H ormander’s L2 estimate and the Ohsawa-Takegoshi ...x10001c/L2estimate.pdf · H ormander’s L2...

Hormander’s L2−estimate and the

Ohsawa-Takegoshi extension theorem

Bo-Yong Chen

Young mathematician workshop on SCVJuly 24-July 27, 2013, Nagoya

Hormander’s L2−estimate

We start with the following fundamental

Hormander’s Theorem.Let Ω be a pseudoconvex domain in Cn and ϕ a

smooth strictly psh function on Ω. Then for any∂−closed (0, 1) form v with

∫Ω |v|

2i∂∂ϕ

e−ϕ <∞, ∃solution u to ∂u = v s.t.∫

Ω

|u|2e−ϕ ≤∫

Ω

|v|2i∂∂ϕe−ϕ. (0.1)


Notations:

|v|2i∂∂ϕ =∑

ϕj,kvj vk

where v =∑vjdzj and (ϕj,k) is the matrix inverse

to (∂2ϕ/∂zj∂zk).Equivalently,

|v|2i∂∂ϕ = sup|〈v,X〉|2 : X ∈ T 0,1(Ω), |X|i∂∂ϕ ≤ 1

where |X|2i∂∂ϕ

=∑ ∂2ϕ

∂zj∂zkXjXk. In particular,

i∂∂ϕ ≥ i∂ϕ ∧ ∂ϕ⇒ |∂ϕ|i∂∂ϕ ≤ 1.


The proof of Hormander’s theorem is based on

Morrey-Kohn-Hormander (MKH) Formula.Let Ω = ρ < 0, ρ : C2 defining function,

u ∈ C1(0,1)(Ω) s.t. ∂ρ · u = 0 on ∂Ω, ϕ ∈ C2(Ω,R)

⇒

‖∂u‖2ϕ + ‖∂∗ϕu‖2

ϕ =

∫∂Ω

n∑j,k=1

∂2ρ

∂zj∂zkujuke

−ϕ dS

|∇ρ|

+n∑j=1

∫Ω

∣∣∣∣∂uj∂zj

∣∣∣∣2 e−ϕ+

∫Ω

n∑j,k=1

∂2ϕ

∂zj∂zkujuke

−ϕ.

Donnelly-Fefferman type estimate

Ohsawa and Berndtsson generalized the originalL2 estimate of Donnelly-Fefferman as follows

Donnelly-Fefferman Type Theorem.Suppose furthermore ∃ ψ : C2 psh on Ω, and

0 < α < 1 s.t. iα∂∂ψ ≥ i∂ψ ∧ ∂ψ, then theL2(Ω, ϕ)−minimal solution of ∂u = v satisfies∫

Ω

|u|2eψ−ϕ ≤ Cα

∫Ω

|v|2i∂∂(ϕ+ψ)eψ−ϕ (0.2)

provided the right-hand side is finite.

Donnelly-Fefferman type estimate

There are two important candidates for theDonnelly-Fefferman condition:

a) Ω is hyperconvex, i.e., ∃ρ ∈ PSH−(Ω) s.t.ρ < c is relatively compact in Ω for each c < 0,

ψ := −α log(−ρ);

b) fj ∈ O(Ω),∑

j |fj|2 < 1,

ψ := −α log(− log

∑|fj|2

).

Berndtsson-Charpentier ’00 observed thatHormander ⇒ Donnelly-Fefferman.

Ohsawa-Takegoshi L2 extension theorem

One of the deepest result in complex analysis isthe following

Ohsawa-Takegoshi Extension Theorem.Ω ⊂ Cn pseudoconvex with supΩ |zn|2 < 1.⇒ ∃ constant Cn > 0 s.t. ∀ϕ ∈ PSH(Ω), ∀ f

holomorphic on Ω′ := Ω ∩ zn = 0 with∫Ω′|f |2e−ϕ <∞,

∃ holomorphic extension F of f to Ω s.t.∫Ω

|F |2e−ϕ ≤ Cn

∫Ω′|f |2e−ϕ.

