Some aspects of Ohsawa-Takegoshi type theorems

Some aspects of Ohsawa-Takegoshitype theorems

Bo-Yong Chen

KIAS, April 23–27, 2018

Prologue

We first recall the fundamentalTheorem 1.1 (Hormander 1965)

Ω ⊂ Cn : pseudoconvex, ϕ ∈ PSH(Ω) s.t.

i∂∂ϕ ≥ Θ,

Θ : Kahler form on Ω. ⇒ ∀ ∂−closed (0, 1) form vwith

∫Ω|v|2

Θe−ϕ < ∞, ∃ a solution u of ∂u = v s.t.∫

Ω

|u|2e−ϕ ≤∫

Ω

|v|2Θe−ϕ.

Notion: |v|2Θ

=∑

Θjkvjvk where v =∑

vjdzj,Θ = i

∑Θjkdzj ∧ dzk and (Θjk) = (Θjk)−1.

Prologue

In case when Θ is only semipositive, we needto define |v|Θ via duality

|v|Θ = sup|〈v,X〉| : X ∈ T0,1(Ω), |X|Θ ≤ 1

where for X =

∑j Xj∂/∂zj,

|X|2Θ :=∑j,k

ΘjkXjXk.

For general ϕ ∈ PSH(Ω), Blocki suggested to view|v|2

i∂∂ϕas the infimum of all 0 ≤ H ∈ L∞loc s.t.

iv ∧ v ≤ Hi∂∂ϕ.

Prologue

Among numerous applications of Theorem 1.1,the following one is truly deep.

Theorem 1.2 (Bombieri 1970)

ϕ ∈ PSH(Bn), not identically −∞ ⇒ the set E ofpoints in every neighborhood of which e−ϕ is notintegrable is analytic subset of Bn.

Proof. Set

A2ϕ =

f ∈ O(Bn) :

∫Bn|f |2e−ϕ < ∞

and S =

⋂f∈A2

ϕf −1(0). It suffices to verify E = S.

PrologueE ⊂ S : a ∈ E⇒ f (a) = 0, ∀ f ∈ A2

ϕ.S ⊂ E : for a ∈ Bn\E, choose 0 < ε 1 s.t.

e−ϕ ∈ L1(Bε(a)). Set ϕ = ϕ + |z|2 + 2n log |z − a|,

χ ∈ C∞0 (Bε(a)), χ|B ε2

(a) = 1, |∂χ| ≤ 3ε−1.

By Theorem 1.1, ∃ solution of ∂u = ∂χ s.t.∫Bn|u|2e−ϕ ≤

∫Bn|∂χ|2e−ϕ ≤ Cnε

−2?

Bε(a)e−ϕ.

Thus f := χ − u ∈ O(Bn), f (a) = 1,∫Bn|f |2e−ϕ ≤ Cnε

−2?

Bε(a)e−ϕ. (0.1)

⇒ a < S. Q.E.D.

Prologue

A natural question is whether the term ε−2 in(0.1) can be removed?

A positive answer is essentially the core of theproof of Ohsawa-Takegoshi L2 extension theorem,which will be given later. We first recall

Theorem 1.3 (Demailly 1992)

Km(z) := sup|f (z)|2/‖f ‖2mϕ : f ∈ A2mϕ, i.e. Bergman

kernel of A2mϕ, ϕ ∈ PSH(Bn). Then for ε < d(z, ∂Bn),

C−1n

[?Bε(z)

e−mϕ]−1

≤ Km(z) ≤1

|Bε(z)|

?Bε(z)

emϕ.

(0.2)

Prologue

Proof. Upper bound: f ∈ A2mϕ ⇒

|f (z)| ≤?

Bε(z)|f | ≤

[?Bε(z)

emϕ]1/2 [?

Bε(z)|f |2e−mϕ

]1/2

.

Lower bound: for fixed z, take f ∈ A2mϕ s.t. f (z) = 1,∫

Bn|f |2e−mϕ ≤ Cn

?Bε(z)

e−mϕ.

As Km(z) ≥ |f (z)|2/‖f ‖2mϕ, we are done. Q.E.D.

Corollary 1.4

limm→∞

1m

log Km(z) = ϕ(z). (0.3)

PrologueProof. Note that ζ ∈ Bε(z)⇒ ϕ(ζ) ≤ ϕ(z) + o(1)

as ε→ 0. Letting m→ ∞ and ε→ 0 in (0.2)

lim supm→∞

1m

log Km(z) ≤ ϕ(z).

