H¨ ormander’s ¯ ∂-estimate,

H¨ ormander’s ¯ ∂-estimate,

Some Generalizations, and New Applications

Zbigniew Błocki

(Uniwersytet Jagielloński, Kraków, Poland) http://gamma.im.uj.edu.pl/eblocki

Abel Symposium

in honor of Professor Yum-Tong Siu Trondheim, July 4, 2013

We will discuss applications of H¨ormander’s L²-estimate for ¯∂ in the following problems:

1. Suita Conjecture (1972) from potential theory

2. Optimal constant in the Ohsawa-Takegoshi extension theorem (1987) 3. Mahler Conjecture (1938) from convex analysis

Suita Conjecture

Suita Conjecture

Green function for bounded domain D in C:

(∆GD(·, z) = 2πδz

GD(·, z) = 0 on ∂D (if D is regular)

Suita Conjecture

Green function for bounded domain D in C:

(∆GD(·, z) = 2πδz

GD(·, z) = 0 on ∂D (if D is regular) cD(z) := exp lim

ζ→z(GD(ζ, z) − log |ζ − z|)

(logarithmic capacity of C \ D w.r.t. z)

Suita Conjecture

Green function for bounded domain D in C:

(∆GD(·, z) = 2πδz

GD(·, z) = 0 on ∂D (if D is regular) cD(z) := exp lim

ζ→z(GD(ζ, z) − log |ζ − z|)

(logarithmic capacity of C \ D w.r.t. z) cD|dz| is an invariant metric (Suita metric)

Suita Conjecture

Green function for bounded domain D in C:

(∆GD(·, z) = 2πδz

GD(·, z) = 0 on ∂D (if D is regular) cD(z) := exp lim

ζ→z(GD(ζ, z) − log |ζ − z|)

(logarithmic capacity of C \ D w.r.t. z) cD|dz| is an invariant metric (Suita metric)

Curv_cD|dz|= −(log cD)z ¯z

c²_D

Suita Conjecture

Green function for bounded domain D in C:

(∆GD(·, z) = 2πδz

GD(·, z) = 0 on ∂D (if D is regular) cD(z) := exp lim

ζ→z(GD(ζ, z) − log |ζ − z|)

(logarithmic capacity of C \ D w.r.t. z) cD|dz| is an invariant metric (Suita metric)

Curv_cD|dz|= −(log cD)z ¯z

c²_D

Suita Conjecture (1972): Curv_cD|dz|≤ −1

Suita Conjecture

Green function for bounded domain D in C:

(∆GD(·, z) = 2πδz

GD(·, z) = 0 on ∂D (if D is regular) cD(z) := exp lim

ζ→z(GD(ζ, z) − log |ζ − z|)

(logarithmic capacity of C \ D w.r.t. z) cD|dz| is an invariant metric (Suita metric)

Curv_cD|dz|= −(log cD)z ¯z

c²_D

Suita Conjecture (1972): Curv_cD|dz|≤ −1

• “=” if D is simply connected

• “<” if D is an annulus (Suita)

• Enough to prove for D with smooth boundary

• “=” on ∂D if D has smooth boundary

2 4 6 8 10

-7 -6 -5 -4 -3 -2 -1

Curv_cD|dz|for D = {e⁻⁵< |z| < 1} as a function of t = −2 log |z|

5 10 15 20

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5

Curv_KD|dz|2 for D = {e⁻¹⁰< |z| < 1} as a function of t = −2 log |z|

1 2 3 4 5

-6 -5 -4 -3 -2 -1

Curv_{(log KD)z ¯z|dz|}2 for D = {e⁻⁵< |z| < 1} as a function of t = −2 log |z|

∂²

∂z∂ ¯z(log cD) = πKD (Suita) where KDis the Bergman kernel on the diagonal:

KD(z) := sup{|f (z)|²: f ∈ O(D), Z

D

|f |²dλ ≤ 1}.

∂²

∂z∂ ¯z(log cD) = πKD (Suita) where KDis the Bergman kernel on the diagonal:

KD(z) := sup{|f (z)|²: f ∈ O(D), Z

D

|f |²dλ ≤ 1}.

