The Cauchy Integral in Cn - University of Minnesota

The Cauchy Integral in Cn

Loredana Lanzani

L. Lanzani “Cauchy Integral”

Credits

I E. M. Stein

ArXiv: http://arxiv.org/abs/1201.4148


Overview.

I Develop Lp-theory of Singular Integrals on

D b Cn

I for D with minimal boundary regularity

I modeled after Cauchy Integral for a Lipschitz Curve Γ ⊂ C:

Hf (z) =1

2πi

∫w∈Γ

f (w)dw

w − z, z ∈ D

z ∈ bD: Calderon (1978); Coifman-McIntosh-Meyer (1980s);David (1980s); David-Semmes (1990s); Semmes (1990s).

Also: Jones; Wolfe; Melnikov-Verdera; Nazarov-Volberg-Treil;Tolsa;....

I Focus on Singular Integrals with Holomorphic kernelI Applications to Complex Function theory for D b Cn


Complex Function Theory: Objects

D b Cn = {z = (z1, . . . , zn) , zj = xj + iyj}, n ≥ 1

ϑ(D) :=

{F : D → C,

∂F

∂z j:=

(∂

∂xj+ i

∂

∂yj

)F = 0 , j = 1, . . . , n

}1 ≤ p < +∞

I Bergman Space

ϑLp(D) :=

{F ∈ ϑ(D),

∫w∈D

|F (w)|pdV (w) < +∞}

I D rectifiable: Hardy Space (aka Smirnov Class)

Hp(bD) :=

{F ∈ ϑ(D), sup

ε>0

∫w∈bDε

|F (w)|pdσε(w) < +∞}


Background

LemmaThe Bergman Space

ϑLp(D) =

{F ∈ ϑ(D),

∫w∈D

|F (w)|pdV (w) < +∞}

is a closed subspace of Lp(D, dV ).

Proof.Cauchy formula on (poly)disc:

for any compact subset K ⊂ D, for any F ∈ ϑ(D), for any z ∈ K:

|F (z)| ≤ C (K) ‖F‖L1(P(z,δ)) ≤ C (K) ‖F‖Lp(D)

δ :=1

2dist(K, bD)


Background

LemmaThe Hardy Space

Hp(bD) :=

{F ∈ ϑ(D), sup

ε>0

∫w∈bDε

|F (w)|pdσε(w) < +∞}

is a closed subspace of Lp(bD, σ).

Proof.Cauchy formula on polydisc + Co-Area Formula:

|F (z)| ≤ C (K) ‖F‖Lp(P(z,δ)) ≤ C (K) ‖F‖Lp(Dε)

≤ C (K) supε>0

∫w∈bDε

|F (w)|pdσε(w)

1/p

for any compact subset K ⊂ D, for any F ∈ ϑ(D), for anyz ∈ K.


Complex Function Theory for D b Cn: Objects

D b Cn, n ≥ 1; p = 2

I Bergman Projection:

B : L2(D, dV ) � ϑL2(D), B2 = B, B∗ = B, ‖B‖L2→L2 = 1

I Szego Projection:

S : L2(bD, σ) � H2(bD), S2 = S, S∗ = S, ‖S‖L2→L2 = 1


Complex Function theory for D b Cn: Objectives

given D b Cn, n ≥ 1

Lq-Regularity problem for Szego Projection:

find (largest) Q = Q(D) ∈ [2,+∞] such that

S : Lq(bD, σ)→ Lq(bD, σ) is bounded for Q ′ < q < Q

Lp-Regularity problem for Bergman Projection:

find (largest) P = P(D) ∈ [2,+∞] such that

B : Lp(D, dV )→ Lp(D, dV ) is bounded for P ′ < p < P


Motivation

I Hormander; Kohn: The Bergman Projection is closelyrelated to the canonical solution of

∂u = f

I Bell (n=1): The Szego projection is closely related to thesolution of

∆u = 0, u

∣∣∣∣bD

= f

I Size of each of maximal (P ′,P) and (Q ′,Q) appears to berelated to geometry and regularity of ambient domain D


Focus on Bergman:

Lp-Regularity problem for

B : L2(D, dV )→ ϑL2(D, dV ), B2 = B, B∗ = B, ‖B‖L2→L2 = 1


History D b C (n = 1)

n = 1 D b C simply connected

I L.L. – Stein (2004):