Choose smooth psh functions ϕj ↓ ϕ. Using (0.2)with ϕ replaced by ϕj and let ε→ 0, we have

1m

log Km,j(z) ≥ ϕj(z) −log Cn

m≥ ϕ(z) −

log Cn

m.

A normal family argument implies Km,j(z) ↓ Km(z)as j→ ∞. Q.E.D.

Theorem 1.3 also implies a surprisingly shortproof of a very deep result of Siu.

Prologue

Theorem 1.5 (Siu 1974)

ϕ ∈ PSH(Bn), ν(ϕ, z) : Lelong number of ϕ at a.

⇒ Ec(ϕ) = z ∈ Bn : ν(ϕ, z) ≥ c

is an analytic subset of Bn, ∀ c > 0.

Proof. Set ϕm = 1m log Km. We have

supBε(z)

ϕ − C1m−1 ≤ supBε(z)

ϕm ≤ supB2ε(z)

ϕ − m−1 log(C2ε2n).

Since ν(ϕ, z) = limε→0supBε(z) ϕ

log ε , we have

ν(ϕ, z) − 2n/m ≤ ν(ϕm, z) ≤ ν(ϕ, z).

Prologue

It follows that

Ec(ϕ) = ∩m≥m0Ec−2n/m(ϕm).

One can show that Ec−2n/m(ϕm) is an analytic set,so is Ec(ϕ). Q.E.D.

The above results showed the power of theOhsawa-Takegoshi extension theorem.

Nevertheless, we shall see that techniques ofproving the OT extension theorem is even moreimpressive.

Ohsawa-Takegoshi extension theorem

Theorem 2.1 (Ohsawa-Takegoshi 1987)

Ω ⊂ Cn : pseudoconvex, H : complex hyperplanes.t. supz∈Ω d(z,H) < ∞. Then for every ϕ ∈ PSH(Ω),every f ∈ O(Ω ∩ H) with

∫Ω∩H |f |

2e−ϕ < ∞, thereexists F ∈ O(Ω) s.t. F|Ω∩H = f and∫

Ω

|F|2e−ϕ ≤ C∫

Ω∩H|f |2e−ϕ

where C depends only on supz∈Ω d(z,H).

Remark. By induction, one gets an analogousextension theorem when Ω is bounded and H is acomplex affine subspace.


The original approach of OT is built on thegeneral framework of Kahler geometry. The mainidea is to use a twisted Bochner-Kodaira-Nakanoinequality, inspired by Donnelly-Fefferman (1983)and Donnelly-Xavier (1984).

By using a twisted Morrey-Kohn-Hormanderinequality, Siu (1996) and Berndtsson (1996) areable to give simplified and more accessible proofs.

Chen (2011) observed that OT can be deriveddirectly from Theorem 1.1. We shall explain thisapproach in detail here.


One may assume H = zn = 0, supΩ |zn|2 < e−1.

Lemma 2.2

Suppose furthermore Ω ⊂⊂ Cn, ∂Ω ∈ C∞, ϕ : C∞

strictly psh on Ω, f ∈ O(V), V : a neighborhood ofΩ ∩ H in H. Then ∃ F ∈ O(Ω) s.t. F|Ω∩H = f and∫

Ω

|F|2e−ϕ ≤ C0

∫V|f |2e−ϕ (0.4)

where C0 is a universal constant.

Lemma 2.2⇒ OT: Choose smooth boundedpseudoconvex domain Ωj ↑ Ω, smooth strictly pshfunction ϕj on Ωj+1 s.t. ϕj ↓ ϕ.

Ohsawa-Takegoshi extension theoremWe have Fj ∈ O(Ωj) s.t. Fj|Ωj∩H = f and∫

Ωj

|Fj|2e−ϕj ≤ C0

∫Ωj+1∩H

|f |2e−ϕj ≤ C0

∫Ω∩H|f |2e−ϕ.

Thus Fj forms a normal family and it suffices tochoose a weak limit of Fj. Q.E.D.

Since for ε 1

(∂V × |zn| < ε) ∩Ω = ∅,

so χ(|zn|2/ε2)f gives a smooth extension of f on Ω,

where χ : R→ [0, 1] s.t. χ|[1,∞) = 0, χ|(−∞,1/2] = 1.In order that F = χ(|zn|

2/ε2)f − u is a holomorphicL2ϕ extension of f , it suffices to solve


∂u = ∂[χ(|zn|2/ε2)f ] =: vε, u|Ω∩H = 0∫

Ω

|u|2e−ϕ ≤ C0

∫V|f |2e−ϕ.