Therefore the Suita conjecture is equivalent to c²_D≤ πK_D.

∂²

∂z∂ ¯z(log cD) = πKD (Suita) where KDis the Bergman kernel on the diagonal:

KD(z) := sup{|f (z)|²: f ∈ O(D), Z

D

|f |²dλ ≤ 1}.

Therefore the Suita conjecture is equivalent to c²_D≤ πK_D.

It is thus an extension problem: for z ∈ D find holomorphic f in D such that f (z) = 1 and

Z

D

|f |²dλ ≤ π (c_D(z))².

∂²

∂z∂ ¯z(log cD) = πKD (Suita) where KDis the Bergman kernel on the diagonal:

KD(z) := sup{|f (z)|²: f ∈ O(D), Z

D

|f |²dλ ≤ 1}.

Therefore the Suita conjecture is equivalent to c²_D≤ πK_D.

It is thus an extension problem: for z ∈ D find holomorphic f in D such that f (z) = 1 and

Z

D

|f |²dλ ≤ π (c_D(z))².

Ohsawa (1995), using the methods of the Ohsawa-Takegoshi extension theorem, showed the estimate

c²_D≤ CπK_D with C = 750.

∂²

∂z∂ ¯z(log cD) = πKD (Suita) where KDis the Bergman kernel on the diagonal:

KD(z) := sup{|f (z)|²: f ∈ O(D), Z

D

|f |²dλ ≤ 1}.

Therefore the Suita conjecture is equivalent to c²_D≤ πK_D.

It is thus an extension problem: for z ∈ D find holomorphic f in D such that f (z) = 1 and

Z

D

|f |²dλ ≤ π (c_D(z))².

Ohsawa (1995), using the methods of the Ohsawa-Takegoshi extension theorem, showed the estimate

c²_D≤ CπK_D with C = 750.

C = 2 (B., 2007)

C = 1.95388 . . . (Guan-Zhou-Zhu, 2011)

Ohsawa-Takegoshi Extension Theorem (1987)

Ω - bounded pseudoconvex domain in Cⁿ, ϕ - psh in Ω H - complex affine subspace of Cⁿ

f - holomorphic in Ω⁰:= Ω ∩ H

Then there exists a holomorphic extension F of f to Ω such that Z

Ω

|F |²e^−ϕdλ ≤ C Z

Ω⁰

|f |²e^−ϕdλ⁰, where C depends only on n and the diameter of Ω.

Ohsawa-Takegoshi Extension Theorem (1987)

Ω - bounded pseudoconvex domain in Cⁿ, ϕ - psh in Ω H - complex affine subspace of Cⁿ

f - holomorphic in Ω⁰:= Ω ∩ H

Then there exists a holomorphic extension F of f to Ω such that Z

Ω

|F |²e^−ϕdλ ≤ C Z

Ω⁰

|f |²e^−ϕdλ⁰, where C depends only on n and the diameter of Ω.

Siu / Berndtsson (1996): If Ω ⊂ Cⁿ⁻¹× {|zn< 1} and H = {zn= 0}

then C = 4π.

Ohsawa-Takegoshi Extension Theorem (1987)

Ω - bounded pseudoconvex domain in Cⁿ, ϕ - psh in Ω H - complex affine subspace of Cⁿ

f - holomorphic in Ω⁰:= Ω ∩ H

Then there exists a holomorphic extension F of f to Ω such that Z

Ω

|F |²e^−ϕdλ ≤ C Z

Ω⁰

|f |²e^−ϕdλ⁰, where C depends only on n and the diameter of Ω.

Siu / Berndtsson (1996): If Ω ⊂ Cⁿ⁻¹× {|zn< 1} and H = {zn= 0}

then C = 4π.

Problem.Can we improve to C = π?