I If D is of class C 1, then P = +∞

(Also true for Vanishing Chord-Arc)

(“VCA” ⇐⇒ σ(W ,Z ) = (1 + o(1))|W − Z |, W , Z ∈ bD)

I If D Lipschitz with constant M, then

P ≥ 2(

1 +π

2 arctanM

)> 4

I If D rectifiable local graph, then P ≥ 4

I Hedenmalm (2002): P = 2 + ε(D) for any D


Main tools in C (n=1)

D ⊂ C simply connected

I D = {ρ(w) < 0} ∈ C 1:

Solid Cauchy: Hf (z) =1

2πi

∫w∈D

f (w)∂ρ(w) ∧ dw

[w − z − ρ(w)]2z ∈ D

I D ∈ { Lipschitz; Rectifiable graph; “any”}:

Conformal Map: ϕ : D1(0)→ D


Obstacles in Cn n ≥ 2

Example: a dimension-induced phenomenon:

“Every connected open set S ⊂ X is convex”

I True for X = RI False for X = RN , N ≥ 2

The Lp-theory for the Bergman projection for D b Cn, n ≥ 2 ismuch less developed than corresponding theory for D b C due to

I Dimension-induced obstructions (Cn vs. C)

I Complex-Structure-induced obstructions (Cn vs. R2n)

These obstructions ultimately lead to the requirement that

D b Cn be “pseudoconvex”

(Note: every D b C is pseudoconvex)L. Lanzani “Cauchy Integral”

History D b Cn, n ≥ 2

Definition:

D = {λ(w) < 0} b Cn is Strongly Levi-Pseudoconvex iff

D is of class C 2, i.e. λ ∈ C 2(Cn) and

n∑j ,k=1

∂2λ

∂ζj∂ζk(ζ) vjvk ≥ c0|v |2, ζ ∈ bD, v ∈ TC

ζ (bD)

for any defining function λ. Here

TCζ (bD) = TR

ζ (bD) ∩ iTRζ (bD)

Fact: If D is strongly Levi-pscvx then there is a special definingfunction ρ that is strictly plurisubharmonic in D i.e.

n∑j ,k=1

∂2ρ

∂ζj∂ζk(ζ) vjvk ≥ c0|v |2, ζ ∈ bD, v ∈ Cn


Example: the Siegel Upper Half Space:

D := {z = (z1, z2) ∈ C2 | Im z2 < |z1|2}

is strongly Levi-pseudoconvex but is not strongly convex becausebD contains the real line

{(0, x2 + i0), x2 ∈ R}


(Some) History D b Cn, n ≥ 2

Bekolle’-Bonami; Bell; Bonami-Lohoue’; Charpentier-DuPain;Fefferman; Halfpap-Nagel-Wainger; Krantz-Peloso; McNeal;McNeal-Stein; Nagel-Pramanik; Nagel-Rosay-Stein-Wainger;Phong-Stein.....

I Ligocka (1982): If is D strongly Levi-pseudoconvex andD ∈ C 3, then:

B is bounded : Lp → Lp for 1 < p < +∞

I Zeytuncu (2011): there is D0 b C2 such that

B is bounded : Lp → Lp only for p = 2

(!!! see Hedenmalm (n = 1)!!!)


Focus on Ligocka (1982):

If is D strongly Levi-pseudoconvex and D ∈ C 3, then

B is bounded : Lp → Lp for 1 < p < +∞


Ligocka’s Strategy for B, n ≥ 1

Compare B with suitable Cauchy-Fantappie’ Integral H:

B : L2(D, dV ) � ϑL2(D), B2 = B, B∗ = B

H : L2(D, dV )→ ϑL2(D), H2 = H, (H∗ 6= H):

BH = H

HB = B ⇒ BH∗ = B

H = B [1− (H∗ −H)]

A := H∗ −H

Goal:

1. Prove H bounded: Lp(D, dV )→ Lp(D, dV ), 1 < p <∞2. Invert 1− A in Lp(D, dV )

I conclude B = H (1− A)−1 is bounded Lp → Lp

Note: this strategy requires H with holomorphic kernelL. Lanzani “Cauchy Integral”

Obstacles in Cn, n ≥ 2.