Similar as the proof of the Bombieri theorem, oneneeds to estimate integral of type∫

Ω

|vε|2Θe−ϕ, provided i∂∂ϕ ≥ Θ.

If ϕ = ϕ + |zn|2 + 2 log |zn| and Θ = idzn ∧ dzn, then∫Ω

|vε|2Θe−ϕ ≤ C0ε−2

∫V|f |2e−ϕ.

To get rid of the term ε−2, one needs to replace |zn|2

by some ψ(zn) with slow growth and large Hessian


near zn = 0, e.g. Poincare type potentials:

ψ(zn) = −r log[− log(ε2 + |zn|2)], r > 0

i∂∂ψ =ε2| log(ε2 + |zn|

2)| + |zn|2

(ε2 + |zn|2)2| log(ε2 + |zn|

2)|2ridzn∧dzn (0.5)

In this case, we have for ε 1∫Ω

|vε|2i∂∂ψe−ϕ ≤ Cr| log ε|1+r∫

V|f |2e−ϕ,

which is useless unless r = −1! Thus we have todeal with L2−estimates with weight φ − ψ, whereφ, ψ are psh. Such estimates originated from thework of Donnelly-Fefferman (1983).


Lemma 2.3

φ, ψ ∈ C2(Ω) s.t. φ + ψ is psh on Ω, ϕ ∈ PSH(Ω)⇒the L2

φ+ϕ(Ω) minimal solution of ∂u = v satisfies∫Ω

|u|2eψ−φ−ϕ[1 − |∂ψ|2i∂∂(φ+ψ) − κχsupp v|∂ψ|

2i∂∂(φ+ψ)

]≤

∫Ω

(1 + κ−1)|v|2i∂∂(φ+ψ)eψ−φ−ϕ (0.6)

for every continuous function κ ≥ 0 on Ω.

Proof. We use a trick due to Berndtsson. Sinceu⊥A2

φ+ϕ(Ω), so ueψ⊥A2φ+ψ+ϕ. Applying Theorem 1.1

with Θ = i∂∂(φ + ψ), we have


∫Ω

|u|2eψ−φ−ϕ ≤∫

Ω

|∂(ueψ)|2i∂∂(φ+ψ)e−ψ−φ−ϕ

≤

∫Ω

(1 + κ−1)|v|2i∂∂(φ+ψ)eψ−φ−ϕ

+

∫Ω

|∂ψ|2i∂∂(φ+ψ)|u|2eψ−φ−ϕ

+

∫supp v

κ|∂ψ|2i∂∂(φ+ψ)|u|2eψ−φ−ϕ Q.E.D.

We apply Lemma 2.3 with κ ≡ r, φ = 0 and ϕreplaced by ϕ + 2 log |zn|. The key is choice of ψ.

Ohsawa-Takegoshi extension theoremAs I mentioned before, a natural choice is

ψ = − log[− log(|zn|

2 + ε2)].

But (0.5) would imply that

1 − |∂ψ|2i∂∂ψ =ε2| log(ε2 + |zn|

2)||zn|

2 + ε2| log(ε2 + |zn|2)|

so that the LHS of (0.6) is bounded above by∫Ω

|u|2e−ϕ ·ε2

|zn|2[|zn|

2 + ε2| log(ε2 + |zn|2)|]

⇒ LHS of (0.6) can not be bounded below by

const.∫

Ω

|u|2e−ϕ!

Ohsawa-Takegoshi extension theoremThus one needs to modify the choice of ψ, e.g.

ψ = − log η

where

η = −ρ + log(−ρ), ρ = log(|zn|2 + ε2).

Then

i∂∂ψ =i∂∂ρη

+i∂∂[− log(−ρ)]

η+

i∂η ∧ ∂ηη2

⇒ i∂∂ψ ≥[1 +

η

(−ρ + 1)2

]i∂η ∧ ∂η

η2

⇒ |∂ψ|2i∂∂ψ ≤1

1 +η

(−ρ+1)2

.


i∂∂ψ ≥i∂∂ρη

=ε2idzn ∧ dzn

η(|zn|2 + ε2)2 ,

⇒ |∂ψ|2i∂∂ψ ≤4η

on supp vε, for ε 1.

and ⇒

∫Ω

|vε|2i∂∂ψeψ−ϕ−2 log |zn|

≤

∫ε2/2≤|zn|

2≤ε2|χ′|2|zn|

2

ε4

η(|zn|2 + ε2)2

ε2

|f |2

η|zn|2 e−ϕ

≤ C0

∫V|f |2e−ϕ, for ε 1.