Ohsawa-Takegoshi Extension Theorem (1987)

Ω - bounded pseudoconvex domain in Cⁿ, ϕ - psh in Ω H - complex affine subspace of Cⁿ

f - holomorphic in Ω⁰:= Ω ∩ H

Then there exists a holomorphic extension F of f to Ω such that Z

Ω

|F |²e^−ϕdλ ≤ C Z

Ω⁰

|f |²e^−ϕdλ⁰, where C depends only on n and the diameter of Ω.

Siu / Berndtsson (1996): If Ω ⊂ Cⁿ⁻¹× {|zn< 1} and H = {zn= 0}

then C = 4π.

Problem.Can we improve to C = π?

B.-Y. Chen (2011): Ohsawa-Takegoshi extension theorem can be deduced directly from H¨ormander’s estimate for ¯∂-equation!

Mahler Conjecture

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:



@@

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:



@@

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:



@@

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:



@@

Bourgain-Milman (1987): There exists c > 0 such that λ(K)λ(K⁰) ≥ cⁿ4ⁿ

n!. Mahler Conjecture: c = 1

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:



@@

Bourgain-Milman (1987): There exists c > 0 such that λ(K)λ(K⁰) ≥ cⁿ4ⁿ

n!.

Mahler Conjecture: c = 1, G. Kuperberg (2006):c = π/4

Mahler Conjecture

K - convex symmetric body in Rⁿ

K⁰:= {y ∈ Rⁿ: x · y ≤ 1 for every x ∈ K}

Mahler volume := λ(K)λ(K⁰)

Santaló Inequality (1949): Mahler volume ismaximizedby balls.

Mahler Conjecture (1938):Mahler volume isminimizedby cubes.

True for n = 2:



@@

Bourgain-Milman (1987): There exists c > 0 such that λ(K)λ(K⁰) ≥ cⁿ4ⁿ

n!.

Mahler Conjecture: c = 1, G. Kuperberg (2006):c = π/4

Nazarov (2012): One can show the Bourgain-Milman inequality with c = (π/4)³using H¨ormander’s estimate.

H¨ormander’s Estimate (1965)

Ω - pseudoconvex in Cⁿ, ϕ - smooth, strongly psh in Ω α =P

jαjd¯zj∈ L²_loc,(0,1)(Ω), ¯∂α = 0

Then one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^−ϕdλ ≤ Z

Ω

|α|²

i∂ ¯∂ϕe^−ϕdλ.

Here |α|²

i∂ ¯∂ϕ=P

j,kϕ^{j ¯k}α¯jαk, where (ϕ^{j ¯k}) = (∂²ϕ/∂zj∂ ¯zk)⁻¹is the length of α w.r.t. the K¨ahler metric i∂ ¯∂ϕ.

H¨ormander’s Estimate (1965)

Ω - pseudoconvex in Cⁿ, ϕ - smooth, strongly psh in Ω α =P

jαjd¯zj∈ L²_loc,(0,1)(Ω), ¯∂α = 0

Then one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^−ϕdλ ≤ Z

Ω

|α|²

i∂ ¯∂ϕe^−ϕdλ.

Here |α|²

i∂ ¯∂ϕ=P

j,kϕ^{j ¯k}α¯jαk, where (ϕ^{j ¯k}) = (∂²ϕ/∂zj∂ ¯zk)⁻¹is the length of α w.r.t. the K¨ahler metric i∂ ¯∂ϕ.

The estimate also makes sense for non-smooth ϕ: instead of |α|²

i∂ ¯∂ϕone has to take any nonnegative H ∈ L^∞_loc(Ω) with

i ¯α ∧ α ≤ H i∂ ¯∂ϕ (B., 2005).

Donnelly-Fefferman (1982) Ω, α, ϕ as before ψ psh in Ω s.th. | ¯∂ψ|²

i∂ ¯∂ψ≤ 1 (that is i∂ψ ∧ ¯∂ψ ≤ i∂ ¯∂ψ) Then one can find u ∈ L²_loc(Ω) with ¯∂u = α and

Z

Ω

|u|²e^−ϕdλ ≤ C Z

Ω

|α|²_{i∂ ¯∂ψ}e^−ϕdλ, where C is an absolute constant.