I “Canonical” Kernel for D = {ρ(w) < 0}:

H(w , z) =

∂w n∑

j=1

(w j − z j)

|w − z |2 − ρ(w)dwj

n

Bochner-Martinelli-Ligocka kernel

I n = 1: H(w , z) = ∂w

(dw

w − z − ρ(w)

)Cauchy-Ligocka

I n ≥ 2: B-M-L not holomorphic w.r.t. z ∈ D:

need ad-hoc H


Classical Approach to implementing Ligocka’s Strategy

n ≥ 2 D b Cn

I Find ad-hoc H : L2(D, dV )→ ϑL2(D), H2 = H

I Show: H bounded Lp(D, dV )→ Lp(D, dV ), 1 < p <∞

I Show: A := H∗ −H compact in L2(D, dV );

I Obtain: B = H(1− A)−1 in L2(D, dV )

I Conclude: B : Lp(D, dV )→ Lp(D, dV ), 1 < p <∞


Ligocka: D b Cn, strongly ψ-cvx, D ∈ C 3

Step 1: Construct ad-hoc H: an integral operator with kernel

H(w , z) := H1(w , z) + H2(w , z) ∈ Λ(w)n,n (D)× Λ

(z)0,0(D)

I H1(w , z) = locally holomorphic w.r.t. z

I H2(w , z) ∈ C (D × D) a correction term s.t.

∂z(H1(w , z) + H2(w , z)) = 0, (w , z) ∈ D × U(D)

(H2 : Lp → Lp; H∗2 −H2 compact on L2 ... – so ignore)



Step 1: Construction of H1(w , z):

Let

I D = {ρ(w) < 0}, ρ ∈ C 3(Cn) strictly plurisubharmonic.

I Pw (z) := Levi polynomial of ρ:

Pw (z) :=∑j

∂ρ(w)

∂ζj(wj−zj)−

1

2

∑j ,k

∂2ρ(w)

∂ζj∂ζk(wj−zj)(wk−zk)

I χ(w , z) = χ(|w − z |) smooth cutoff on {|w − z | < µ}

I g(w , z) := Pw (z)χ(w , z) + |w − z |2(1− χ(w , z))− ρ(w)


Ligocka: D b Cn, strongly ψ-cvx, D = {ρ < 0} ∈ C 3

If

g(w , z) := Pw (z)χ(w , z) + |w − z |2(1− χ(w , z))− ρ(w)

Then

I basic inequality:

2Re g(w , z) ≥{−ρ(w)− ρ(z) + c |w − z |2, if |w − z | < µ

c > 0, if |w − z | ≥ µ

I size estimate:

|g(w , z)| ≈ |ρ(w)|+ |ρ(z)|+ |Im〈∂ρ(w),w − z〉|+ |w − z |2

I symmetry estimate: |g(w , z)| ≈ |g(z ,w)|

I cancellation: |g(w , z)− g(z ,w)| . |w − z |3


Ligocka: D b Cn, strongly ψ-cvx, D = {ρ < 0} ∈ C 3

Step 1: Construction of H1(w , z):

I H1(w , z) :=

[∂w

(η(w , z)

g(w , z)

)]nI η(w , z) :=∑j

[(∂ρ(w)

∂ζj− 1

2

∑k

∂2ρ(w)

∂ζj∂ζk(wk − zk)

)χ+ (w j − z j)(1− χ)

]dwj

H1f (z) :=

∫w∈D

f (w)

[∂w

(η(w , z)

g(w , z)

)]n



Step 2: show H1 bounded: Lp(D, dV )→ Lp(D, dV )

I Comparison operator:

Γ (f )(z) =

∫D

|g(w , z)|−n−1f (w) dV (w), z ∈ D

I Theorem: For 1 < p <∞, we have

‖Γ (f )‖Lp(D) ≤ cp‖f ‖Lp(D)

Proof: size estimates and symmetry estimates for g .

I Theorem: H1 is bounded: Lp → Lp, 1 < p <∞

Proof: |H1f (z)| . Γ (|f |)(z), z ∈ D



Step 3: show A1 := H∗1 −H1 compact in L2(D, dV ).