By (0.6), we have∫Ω

η(−ρ+1)2

1 +η

(−ρ+1)2

−4rη

|u|2|zn|

2 eψ−ϕ ≤ (1+r−1)C0

∫V|f |2e−ϕ.

Since η ∼ −ρ (ε→ 0), we may choose r 1 s.t.

LHS ≥ c0

∫Ω

|u|2e−ϕ

|zn|2[log(ε2 + |zn|

2)]2 .

In order to get the extension, it suffices to takea weak limit of

Fε = χ(|z|2/ε2)f − u. Q.E.D.

Ohsawa-Takegoshi extension theoremWe conclude this section by giving a lovely

consequence of OT.

Corollary 2.4

Ω ⊂⊂ Cn : pseudoconvex, ∂Ω ∈ Lip⇒

KΩ(z) ≥ const.δ(z)−2,

where KΩ : Bergman kernel, δ(z) = d(z, ∂Ω).

Proof. For (fixed) z close to ∂Ω, ∃ a cone Λ withvertex at z∗ ∈ ∂Ω and apical angel α0 > 0 s.t. z lieson the axis (⇒ |z − z∗| δ(z)). One can show

KΩ∩H(z) ≥ const.|z − z∗|−2.


OT⇒ ∀ f ∈ A2(Ω ∩ H), ∃ F ∈ A2(Ω) s.t.F(z) = f (z) and ‖F‖L2 ≤ const.‖f ‖L2. Thus

KΩ(z) ≥|F(z)|2

‖F‖2L2

≥ const.|f (z)|2

‖f ‖2L2

,

so that KΩ(z) ≥ const.KΩ∩H(z). Q.E.D.

Problem 2.5

Does one have SΩ(z) ≥ const.δ(z)−1?

Here SΩ is the Szego kernel, i.e. reproducingkernel of the Hardy space H2(Ω). It is even notknown where SΩ(z) is an exhaustion function.

Ohsawa-Takegoshi type theorems

After the paper of OT, a number of variations,refinements and generalisations appeared. Wecall them OT type theorems.

Here we shall focus on domains in Cn, andhave to miss the rather important generalisationsof OT to complex manifolds.

In particular, I could not introduce Siu’s proof(simplified by Paun) of invariance of plurigenera,which from my viewpoint is the most significantapplication of OT.

We refer the audience to those beautifularticles/monographs of Demailly/Berndtsson.


Theorem 3.1 can also be used to improve theestimate in Corollary 2.4 for domains of finite type.

Corollary 3.2

Ω ⊂ Cn : bounded smooth pseudoconvex domainof finite type⇒ ∃ α > 0 s.t.

KΩ(z) ≥ const.δ(z)−2−α.

A major subject in SCV is to understand theboundary behavior of the Bergman kernel forsmooth pseudoconvex domains.


Theorem 3.3 (Ohsawa 2002)

Ω ⊂ Cn : bounded, strongly pseudoconvex, H :complex hyperplane intersecting ∂Ω transversally.Then ∃ C > 0 s.t. ∀ϕ ∈ PSH(Ω) which extendscontinuously to ∂Ω and f ∈ O(Ω ∩ H) with∫

Ω∩H|f |2e−ϕ < ∞,

∃ F ∈ H2(Ω) s.t. F|Ω∩H = f and∫∂Ω

|F|2e−ϕ ≤ C∫

Ω∩H|f |2e−ϕ.

Ohsawa-Takegoshi type theoremsAlthough the result above is quite impressive, it

brought nothing new on the Szego kernel, since itonly implies that if z ∈ Ω ∩ H then

SΩ(z) ≥ const.KΩ∩H(z) ≥ const.δ(z)−n,

which is well-known.Problem 3.4

Ω ⊂ Cn : bounded, smooth pseudoconvex, H :complex hyperplane intersecting ∂Ω transversally.Does there exist C > 0 s.t. ∀ f ∈ H2(Ω ∩ H),∃F ∈ H2(Ω) s.t.∫

∂Ω

|F|2 ≤ C∫∂(Ω∩H)

|f |2?

Ohsawa-Takegoshi type theoremsIn April 2, 2012, many experts of SCV received

an e-mail from Zbigniew Blocki, who proved thefollowing version of OT with optimal constant, in apreprint posed at his homepage.

