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The Cauchy Integral in Cn
Loredana Lanzani
L. Lanzani “Cauchy Integral”
Credits
I E. M. Stein
ArXiv: http://arxiv.org/abs/1201.4148
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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) < +∞}
L. Lanzani “Cauchy Integral”
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)
L. Lanzani “Cauchy Integral”
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.
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
Focus on Bergman:
Lp-Regularity problem for
B : L2(D, dV )→ ϑL2(D, dV ), B2 = B, B∗ = B, ‖B‖L2→L2 = 1
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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}
L. Lanzani “Cauchy Integral”
(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)!!!)
L. Lanzani “Cauchy Integral”
Focus on Ligocka (1982):
If is D strongly Levi-pseudoconvex and D ∈ C 3, then
B is bounded : Lp → Lp for 1 < p < +∞
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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 <∞
L. Lanzani “Cauchy Integral”
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)
L. Lanzani “Cauchy Integral”
Ligocka: D b Cn, strongly ψ-cvx, D ∈ C 3
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)
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
Ligocka: D b Cn, strongly ψ-cvx, D ∈ C 3
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
L. Lanzani “Cauchy Integral”
Ligocka: D b Cn, strongly ψ-cvx, D ∈ C 3
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
L. Lanzani “Cauchy Integral”
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)
L. Lanzani “Cauchy Integral”
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 <∞
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
D b Cn, strongly ψ-cvx, D ∈ C 2
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)
L. Lanzani “Cauchy Integral”
D b Cn, strongly ψ-cvx, D ∈ C 2
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”
D b Cn, strongly ψ-cvx, D ∈ C 2
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
L. Lanzani “Cauchy Integral”
D b Cn, strongly ψ-cvx, D ∈ C 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
L. Lanzani “Cauchy Integral”
D b Cn, strongly ψ-cvx, D ∈ C 2
Note: A1,ε := H∗1,ε−H1,ε fails compactness criteria for L2(D, dV ).
I Proof: Cancellation: if |w − z | ≤ µ then
|A1,ε(w , z)| = |H1,ε(z ,w)− H1,ε(w , z)| . ε
|gε(w , z)|n+1
I A1,ε,λf (z) :=
∫|gε(w ,z)|>λ
f (w)
|gε(w , z)|n+1dV (w), λ < µ
I A1,ε,λ is compact in L2(D, dV ) for any λ > 0, but
I ‖A1,ε − A1,ε,λ‖L2→L2 . λ0 = 1
I A1,ε,λ 6→ A1,ε as λ→ 0
L. Lanzani “Cauchy Integral”
D b Cn, strongly ψ-cvx, D ∈ C 2
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
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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.
L. Lanzani “Cauchy Integral”
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
L. Lanzani “Cauchy Integral”
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).
L. Lanzani “Cauchy Integral”