Donnelly-Fefferman (1982) Ω, α, ϕ as before ψ psh in Ω s.th. | ¯∂ψ|²

i∂ ¯∂ψ≤ 1 (that is i∂ψ ∧ ¯∂ψ ≤ i∂ ¯∂ψ) Then one can find u ∈ L²_loc(Ω) with ¯∂u = α and

Z

Ω

|u|²e^−ϕdλ ≤ C Z

Ω

|α|²_{i∂ ¯∂ψ}e^−ϕdλ, where C is an absolute constant.

Berndtsson (1996) Ω, α, ϕ, ψ as before

Then, if 0 ≤ δ < 1, one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^δψ−ϕdλ ≤ 4 (1 − δ)²

Z

Ω

|α|²

i∂ ¯∂ψe^δψ−ϕdλ.

Donnelly-Fefferman (1982) Ω, α, ϕ as before ψ psh in Ω s.th. | ¯∂ψ|²

i∂ ¯∂ψ≤ 1 (that is i∂ψ ∧ ¯∂ψ ≤ i∂ ¯∂ψ) Then one can find u ∈ L²_loc(Ω) with ¯∂u = α and

Z

Ω

|u|²e^−ϕdλ ≤ C Z

Ω

|α|²_{i∂ ¯∂ψ}e^−ϕdλ, where C is an absolute constant.

Berndtsson (1996) Ω, α, ϕ, ψ as before

Then, if 0 ≤ δ < 1, one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^δψ−ϕdλ ≤ 4 (1 − δ)²

Z

Ω

|α|²

i∂ ¯∂ψe^δψ−ϕdλ.

The above constant was obtained in B. 2004 and is optimal (B. 2012).

Therefore C = 4 is optimal in Donnelly-Fefferman.

Donnelly-Fefferman (1982) Ω, α, ϕ as before ψ psh in Ω s.th. | ¯∂ψ|²

i∂ ¯∂ψ≤ 1 (that is i∂ψ ∧ ¯∂ψ ≤ i∂ ¯∂ψ) Then one can find u ∈ L²_loc(Ω) with ¯∂u = α and

Z

Ω

|u|²e^−ϕdλ ≤ C Z

Ω

|α|²_{i∂ ¯∂ψ}e^−ϕdλ, where C is an absolute constant.

Berndtsson (1996) Ω, α, ϕ, ψ as before

Then, if 0 ≤ δ < 1, one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^δψ−ϕdλ ≤ 4 (1 − δ)²

Z

Ω

|α|²

i∂ ¯∂ψe^δψ−ϕdλ.

The above constant was obtained in B. 2004 and is optimal (B. 2012).

Therefore C = 4 is optimal in Donnelly-Fefferman.

Berndtsson’s estimate is not enough to obtain Ohsawa-Takegoshi (it would be if it were true for δ = 1).

Berndtsson’s Estimate Ω - pseudoconvex α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0 ϕ, ψ - psh, | ¯∂ψ|²_{i∂ ¯∂ψ}≤ 1

Then, if 0 ≤ δ < 1, one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^δψ−ϕdλ ≤ 4 (1 − δ)²

Z

Ω

|α|²

i∂ ¯∂ψe^δψ−ϕdλ.

Berndtsson’s Estimate Ω - pseudoconvex α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0 ϕ, ψ - psh, | ¯∂ψ|²_{i∂ ¯∂ψ}≤ 1

Then, if 0 ≤ δ < 1, one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^δψ−ϕdλ ≤ 4 (1 − δ)²

Z

Ω

|α|²

i∂ ¯∂ψe^δψ−ϕdλ.

Theorem. Ω, α, ϕ, ψ as above

Assume in addition that | ¯∂ψ|²_{i∂ ¯∂ψ}≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) solving ¯∂u = α with Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ψ})e^ψ−ϕdλ ≤ 1 (1 −√

δ)² Z

Ω

|α|²_{i∂ ¯∂ψ}e^ψ−ϕdλ.