Theorem 3.5

Ω ⊂ Cn−1 × D : pseudoconvex, 0 ∈ D ⊂⊂ C⇒∀ϕ ∈ PSH(Ω) and f ∈ O(Ω′) with

∫Ω′|f |2e−ϕ < ∞,

where Ω′ = Ω ∩ zn = 0, ∃ F ∈ O(Ω) s.t. F|Ω′ = f ,∫Ω

|F|2e−ϕ ≤π

(cD(0))2

∫Ω′|f |2e−ϕ.

Here cD(z) = exp[limw→z(gD(w, z) − log |w − z|)],gD : Green function of D.

Ohsawa-Takegoshi type theoremsA direct consequence of Theorem 3.5 is the

following result conjectured by Suita (1972).

Corollary 3.6

D ⊂ C : bounded domain⇒

c2D ≤ πKD. (0.7)

Proof. Theorem 3.5⇒ given z ∈ D, ∃ f ∈ O(D)s.t. f (z) = 1 and∫

D|f |2 ≤

π

cD(z)2 . Q.E.D.

The relation between Suita’s conjecture andOT was first noticed by Ohsawa (1995).


Now we discuss about Blocki’s proof. His newL2−estimate for the ∂−equation is as follows.

Lemma 3.7

ϕ ∈ PSH(Ω), φ, ψ ∈ C2(Ω) with φ ∈ PSH(Ω) s.t.

|∂ψ|2i∂∂φ < 1, |∂ψ|2i∂∂φ ≤ γ < 1 on supp v.

Then ∃ a solution of ∂u = v s.t.∫Ω

|u|2e2ψ−φ−ϕ[1 − |∂ψ|2i∂∂φ

]≤

1 +√γ

1 −√γ

∫Ω

|v|2i∂∂φe2ψ−φ−ϕ.

Proof. With φ replaced by φ − ψ in Lemma 2.3,we have

Ohsawa-Takegoshi type theorems∫Ω

|u|2e2ψ−φ−ϕ[1 − |∂ψ|2i∂∂φ − κχsupp v|∂ψ|

2i∂∂φ

]≤

∫Ω

(1 + κ−1)|v|2i∂∂φe2ψ−φ−ϕ.

It suffices to choose

κ =1 − |∂ψ|2

i∂∂φ√γ(1 +

√γ). Q.E.D.

Set g := gD(·, 0). We choose

φ = η(−2g), ψ = ζ(−2g)

vε = ∂[fχ(−2g)]

where η ∈ C1,1(R+), ζ ∈ C0,1(R+) and χ ∈ C2(R) s.t.


χ(∞) = 1, χ|(−∞,−2 log ε] = 0 (to be determinedlater). We need that η is convex decreasing s.t.

|∂ψ|2i∂∂φ ≤(ζ′)2

η′′ (−2g). (0.8)

To apply Lemma 3.7 with ϕ replaced by ϕ + 2g, weneed the conditions:

|∂ψ|2i∂∂φ ≤ γ on supp vε (0.9)[1 − |∂ψ|2i∂∂φ

]e2ψ−φ−2g ≥ 1 on Ω (0.10)


for they would imply ∃ a solution of ∂u = vε s.t.∫Ω

|u|2e−ϕ ≤∫

Ω

|u|2[1 − |∂ψ|2i∂∂φ

]e2ψ−φ−2g−ϕ

≤1 +√γ

1 −√γ

∫Ω

|vε|2i∂∂φe2ψ−φ−2g−ϕ.

We need to choose η, ζ, χ satisfying (0.9)∼(0.10),γ = γε → 0 and

lim supε→0

∫Ω

|vε|2i∂∂φe2ψ−φ−2g−ϕ ≤

π

cD(0)2

∫V|f |2e−ϕ.

(0.11)


From (0.8), we see that if[1 −

(ζ′)2

η′′

]e2ζ−η+t ≥ 1,

then (0.10) is satisfied. Thus it suffices to solve thefollowing ODE problem:[

1 −(ζ′)2

η′′

]e2ζ−η+t = 1. (0.12)

With ζ = log(−η′), one reduces (0.12) to

(e−η)′′ = e−t.

Ohsawa-Takegoshi type theoremsOne may choose

η(t) = − log(t + e−t − 1)ζ(t) = − log(t + e−t − 1) + log(1 − e−t).