I Proof: Cancellation: If |w − z | ≤ µ then

|A1(w , z)| = |H1(z ,w)− H1(w , z)| . |w − z ||g(w , z)|n+1

I Size estimates:

|w − z ||g(w , z)|n+1

.1

|g(w , z)|n+1−1/2

I A1,λf (z) :=

∫|g(w ,z)|>λ

f (w)

|g(w , z)|n+1−1/2dV (w), λ < µ

I A1,λ is compact in L2(D, dV ) for any λ > 0.

I ‖A1 − A1,λ‖L2→L2 . λ1/2

I A1,λ → A1 as λ→ 0


Objective

Ligocka: Why D ∈ C 3?

I η has∂2ρ

∂ζi∂ζj(w) (so D ∈ C 2 would seem OK), but

I H1(w , z) ≈ (∂w η)n uses three derivatives of ρ.

I Cancellation: |g(w , z)− g(z ,w)| . |w − z |3 uses ρ ∈ C 3.

Our current goal: deal with D ∈ C 2

D ∈ C 2 is optimal (minimal) in strongly Levi-pseudconvex category

(D strongly Levi-pseudconvex ⇐⇒ D ∈ C 2 and strongly pseudoconvex)


Results

D b Cn, strongly ψ-cvx, D ∈ C 2

Bergman : Bf (z) =

∫w∈D

f (w)B(w , z) dV (w) z ∈ D

Absolute Bergman : |B|f (z) :=

∫w∈D

f (w) |B(w , z)| dV (w) z ∈ D

I Theorem (L. - Stein, 2011) B bdd: Lp → Lp

I Theorem (L. - Stein, 2011): ϑ(D)Lp

= ϑLp(D)

I Theorem (L. - Stein, 2011) : |B| bdd: Lp → Lp

1 < p <∞


Also, analogous results for

Bσ : L2σ(D) � ϑL2

σ(D), (f , g)σ :=

∫w∈D

f (w)g(w)σ(w)dV (w)

σ ∈ C (D), σ(w) > 0, w ∈ D



A single H will no longer do. Instead:

Step 1: Construct ad-hoc family {Hε}ε>0 with kernel

Hε(w , z) := H1,ε(w , z) + H2,ε(w , z) ∈ Λ(w)n,n (D)× Λ

(z)0,0(D)

I H1,ε(w , z) =

(∂w

(ηεgε

))nlocally holomorphic w.r.t. z

I H2,ε(w , z) ∈ C (D × D) a correction term s.t.

∂z(H1,ε(w , z) +H2,ε(w , z)) = 0, ε > 0, z ∈ Uw (D), w ∈ D

(H2,ε : Lp → Lp; H∗2,ε −H2,ε compact on L2 – so ignore)



Step 1: Construction of H1,ε(w , z):

Let

I D = {ρ(w) < 0}, ρ ∈ C 2(Cn) strictly plurisubharmonic.

I Given ε > 0 choose τ εj ,k ∈ C 2(D) s.t.

supw∈D

∣∣∣∣∂2ρ(w)

∂ζj∂ζk− τ εj ,k(w)

∣∣∣∣ ≤ ε for all 1 ≤ j , k ≤ n

Use τ εj ,k(w) in place of∂2ρ(w)

∂ζj∂ζk

I H1,ε(w , z) :=

(∂w

(ηε(w , z)

gε(w , z)

))nL. Lanzani “Cauchy Integral”


If

gε(w , z) := Pw ,ε(z)χ(w , z) + |w − z |2(1− χ(w , z))− ρ(w)

Then for any 0 < ε < ε0

I basic inequality:

2Re gε(w , z) ≥{−ρ(w)− ρ(z) + c |w − z |2, if |w − z | < µ

c > 0, if |w − z | ≥ µ

I size estimate:

|gε(w , z)| ≈ |ρ(w)|+ |ρ(z)|+ |Im〈∂ρ(w),w − z〉|+ |w − z |2

I symmetry estimate: |gε(w , z)| ≈ |gε(z ,w)|

I cancellation: |gε(w , z)− gε(z ,w)| ≤ cε|w − z |2



Step 2: show H1,ε bounded: Lp(D, dV )→ Lp(D, dV ) unif. in ε

I Comparison operators:

Γε (f )(z) =

∫D

|gε(w , z)|−n−1f (w) dV (w), z ∈ D

I Theorem: For 1 < p <∞, for any 0 < ε < ε0 we have

‖Γε(f )‖Lp(D) ≤ cp‖f ‖Lp(D)

Proof: size estimates and symmetry estimates for gε.

