁡(n,α,A)\left\lVert\rho\right\rVert_{C^{0}}\leq C(n,\alpha,A), since the RHS measure satisfies a Skoda estimate. We choose a=max⁡(C⁡(n,α,A),A′)a=\max(C(n,\alpha,A),A^{\prime}), so that

−a≤ρ≤0,ϕ≥−a,-a\leq\rho\leq 0,\quad\phi\geq-a,

implying the set inclusion

E′={ϕ>2as+smaxϕ}⊂E={(1−s)ϕ+sρ−as>0}⊂E0.E^{\prime}=\{\phi>2as+s\max\phi\}\subset E=\{(1-s)\phi+s\rho-as>0\}\subset E_{0}.

Our next goal is to show d​μ​(E′)d\mu(E^{\prime}) is small.

Let G={1−s2≥d​νd​μ}⊂Y∖SG=\{1-s^{2}\geq\frac{d\nu}{d\mu}\}\subset Y\setminus S. On the open set E0∖(S∪G)E_{0}\setminus(S\cup G), by the concavity Lemma 2.5,

ωs​ρ+(1−s)​ϕnVol​(Y)≥(sdμ(E0)−1/n+(1−s)(1−s2)1/n)ndμ.\frac{\omega_{s\rho+(1-s)\phi}^{n}}{\text{Vol}(Y)}\geq\left(sd\mu(E_{0})^{-1/n}+(1-s)(1-s^{2})^{1/n}\right)^{n}d\mu.

Now dμ(E0)−1/n≥(1−λ)−1/nd\mu(E_{0})^{-1/n}\geq(1-\lambda)^{-1/n} by assumption. Choose 0<q<n{(1−λ)−1/n−1}0<q<n\{(1-\lambda)^{-1/n}-1\}, so for 0<s≪10<s\ll 1 depending on λ,n\lambda,n, by Taylor expansion in ss,

(sdμ(E0)−1/n+(1−s)(1−s2)1/n)n≥1+qs.\left(sd\mu(E_{0})^{-1/n}+(1-s)(1-s^{2})^{1/n}\right)^{n}\geq 1+qs.

Combining this with the comparison principle Lemma 2.4, and the assumption d​μ​(S)=0d\mu(S)=0,

(1+q​s)​∫E∖G𝑑μ≤∫Eωs​ρ+(1−s)​ϕnVol​(Y)≤∫EωnVol​(Y)=∫E𝑑μ.(1+qs)\int_{E\setminus G}d\mu\leq\int_{E}\frac{\omega_{s\rho+(1-s)\phi}^{n}}{\text{Vol}(Y)}\leq\int_{E}\frac{\omega^{n}}{\text{Vol}(Y)}=\int_{E}d\mu.

On the other hand, by the definition of GG and the L1L^{1}-stability assumption,

d​μ​(G)≤s−2​∫G(𝑑μ−𝑑ν)≤s−2​∫Y|𝑑μ−𝑑ν|≤s2​n+1,d\mu(G)\leq s^{-2}\int_{G}(d\mu-d\nu)\leq s^{-2}\int_{Y}|d\mu-d\nu|\leq s^{2n+1},

hence

(1+q​s)​∫E∖G𝑑μ≤∫E∖G𝑑μ+s2​n+1.(1+qs)\int_{E\setminus G}d\mu\leq\int_{E\setminus G}d\mu+s^{2n+1}.

We conclude ∫E∖G𝑑μ≤q−1​s2​n\int_{E\setminus G}d\mu\leq q^{-1}s^{2n}, so

∫E′𝑑μ≤∫E𝑑μ≤q−1​s2​n+s2​n+1≤(q−1+1)​s2​n.\int_{E^{\prime}}d\mu\leq\int_{E}d\mu\leq q^{-1}s^{2n}+s^{2n+1}\leq(q^{-1}+1)s^{2n}.

We now apply the stability estimate Cor. 2.3 to compare the potentials ϕ\phi and 00, to see for ss sufficiently small depending on n,A,A′,α,λn,A,A^{\prime},\alpha,\lambda,

minY⁡(−ϕ)≥−2​a​s−s​maxY​ϕ−4​B​(∫E′𝑑μ)1/2​n≥−2​a​s−s​maxY​ϕ−4​B​(1+q−1)1/2​n​s,\min_{Y}(-\phi)\geq-2as-s\max_{Y}\phi-4B(\int_{E^{\prime}}d\mu)^{1/2n}\geq-2as-s\max_{Y}\phi-4B(1+q^{-1})^{1/2n}s,

whence maxY⁡ϕ≤C⁡(n,A,A′,α,λ)​s\max_{Y}\phi\leq C(n,A,A^{\prime},\alpha,\lambda)s as required. ∎

009P

Remark 2.7. It is not clear to the author why in Kolodziej’s original argument [30, Lemma 1.2] applies in the proof of [30, Thm. 4.1], as E0∖GE_{0}\setminus G is not an open domain if one considers general LpL^{p}-densities.

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