We need to adjust the definition of η, ζ on the partt : t ≥ M := −2 log ε by simpler ones

η(t) = −γ log(t −M + a) + bζ(t) = −γ log(t −M + a) + c

in order to match (0.9) and (0.11). Then

a = −γ

η′(M), b = η(M) + γ log a, c = ζ(M) + γ log a.


For t ≥ M, one has

(ζ′)2

η′′= γ,

which implies (0.9).We may choose γ := M−1/2 → 0 s.t. a→ ∞ as

ε→ 0. Thus 2ζ − η + t ∼ t as t → +∞, so that fort ≥ M 1, [

1 −(ζ′)2

η′′

]e2ζ−η+t ≥ 1

and (0.10) is still satisfied.


We are left to verify (0.11). Recall that∫Ω

|vε|2i∂∂φe2ψ−φ−2g−ϕ ≤

∫g≤log ε

|f |2e−ϕ[(χ′)2

η′′e2ζ−η+t

](−2g).

Since g(zn) = log |zn| + log cD(0) + o(1) as zn → 0,the RHS is bounded by

(Bε + o(1))[∫

V|f |2e−ϕ + o(1)

]where

Bε :=1

cD(0)2

∫log |zn|≤log ε

[(χ′)2

η′′e2ζ−η+t

] (−2 log |zn|)


Let t = −2 log |zn|. Then

Bε =π

cD(0)2

∫ ∞

M

(χ′)2e2ζ−η

η′′.

Now choose χ s.t. χ|(−∞,M] = 0 and

χ(t) =

∫ t

Mη′′eη−2ζ

[∫ ∞

Mη′′eη−2ζ

]−1

which implies χ(∞) = 1. Thus

Bε =π

cD(0)2

[∫ ∞

Mη′′eη−2ζ

]−1

.


Note that∫ ∞

Mη′′eη−2ζ = γeb−2c

∫ ∞

0(t + a)γ−2

=1

1 − γeη(M)−2ζ(M)+log(−η′(M)).

Since

η(M) − 2ζ(M) + log(−η′(M)) = η(M) − ζ(M)→ 0

as M → ∞, i.e. ε→ 0, it follows that

Bε →π

cD(0)2 . Q.E.D.


After Blocki’s work, Guan-Zhou obtained someresults analogous to Theorem 3.5. Here we onlymention one of them.Theorem 3.8 (Guan-Zhou 2013)

Ω ⊂ Cn : pseudoconvex, ϕ, ψ ∈ PSH(Ω) with

supz∈Ω

[ψ(z) + 2 log |zn|] ≤ 0.

Then ∀ f ∈ O(Ω′) with∫

Ω′|f |2e−ϕ < ∞ where

Ω′ = Ω ∩ zn = 0, ∃ F ∈ O(Ω) s.t. F|Ω′ = f and∫Ω

|F|2e−ϕ+ψ ≤ π

∫Ω′|f |2e−ϕ.

Ohsawa-Takegoshi type theoremsBased on Theorem 3.8, Guan-Zhou found an

interesting new proof of the following

Theorem 3.9 (Berndtsson 2006)

Ω ⊂ Cnz × C

mt : pseudoconvex domain, ϕ ∈ PSH(Ω).

Set Ωt = Ω ∩ w = t, ϕt = ϕ|Ωt. Let Kt denote theBergman kernel of A2

ϕt(Ωt). Then

log Kt(z) ∈ PSH(Ω).

Proof. It suffices to consider the case m = 1.One needs to get a mean value inequality nearany fixed (z0, t0) ∈ Ω. Assume t0 = 0 and set

f (z) = K0(z, z0)/√

K0(z0).

Ohsawa-Takegoshi type theoremsTheorem 3.8 (with ψ := −2 log r)⇒ ∃ F ∈ O(Ωr)

where Ωr := Ω ∩ (z, t) : |t| < r, s.t. F|Ω0 = f and∫Ωr|F|2e−ϕ ≤ |Dr| (0.13)

where Dr = t ∈ C : |t| < r. Since

Kt(z) ≥ |F(z, t)|2/‖F(·, t)‖2L2ϕt (Ωt)

,

it follows from (0.13) and Fubini’s theorem that

1 ≥1|Dr|

∫Dr

‖F(·, t)‖2L2ϕt (Ωt)

≥1|Dr|

∫Dr

|F(z0, t)|2/Kt(z0)

≥ exp[

1|Dr|

∫Dr

log |F(z0, t)|2/Kt(z0)]

in view of Jensen’s inequality.