I Theorem: H1,ε is bounded: Lp → Lp, 1 < p <∞

Proof: |H1,εf (z)| . Γε(|f |)(z), z ∈ D



What works instead:

Lemma For each ε > 0, we have

H∗1,ε −H1,ε = Eε + Rε = Essential Part + Remainder

where

(a) ‖Eε‖Lp→Lp ≤ ε cp , for 1 < p <∞

(b) Each Remainder has continuous kernel on D × D – hence

Rε : L1(D)→ C (D)Proof:

I by cancellation: |Eε(w , z)| . ε|gε(w , z)|−n−1

I |Eεf (z)| . εΓε(|f |)(z)

Caveat: the norm of Rε may increase as ε→ 0


D b Cn strongly ψ-cvx, D ∈ C 2: Kerzman-Stein-Ligocka,revisited

Theorem: B is bounded: Lp → Lp, 1 < p <∞.

Proof:

I B (1− (H∗ε −Hε)) = Hε

I H∗ε −Hε = Eε + Rε

I B (1− Eε) = Hε − BRε

I fix 1 < p <∞; choose ε = ε(p) s.t. ‖Eε‖Lp→Lp < 1. Then

I B = (Hε − BRε) (1− Eε)−1

I Claim: BRε : Lp → Lp:I Proof. wlog: 1 < p ≤ 2. Then

Lp ↪→ L1 → C (D) ↪→ L∞ ↪→ L2 → L2 ↪→ Lp


Absolute Bergman: Positive majorants.

I Definition. T is a bounded linear operator on Lp, we saythat T has a positive majorant T , if T is bounded linearoperator on Lp s.t.{

T (f ) ≥ 0 if f ≥ 0, and

|T (f )(z)| ≤ T (|f |)(z), for a.e. z .

I (T1 + T2) = T1 + T2

I (T1 ◦ T2) = T1 ◦ T2

I ‖Tn − T‖p → 0 and ‖Tn − S‖p → 0⇒ S = T

I All of these grant

(1− T )−1 = (1− T )−1

I If T : Lp → C (D) then T has a positive majorant.


Absolute Bergman

I Hε = cΓεI H∗ε = c ′ΓεI Eε = c ′′ΓεI (1− Eε)−1 = (1− Eε)

−1

I BRε has a positive majorant.

As a result: B has a positive majorant B

And so does |B|:

||B|(f )(z)| ≤ M(B(|f |))(z), z ∈ D

M(F )(z) :=1

V (Bz)

∫Bz

F (w) dV (w), Bz = {|w−z | < 1

2d(z , bD)}

So |B| is bounded: Lp → Lp, 1 < p <∞.L. Lanzani “Cauchy Integral”

Density for Bergman Space.

Theorem ϑ(D)Lp

= ϑLp(D)

Proof

I f ∈ ϑLp(D)⇒ Hε(f ) = f

fn(w) :=

{f (w) if w ∈ D1/n := {ρ < −1/n}

0 if w ∈ D \ D1/n

I ‖fn − f ‖p → 0

I Fn := Hεfn ∈ ϑ(D−1/n)

I D1/n b D b D−1/n

I ‖Fn − f ‖p = ‖Hε(fn − f )‖p . ‖fn − f ‖p → 0


Szego Projection

Suppose

D b Cn strongly Levi-pseudoconvex ,D ∈ C 2

Then

1. Theorem (L. - Stein, 2011) S bdd: Lp(bD)→ Lp(bD)

2. Theorem (L. - Stein, 2011): ϑ(D)Lp

= Hp(bD)

1 < p <∞

I Proofs for S are more difficult than proofs for B because

I There is no comparison operator Γ that will work for S.What works instead:

I T (1)-theorem for suitable space of homogeneous type:{bD; µ; d}.

I Also, |S| not bounded: Lp(bD, dσ)→ Lp(bD, dσ).


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