Berndtsson’s Estimate Ω - pseudoconvex α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0 ϕ, ψ - psh, | ¯∂ψ|²_{i∂ ¯∂ψ}≤ 1

Then, if 0 ≤ δ < 1, one can find u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²e^δψ−ϕdλ ≤ 4 (1 − δ)²

Z

Ω

|α|²

i∂ ¯∂ψe^δψ−ϕdλ.

Theorem. Ω, α, ϕ, ψ as above

Assume in addition that | ¯∂ψ|²_{i∂ ¯∂ψ}≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) solving ¯∂u = α with Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ψ})e^ψ−ϕdλ ≤ 1 (1 −√

δ)² Z

Ω

|α|²_{i∂ ¯∂ψ}e^ψ−ϕdλ.

From this estimate one can obtain Ohsawa-Takegoshi and Suita with C = 1.95388 . . . (obtained earlier by Guan-Zhou-Zhu).

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.)

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.) By approximation we may assume that ϕ is smooth up to the boundary and strongly psh, and ψ is bounded.

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.) By approximation we may assume that ϕ is smooth up to the boundary and strongly psh, and ψ is bounded.

u - minimal solution to ¯∂u = α in L²(Ω, e^ψ−ϕ)

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.) By approximation we may assume that ϕ is smooth up to the boundary and strongly psh, and ψ is bounded.

u - minimal solution to ¯∂u = α in L²(Ω, e^ψ−ϕ)

⇒ u ⊥ ker ¯∂ in L²(Ω, e^ψ−ϕ)

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.) By approximation we may assume that ϕ is smooth up to the boundary and strongly psh, and ψ is bounded.

u - minimal solution to ¯∂u = α in L²(Ω, e^ψ−ϕ)

⇒ u ⊥ ker ¯∂ in L²(Ω, e^ψ−ϕ)

⇒ v := ue^ψ⊥ ker ¯∂ in L²(Ω, e^−ϕ)

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.) By approximation we may assume that ϕ is smooth up to the boundary and strongly psh, and ψ is bounded.

u - minimal solution to ¯∂u = α in L²(Ω, e^ψ−ϕ)

⇒ u ⊥ ker ¯∂ in L²(Ω, e^ψ−ϕ)

⇒ v := ue^ψ⊥ ker ¯∂ in L²(Ω, e^−ϕ)

⇒ v - minimal solution to ¯∂v = β := e^ψ(α + u ¯∂ψ) in L²(Ω, e^−ϕ)

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Proof. (Some ideas going back to Berndtsson and B.-Y. Chen.) By approximation we may assume that ϕ is smooth up to the boundary and strongly psh, and ψ is bounded.

u - minimal solution to ¯∂u = α in L²(Ω, e^ψ−ϕ)

⇒ u ⊥ ker ¯∂ in L²(Ω, e^ψ−ϕ)

⇒ v := ue^ψ⊥ ker ¯∂ in L²(Ω, e^−ϕ)

⇒ v - minimal solution to ¯∂v = β := e^ψ(α + u ¯∂ψ) in L²(Ω, e^−ϕ) By H¨ormander’s estimate

Z

Ω

|v|²e^−ϕdλ ≤ Z

Ω

|β|²

i∂ ¯∂ϕe^−ϕdλ.

Therefore Z

Ω

|u|²e^2ψ−ϕdλ ≤ Z

Ω

|α + u ¯∂ψ|²_{i∂ ¯∂ϕ}e^2ψ−ϕdλ

Therefore Z

Ω

|u|²e^2ψ−ϕdλ ≤ Z

Ω

|α + u ¯∂ψ|²_{i∂ ¯∂ϕ}e^2ψ−ϕdλ

≤ Z

Ω



|α|²_{i∂ ¯∂ϕ}+ 2|u|

√

H|α|_{i∂ ¯∂ϕ}+ |u|²H



e^2ψ−ϕdλ, where H = | ¯∂ψ|²

i∂ ¯∂ϕ.