Together with the mean value inequality, we get

log K0(z0) ≤1|Dr|

∫Dr

log |F(z0, t)|2 ≤1|Dr|

∫Dr

log Kt(z0).

Thus log Kt(z) is subharmonic in t. Q.E.D.

Remark. The crucial inequality (0.13) can alsobe derived from Theorem 3.5, for

cDr(0) = exp[limz→0

(log |z|/r − log |z|)]

= r−1

⇒π

cDr(0)2 = |Dr|.


Berndtsson’ original proof is based on aningenious application of Theorem 1.1, is alreadyvery beautiful.

More surprisingly, a variant of Theorem 3.9was used by Berndtsson-Lempert (2015) to give aentirely new approach of OT with optimal constant.

Another interesting constructive proof wasposed by Ohsawa (2015).


Guan-Zhou (2013) (see also Lempert (2014),Heip (2014)) discovered that OT can be used toprove the following

Theorem 3.10 (Strong Openness Theorem)∫U |f |

2e−ϕ < ∞ for fixed f ∈ O(U) and ϕ ∈ PSH(U)⇒ ∀V ⊂⊂ U, ∃ p > 1 s.t.

∫V |f |

2e−pϕ < ∞.

The special case f = 1 was first proved byBerndtsson (2013) by using Theorem 3.9, whichsolved the openness conjecture of Demailly-Kollar(2001).

Another approach using only Theorem 1.1 wasposed by Chen (2016).

Ohsawa-Takegoshi type theoremsDefinition 3.11

A domain Ω ⊂ Cn is called complete Kahler (CK) ifit admits a complete Kahler metric.

Lemma 3.12

Pseudoconvex⇒ CK.

Proof.

Take a smooth strictly psh exhaustion functionρ ≥ 1, s.t. ds2 = ∂∂ρ2 gives a CK metric.

On the other hand, the Poincare metric

ds2 = −∂∂ log(− log |z|2)

is CK on Bn\0, which is not pseudoconvex ifn ≥ 2.


CK⇒ pseudoconvex under one of the followingconditions

1 ∂Ω ∈ Cω (Grauert ’56).2 ∂Ω ∈ C1 (Ohsawa ’80).3 Ω

= Ω (Diederich-Pflug 81).

A large class of CK domains are given by Ω\E,where Ω ⊂ Cn : pseudoconvex, E ⊂ Ω : closed andcomplete pluripolar, i.e. ∀ a ∈ E, ∃ψ ∈ PSH(U)where U 3 a : neighborhood, s.t.

E ∩ U = ψ−1(−∞).


Problem 3.13 (Ohsawa 1995)

Ω ⊂ Cn : bounded CK, H : complex hyperplane⇒∀ f ∈ A2(Ω ∩ H), ∃F ∈ A2(Ω) s.t. F|Ω∩H = f ?

Problem 3.14

Can the OT type theorems be generalized tobounded CK domains?

The basic difference between CK domains andpseudoconvex domains is that the former can notbe exhausted by relatively compact CK domains ingeneral, so that one can not first solve extensionproblem on relatively compact ones then passingto a weak limit.


Theorem 3.15 (Chen-Wang-Wu 2015)

Ω ⊂ Bn : bounded CK, ϕ ∈ PSH(Ω)⇒ ∀ a ∈ Ω,∀ c ∈ C with |c|2 ≤ eϕ(a), ∃ f ∈ O(Ω) s.t. f (a) = c,∫

Ω

|f |2e−ϕ ≤ Cn.

Remark. Actually, OT can be generalized tobounded CK domains Ω and complex hyperplanesH s.t. Ω is pseudoconvex in a neighborhood of∂Ω ∩ H.

Ohsawa-Takegoshi type theoremsLet us explain briefly the idea of the proof. The

key point is to generalize Theorem 1.1 to boundedCK domains. Then Theorem 3.15 can be derivedsimilarly as the classical OT.

The case of smooth weight ϕ is only a specialcase of the general theory of Andreotti-Vesentini(1965). For general ϕ ∈ PSH(Ω), one has smoothpsh functions ϕj ↓ ϕ, but only locally on Ω. ThusHormander’s approximation procedure breaksdown.

The difficulty was overcome by Demailly in1982, who used a rather involved approximationargument, however.

Ohsawa-Takegoshi type theoremsA more transparent approach was posed by

Chen-Wang-Wu (2015) as follows.We exhaust Ω by a sequence Ωj of bounded

subdomains with smooth boundaries, and takeϕj ∈ SPSH(Ωj) s.t. ϕj ↓ ϕ, i∂∂ϕj ≥ Θ.