Ohsawa-Takegoshi L2 extension theorem

The original proof relies on

Twisted MKH Formula.Let ρ, u, ϕ be as in MKH formula, and

η ∈ C2(Ω,R+). Then

‖√η∂u‖2ϕ + ‖√η∂∗ϕu‖2

ϕ

=n∑

j,k=1

∫∂Ω

η∂2ρ

∂zj∂zkujuke

−ϕ dσ

|∇ρ|

+n∑

j,k=1

∫Ω

(η∂2ϕ

∂zj∂zk− ∂2η

∂zj∂zk

)ujuke

−ϕ

+n∑j=1

∫Ω

η

∣∣∣∣∂uj∂zj

∣∣∣∣2 e−ϕ + 2Re

∫Ω

(∂η · u)∂∗ϕue−ϕ.

Hormander ⇒ Ohsawa-Takegoshi

The original extension theorem was generalizedby many people, in particular, Blocki obtained theoptimal constant in the L2 extension theorem.

Here I would like to present a proof based onlyon Hormander’s theorem.

1. A standard reduction: It suffices to show thatthere is a holomorphic extension F of f to Ω s.t.∫

Ω

|F |2e−ϕ ≤ Cn

∫V

|f |2e−ϕ

under the stronger hypothesis that Ω is boundedwith ∂Ω ∈ C∞, ϕ is C∞ strictly psh on Ω, f isholomorphic in an open neighborhood V of Ω′ inzn = 0.


To see this, choose an increasing sequence ofbounded smooth pseudoconvex domains Ωj with∪Ωj = Ω, and take a sequence of smooth strictlypsh functions ϕj on Ω with ϕj ↓ ϕ (note that Ω ispseudoconvex).

For each j, there is a holomorphic function Fjon Ωj s.t. Fj = f on Ω′j := Ωj ∩ zn = 0 and∫

Ωj

|Fj|2e−ϕj ≤ Cn

∫Ω′j+1

|f |2e−ϕj ≤ Cn

∫Ω′|f |2e−ϕ.

Thus Fj is a normal family and we only need totake a limit of Fj.


2. Put

φ = ϕ+ log |zn|2, vε = f∂χ(|zn|2/ε2)

where ε 1 and χ is a cut-off function s.t.χ|(−∞,1/2) = 1 and χ|(1,∞) = 0. Note that vε is aC∞ (0, 1) form in a neighborhood of Ω provided εsmall enough.

By Hormander’s theorem, there is a solution of∂u = vε on Ω := Ω\zn = 0 verifying∫

Ω

|u|2e−φ ≤∫

Ω

|vε|2i∂∂φe−φ.

Unfortunately, the RHS depends on ε!


3. To overcome this difficulty, we use a trickgoes back to Berndtsson and Charpentier ’00 asfollows: without loss of generality, we may assumethat u is the L2(Ω, φ)−minimal solution of ∂u = vε,i.e.,

u⊥Ker ∂ in L2(Ω, φ).

Thus for any bounded smooth psh function ψ on Ω,

ueψ⊥Ker ∂ in L2(Ω, φ+ ψ) = L2(Ω, φ),

i.e., ueψ is L2(Ω, φ+ ψ)−minimal solution of

∂u = v := ∂(ueψ).


By Hormander’s theorem again, we have∫Ω

|u|2e−φ−ψ ≤∫

Ω

|v|2i∂∂(φ+ψ)e−φ−ψ,

that is∫Ω

|u|2eψ−φ ≤∫

Ω

|vε + ∂ψ ∧ u|2i∂∂(φ+ψ)eψ−φ

≤ (1 + 1/r)

∫Ω

|vε|2i∂∂(φ+ψ)eψ−φ

+

∫Ω

|∂ψ|2i∂∂(φ+ψ)|u|2eψ−φ

+r

∫supp vε

|∂ψ|2i∂∂(φ+ψ)|u|2eψ−φ.

in view of Schwarz’s inequality.