Therefore Z

Ω

|u|²e^2ψ−ϕdλ ≤ Z

Ω

|α + u ¯∂ψ|²_{i∂ ¯∂ϕ}e^2ψ−ϕdλ

≤ Z

Ω



|α|²_{i∂ ¯∂ϕ}+ 2|u|

√

H|α|_{i∂ ¯∂ϕ}+ |u|²H



e^2ψ−ϕdλ, where H = | ¯∂ψ|²

i∂ ¯∂ϕ. For t > 0 we will get Z

Ω

|u|²(1 − H)e^2ψ−ϕdλ

≤ Z

Ω



|α|²

i∂ ¯∂ϕ



1 + t⁻¹ H 1 − H



+ t|u|²(1 − H)



e^2ψ−ϕdλ

≤



1 + t⁻¹ δ 1 − δ

 Z

Ω

|α|²_{i∂ ¯∂ϕ}e^2ψ−ϕdλ

+ t Z

Ω

|u|²(1 − H)e^2ψ−ϕdλ.

Therefore Z

Ω

|u|²e^2ψ−ϕdλ ≤ Z

Ω

|α + u ¯∂ψ|²_{i∂ ¯∂ϕ}e^2ψ−ϕdλ

≤ Z

Ω



|α|²_{i∂ ¯∂ϕ}+ 2|u|

√

H|α|_{i∂ ¯∂ϕ}+ |u|²H



e^2ψ−ϕdλ, where H = | ¯∂ψ|²

i∂ ¯∂ϕ. For t > 0 we will get Z

Ω

|u|²(1 − H)e^2ψ−ϕdλ

≤ Z

Ω



|α|²

i∂ ¯∂ϕ



1 + t⁻¹ H 1 − H



+ t|u|²(1 − H)



e^2ψ−ϕdλ

≤



1 + t⁻¹ δ 1 − δ

 Z

Ω

|α|²_{i∂ ¯∂ϕ}e^2ψ−ϕdλ

+ t Z

Ω

|u|²(1 − H)e^2ψ−ϕdλ.

We will obtain the required estimate if we take t := 1/(δ^−1/2+ 1).

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Remarks.1. Setting ψ ≡ 0 we recover the H¨ormander estimate.

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Remarks.1. Setting ψ ≡ 0 we recover the H¨ormander estimate.

2. This theorem implies Donnelly-Fefferman and Berndtsson’s estimates with optimal constants:

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Remarks.1. Setting ψ ≡ 0 we recover the H¨ormander estimate.

2. This theorem implies Donnelly-Fefferman and Berndtsson’s estimates with optimal constants: for psh ϕ, ψ with | ¯∂ψ|²

i∂ ¯∂ψ≤ 1 and δ < 1 set ϕ := ϕ + ψ and ee ψ =^1+δ₂ ψ.

Theorem. Ω - pseudoconvex in Cⁿ, ϕ - psh in Ω α ∈ L²_loc,(0,1)(Ω), ¯∂α = 0

ψ ∈ W_loc^1,2(Ω) locally bounded from above, s.th.

| ¯∂ψ|²_{i∂ ¯∂ϕ}

(≤ 1 in Ω

≤ δ < 1 on supp α.

Then there exists u ∈ L²_loc(Ω) with ¯∂u = α and Z

Ω

|u|²(1 − | ¯∂ψ|²_{i∂ ¯∂ϕ})e^2ψ−ϕdλ ≤1 +√ δ 1 −√

δ Z

Ω

|α|²

i∂ ¯∂ϕe^2ψ−ϕdλ.

Remarks.1. Setting ψ ≡ 0 we recover the H¨ormander estimate.

2. This theorem implies Donnelly-Fefferman and Berndtsson’s estimates with optimal constants: for psh ϕ, ψ with | ¯∂ψ|²

i∂ ¯∂ψ≤ 1 and δ < 1 set ϕ := ϕ + ψ and ee ψ =^1+δ₂ ψ.

Then 2 eψ −ϕ = δψ − ϕ and | ¯e ∂ eψ|²

i∂ ¯∂ϕe≤ ^(1+δ)₄ ² =: eδ.