By using Friedrichs mollifier, we have smooth(0, 1) form vj on Ω s.t.∫

Ω

|∂vj|2ωe−ϕ,

∫Ω

|vj − v|2Θe−ϕ → 0

where ω ≥ Θ is a CK metric. Instead of solving

∂uj = vj

on Ωj, we solve the Laplace-Beltrami equation


ϕjw = vj := dz ∧ vj

on Ωj s.t.

‖∂wj‖2ϕj

+ ‖∂∗ϕjwj‖

2ϕj≤

∫Ω

|vj|2Θe−ϕ.

where ϕj is the Laplace operator w.r.t. weight ϕj

and metric ω. The (unique) solution exists by theBochner-Kodaira-Nakano inequality.

Put uj = ∂∗ϕjwj on Ωj. We may choose a weak

limit u = udz in L2(n,0)(Ω, loc) s.t.∫

Ω

|u|2e−ϕ ≤ lim infj→∞

∫Ω

|vj|2Θe−ϕ =

∫Ω

|v|2Θe−ϕ.


Since vj = ∂uj + ∂∗ϕj∂wj, so

∂u = v := dz ∧ v ⇐⇒ ∂∗ϕj∂wj → 0

weakly in L2(n,0)(Ω, loc).

The conditions ∂v = 0 and ω is complete areused only to verify

∂∗ϕj∂wj → 0.

In such a way, one can show that Theorem 1.1holds for all CK domains. Q.E.D.


As an application of Theorem 3.15, one has

Theorem 3.16 (Chen-Wang-Wu 2015)

E ⊂ Dn × Dr (r < 1), closed complete pluripolar,ϕ ∈ PSH(Dn+1\E). Suppose ∃ A ⊂ Dn with |A|n > 0,

∀ z′ ∈ A, ϕ(z′, ·) ∈ SH(D).

Then ϕ can be extended to a psh function on Dn+1.

The important case when E is an analytic setwas proved by Siu (1974). Such theorems arecalled Thullen type extension theorem for pshfunctions in the literature.


Proof. It suffices to show that ϕ ∈ L∞loc nearevery a ∈ E. Let B ⊂⊂ Dn : ball, z0 ∈ (B × Dr)\E.

Theorem 3.15⇒ ∃ f ∈ O(Dn+1\E), f (z0) = eϕ(z0)

2 ,∫Dn+1\E

|f |2e−ϕ ≤ Cn.

Fubini’s theorem⇒ ∃ Z1 ⊂ Dn, |Z1|n = 0, s.t.∫

D

|f (z′, ·)|2e−ϕ(z′,·) < ∞, z′ ∈ Dn\Z1.

E is pluripolar⇒ ∃ Z2 ⊂ Dn, |Z2|n = 0, s.t.

Ez′ := (z′ × D) ∩ E is polar , z′ ∈ Dn\Z2.


Thus

f (z′, ·) ∈ L2loc, ∀ z′ ∈ A − Z1 − Z2,

which implies f (z′, ·) ∈ O(D), in view of

Theorem 3.17 (Carleson 1967)

Ω ⊂ C : open set, E ⊂ Ω closed. Then

A2(Ω\E) = A2(Ω) ⇐⇒ E is polar.

By the Hartogs theorem, we get

f ∈ O(Dn+1).


Fix r < r′′ < r′ < 1, B ⊂⊂ B′ ⊂⊂ Dn.

|f (z0)|2 ≤ C∫

B′×(Dr′\Dr′′ )|f |2

≤ C supB′×(Dr′\Dr′′ )

eϕ∫

B′×(Dr′\Dr′′ )|f |2e−ϕ

≤ C supB′×(Dr′\Dr′′ )

eϕ

i.e.ϕ(z0) ≤ log C + sup

B′×(Dr′\Dr′′ )ϕ.

Q.E.D.


It is natural to askProblem 3.18

Can the complete pluripolarity be replaced bypluripolarity?

It seems interesting to develop an L2−theoryfor the ∂−equation on domains Ω\E where E is aclosed ”thin” set, e.g. a pluripolar set, and Ω is abounded pseudoconvex domain.

On the other hand, suppose the weighted L2

estimate in Theorem 1.1 holds on Ω\E, then whatkind of analytic structure should E have?

Some aspects of Ohsawa-Takegoshi type theorems

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