How to choose ψ?Donnelly-Fefferman type estimate suggests to

look at

ψ = −α log(− log(|zn|2 + ε2))

with α ≤ 1. In order that the first term in RHS ofthe previous inequality is bounded independent of ε,one has to choose α = 1. But then the role of theLHS of previous inequality would be canceled by thesecond term of RHS!

Thus it is reasonable to add some psh functionto ψ with slower growth.


4. Putρ = log(|zn|2 + ε2)

η = −ρ+ log(−ρ)

ψ = − log η.

A direct calculation shows

i∂∂ψ ≥(

1

η2+

1

η(−ρ+ 1)2

)i∂η ∧ ∂η +

i∂∂ρ

η.

Since

∂ψ = − ∂ηη

=1

η

(1− 1

ρ

)∂ρ,

it follows that


|∂ψ|2i∂∂(φ+ψ) ≤1

1 + η(−ρ+1)2

i∂∂ψ ≥ i∂∂ρ

η=

ε2idzndznη(|zn|2 + ε2)2

and on supp vε ⊂ ε2/2 ≤ |zn|2 ≤ ε2,

|∂ψ|2i∂∂(φ+ψ) ≤1

η2

(1− 1

ρ

)2

|∂ρ|2i∂∂ψ

≤ 1

η2

(1− 1

ρ

)2η(|zn|2 + ε2)2

ε2

|zn|2

(|zn|2 + ε2)2

≤ 4

η(provided ε 1).


Thus∫Ω

|∂ψ|2i∂∂(φ+ψ)|u|2eψ−φ ≤

∫Ω

|u|2

1 + η(−ρ+1)2

eψ−φ∫supp v

|∂ψ|2i∂∂(φ+ψ)|u|2eψ−φ ≤

∫Ω

4

η|u|2eψ−φ.

Fubini’s theorem implies∫Ω

|vε|2i∂∂(φ+ψ)eψ−φ

≤∫

Ω

|χ′|2|f |2 |zn|2

ε4

η(|zn|2 + ε2)2

ε2

1

|zn|21

ηe−ϕ

≤ 2

(∫ ε22 <|zn|2<ε2

|χ′|2 (|zn|2 + ε2)2

ε6

)∫V

|f |2e−ϕ.


It follows that∫Ω

|vε|2i∂∂(φ+ψ)eψ−φ ≤ Cn

∫V

|f |2e−ϕ.

Substituting these inequalities to the inequalityin Step 3, we get∫

Ω

(η

(−ρ+1)2

1 + η(−ρ+1)2

− 4r

η

)|u|2eψ−φ

≤ (1 + r−1)Cn

∫V

|f |2e−ϕ.


Since η −ρ provided ε 1, so that

η(−ρ+1)2

1 + η(−ρ+1)2

1

η,

we may choose r = rn 1 s.t. the LHS is boundedbelow by

cn

∫Ω

|u|2

|zn|2η2e−ϕ,

i.e.,∫Ω

|u|2

|zn|2| log(|zn|2 + ε2)|2e−ϕ ≤ Cn

∫Ω′|f |2e−ϕ.

(0.3)


5. By virtue of (0.3) and removable singularitiesof L2 holomorphic functions, we conclude that

Fε = χ(|zn|2/ε2)f − u

is a holomorphic extension of f to Ω s.t.∫Ω

|Fε|2

|zn|2(− log |zn|2)2e−ϕ ≤ C ′n

∫V

|f |2e−ϕ.

By taking a limit of Fε as ε→ 0, we get the desiredextension.

It is interesting to see that the gain in previousestimate is the Poincare metric of the punctureddisk, which was first noticed by Demailly.

A Hormander type estimate

If we take

ψ = − α

α0log(−ρ), 0 < α < α0

in Donnelly-Fefferman type estimate, where ρ < 0,C2 psh, |ρ| δα0

Ω , then∫Ω

|u|2e−ϕδ−αΩ ≤ Cα,Ω

∫Ω

|v|2i∂∂ϕe−ϕδ−αΩ . (0.4)

This estimate is useful in the study of regularityof the Bergman projection (see e.g., Berndtsson andCharpentier ’00, and Michel-Shaw ’01).


Diederich-Fornaess Theorem.Ω : bounded, pseudoconvex, ∂Ω ∈ C2, then(1) ∃ 0 < α < 1, ρ < 0, C2 psh, s.t. |ρ| δαΩ.(2) ∀ 0 < α < 1, ∀ p ∈ ∂Ω, ∃ neighborhood

U 3 p and ρ < 0, C2 psh on Ω ∩ U , s.t. |ρ| δαΩ.

The Diederich-Fornaess index is defined by

α0 := supα > 0 : ∃ρ ∈ PSH−(Ω), s.t.|ρ| δαΩ

Diederich-Fornaess also constructed so-called wormdomains with α0 arbitrarily small.


On the other side, we still have

Theorem 1 (Chen ’12)Let Ω ⊂⊂ Cn be pseudoconvex with ∂Ω ∈ C2

and ϕ a C2 psh function on Ω. Then ∀α < 1, ∀(0, 1) form v with ∂v = 0 and∫

Ω

|v|2i∂∂ϕe−ϕδ−αΩ <∞,

∃ solution u of ∂u = v satisfying (0.4), i.e.,∫Ω

|u|2e−ϕδ−αΩ ≤ Cα,Ω

∫Ω

|v|2i∂∂ϕe−ϕδ−αΩ .


Now we sketch the idea of the proof as follows

1. Diederich-Fornaess theorem ⇒ ∀ 0 < α < 1,∃ cover Uj0≤j≤mα

of Ω, C2 psh functions ρj < 0on Ω ∩ Uj s.t.

C−1α δΩ(z)

α+12 ≤ −ρj(z) ≤ CαδΩ(z)

α+12 .

Standard procedure: solving first ∂uε = v on

Ωε := z ∈ Ω : δΩ(z) > ε,

then take a weak limit of uε as ε→ 0.


2. Let ϕτ(z) = ϕ(z) + τ |z|2 with τ 1 to bedetermined later. Twisted Kohn-Morrey-Hormanderformula ⇒∫

Ωε

(η + c(η)−1)|∂∗ϕτw|2e−ϕτ +

∫Ωε

η|∂w|2e−ϕτ

≥∑k,l

∫Ωε

(η∂2ϕτ∂zk∂zl

− ∂2η

∂zk∂zl

)wkwl e

−ϕτ

−∫

Ωε

c(η)

∣∣∣∣∣∑k

∂η

∂zkwk

∣∣∣∣∣2

e−ϕτ (0.5)

where w ∈ Dom ∂∗ϕτ ∩ C1(Ωε), η ≥ 0, η ∈ C2(Ω),

c ∈ C(R,R+).


3. Let χj be partition of unity subordinate toUj. Clearly,

wj := χjw ∈ Dom ∂∗ϕτ .

Choose χj ∈ C∞0 (Uj) s.t. χj = 1 on suppχj. Put

φj = − 2α

α + 1log(−ρj), η = e−χjφj ,

c(η) =1− α

2αeχjφj .


4. Applying (0.5) to wj,∑k,l

∫Ωε∩Uj

∂2ϕτ∂zk∂zl

|χj|2wkwl e−ϕτ−φj

≤ Cα

∫Ωε∩Uj

(|∂(χjw)|2 + |∂∗ϕτ (χjw)|2)e−ϕτ−φj .

Since e−φj δαΩ on Ω ∩ Uj,∑k,l

∫Ωε∩Uj

∂2ϕτ∂zk∂zl

|χj|2wkwl e−ϕτδαΩ

≤ Cα

∫Ωε∩Uj

(|∂(χjw)|2 + |∂∗ϕτ (χjw)|2)e−ϕτδαΩ.


5. Thus∑k,l

∫Ωε

∂2ϕτ∂zk∂zl

wkwl e−ϕτδαΩ

=∑k,l

∫Ωε

∂2ϕτ∂zk∂zl

(mα∑j=0

χjwk

)(mα∑j=0

χjwl

)e−ϕτδαΩ

≤ (mα + 1)

mα∑j=0

∑k,l

∫Ωε∩Uj

∂2ϕτ∂zk∂zl

|χj|2wkwl e−ϕτδαΩ

≤ Cα

mα∑j=0

∫Ωε∩Uj

(|∂(χjw)|2 + |∂∗ϕτ (χjw)|2)e−ϕτδαΩ.


6. Since

∂(χjw) = χj∂w + ∂χj ∧ w∂∗ϕτ (χjw) = χj∂

∗ϕτw − ∂χjyw,

thus by Schwarz’s inequality,∑k,l

∫Ωε

∂2ϕτ∂zk∂zl

wkwl e−ϕτδαΩ

≤ Cα

∫Ωε

(|∂w|2 + |∂∗ϕτw|2)e−ϕτδαΩ

+Cα

∫Ωε

|w|2∑j

|∂χj|2e−ϕτδαΩ.


7. Since ∂∂ϕτ = ∂∂ϕ+ τ∂∂|z|2, thus whenτ = τ(α,Ω) 1, the second term in theright-hand side may be absorbed by theleft-hand side and we get the following

Fundamental Inequality.∑k,l

∫Ωε

∂2ϕ

∂zk∂zlwkwl e

−ϕτδαΩ

≤ Cα,Ω

∫Ωε

(|∂w|2 + |∂∗ϕτw|2)e−ϕτδαΩ.

The remaining argument is standard.

Berndtsson’s boundary value estimate

In an attempt to generalize Carleson’s CoronaTheorem to the unit ball in Cn, Berndtsson ’87proved the following

Theorem. Let Ω be a pseudoconvex domainwith C2 boundary, and ρ, ϕ ∈ C2(Ω), ρ < 0, suchthat

Θ := (−ρ)∂∂ϕ+ ∂∂ρ > 0.

If u is the L2(Ω, ϕ)−minimal solution of ∂u = v,then ∀ 0 < α < 1,

(1− α)

∫Ω

|u|2(−ρ)−αe−ϕ ≤ 1

α

∫Ω

|v|2iΘ(−ρ)1−αe−ϕ.

(0.6)

Berndtsson’s boundary value estimate

Let α→ 1− in (0.6), one immediately gets thefollowing boundary value estimate:∫

∂Ω

|u|2

|∇ρ|e−ϕ ≤

∫Ω

|v|2iΘe−ϕ,

which plays a crucial role in the study of theSzego kernel (see e.g. Chen-Fu ’11).

Thus it seems to be interesting to know theprecise estimate of the constant Cα,Ω in Theorem 1.

Essay on reproducing kernels

There are numerous applications of L2 estimatesfor ∂ equation, but I mention only a few, dependingon my personal taste.

One of the most remarkable application ofHormander’s theorem to the Bergman kernel is

Theorem (Berndtsson ’05)Let Ω be a pseudoconvex domain in Cn × Ck

and ϕ ∈ PSH(Ω). Let Kϕt be the Bergman kernelof the slice Ωt with weight ϕt. Then logKϕt(z) ispsh on Ω or identically equal to −∞.


Regularity dependence on the parameter lies inthe intersection of complex and real analysis, a littlebit likes the Hardy space.

Example 1. Let Ωt be an increasing (resp.decreasing) sequence of bounded domains over aninterval I. Let Kt be the Bergman kernel of Ωt.Then logKt is decreasing (resp. increasing) in t, sothat ∂

∂t logKt exists a.e. and belongs to L1loc(I), in

view of Lebesgue’s theorem.

Example 2. If Kϕt > 0 in Berndtsson’s theorem,then logKϕt ∈ W

1,2loc (Ω).


Very strong stability properties of the Bergmankernels under small C∞ perturbations of stronglypseudoconvex domains were obtained by Greene andKrantz ’82.

Put I = [0, 1]. One has

Theorem (Diederich-Ohsawa ’91)(1) Let Ωtt∈I be a C∞ family of hyperconvex

domains. Then Kt(z) is continuous in (t, z).(2) Let Ω be a bounded pseudoconvex domain

and ϕtt∈I a C∞ family of psh functions on Ω s.t.i∂∂ϕt ≥ Ci∂∂|z|2 and supΩ | ∂

k

∂tk∂ϕt| ≤ Ck for all

k ≥ 1. Then Kϕt(z) is C∞ in (t, z).


Donnelly-Fefferman type theorem implies

Proposition 1. Let Ωtt∈I be a C2 family ofbounded pseudoconvex domains with ∂Ωt ∈ C2.Then locally uniformly,

| logKt − logKt0| ≤C

| log |t− t0||.

Proposition 2.Let Ωtt∈I be a nondecreasing C2−family of

bounded pseudoconvex domains with ∂Ωt ∈ C2.Suppose for each w ∈ Ω0,

supz∈Ω0

|K0(z, w)| <∞.


Corollary.(i) If supΩ |ϕt − ϕt′|/| log δ| ≤ C|t− t′| for any

t, t′ ∈ I, then logKt is Holder continuous of order1/2 in t.

(ii) If supΩ |ϕt − ϕt′|/ log | log δ| ≤ C|t− t′| forany t, t′ ∈ I, then logKt is log-Lipschitz continuousin t, i.e.,

| logKt − logKt0| ≤ C|t− t0|| log |t− t0||

holds locally uniformly.


Let Ω be a bounded domain with ∂Ω ∈ C2. For∀α < 1, the Bergman space A2

α(Ω) is the set off ∈ O(Ω) s.t.

‖f‖2α :=

∫Ω

|f |2δ−α <∞.

The Hardy space H2(Ω) is the set of f ∈ O(Ω) s.t.

‖f‖2L2(∂Ω) = lim

α→1−(1− α)

∫Ω

|f |2δ−α <∞.


Recall also that

A2α(Ω) = W α/2(Ω) ∩ O(Ω)

H2(Ω) = W 1/2(Ω) ∩ O(Ω)

where W s(Ω) is the Sobolev space.

These Hilbert spaces induce reproducing kernels:the Bergman kernel Kα and the Szego kernel S.The standard Bergman kernel is K = K0.

By Theorem 1 and Corollary, we conclude thatlogKα is 1/2−Holder continuous in α.


Proposition 4. Let Ω be a bounded domainwith ∂Ω ∈ C2. Then

(1− α)−1Kα(z, w)→ S(z, w)

locally uniformly in z, w as α→ 1−.

Corollary.

∂Kα(z, w)

∂α

∣∣∣∣α=1−

= −S(z, w).


Question 1.Is there an expansion

Kα(z, w) = (1− α)S(z, w) +∞∑j=2

aj(z, w)(1− α)j

as α→ 1−?

A related question is

Question 2.Does Kα(z) depend real analytically on α?

Note that the answer to Question 2 is positivewhen Ω is strongly pseudoconvex (Englis ’10).


In his 1972 book, E. M. Stein asked thefollowing

Stein’s Question. What are the relationsbetween K and S?

Theorem (Chen-Fu ’11)Let Ω be a bounded pseudoconvex domain with

C2 boundary in Cn. Then(1) ∀ 0 < α < 1, ∃Cα > 0 s.t.

S/K ≤ CαδΩ| log δΩ|nα

(2) Suppose furthermore Ω is δ−regular, then∃ 0 < α ≤ 1, C > 0 s.t.

S/K ≥ CδΩ| log δΩ|−1/α.


Definition. A domain Ω is called δ−regular ifthere are ϕ ∈ PSH−(Ω) and a defining function ρs.t.

i∂∂ϕ ≥ ρ−1i∂∂ρ near ∂Ω.

Examples:(1) Pseudoconvex domains of finite D’Angelo

type.(2) Domains admitting a psh defining function.

The proof of relies on Berndtsson’s estimate(0.6).

H ormander’s L2 estimate and the Ohsawa-Takegoshi ...x10001c/L2estimate.pdf · H ormander’s L2...

Documents

Transcript of H ormander’s L2 estimate and the Ohsawa-Takegoshi ...x10001c/L2estimate.pdf · H ormander’s L2...