Weak Factorization and Hankel operators on Hardy spaces H 1 . Aline - - PowerPoint PPT Presentation

Weak Factorization and Hankel operators on Hardy spaces H1.

Aline Bonami

Universit´ e d’Orl´ eans

Bardonecchia, June 16, 2009

Let D be the unit disc (or Bn ∈ Cn) the unit ball ), ∂D its boundary. The Holomorphic Hardy space is Hp := {F holo ; sup

r<1

• ∂D

|F(rξ)|pdσ(ξ) < ∞}.

Let D be the unit disc (or Bn ∈ Cn) the unit ball ), ∂D its boundary. The Holomorphic Hardy space is Hp := {F holo ; sup

r<1

• ∂D

|F(rξ)|pdσ(ξ) < ∞}. Hp identifies with the space of its boundary values Hp = Hp(∂D) ∩ Hol,

Let D be the unit disc (or Bn ∈ Cn) the unit ball ), ∂D its boundary. The Holomorphic Hardy space is Hp := {F holo ; sup

r<1

• ∂D

|F(rξ)|pdσ(ξ) < ∞}. Hp identifies with the space of its boundary values Hp = Hp(∂D) ∩ Hol, Hp(∂D) = {f ;

• ∂D

|Mf (ξ)|pdσ(ξ) < ∞}.

Let D be the unit disc (or Bn ∈ Cn) the unit ball ), ∂D its boundary. The Holomorphic Hardy space is Hp := {F holo ; sup

r<1

• ∂D

|F(rξ)|pdσ(ξ) < ∞}. Hp identifies with the space of its boundary values Hp = Hp(∂D) ∩ Hol, Hp(∂D) = {f ;

• ∂D

|Mf (ξ)|pdσ(ξ) < ∞}. Mf (ξ) := sup

r<1

• ∂D

PS(rξ, η)f (η)dσ(η)

• with PS the Poisson (Szeg¨
• ) kernel, which reproduces holomorphic

functions.

Let Φ : [0, ∞) → [0, ∞) be an increasing homeomorphism. We assume that φ is doubling, and, for some p < 1, Φ(st) ≤ CspΦ(t) for s < 1. Particular interest for Φ concave (and, in particular, sub-additive). For us: Φ(t) :=

t log(e+t).

Let Φ : [0, ∞) → [0, ∞) be an increasing homeomorphism. We assume that φ is doubling, and, for some p < 1, Φ(st) ≤ CspΦ(t) for s < 1. Particular interest for Φ concave (and, in particular, sub-additive). For us: Φ(t) :=

t log(e+t).

LΦ is the space of functions such that f LΦ :=

• D

Φ(|f |)dσ < ∞. The Luxembourg “norm” is f lux

LΦ := inf

• λ > 0 :
• X

Φ |f (x)| λ

• dµ(x) ≤ 1
• .

HΦ := {F holo ; sup

r<1

• ∂D

Φ (|F(rξ)|) dσ(ξ) < ∞.} The space HΦ identifies with the space of its boundary values HΦ = HΦ(∂D) ∩ Hol, HΦ(∂D) := {f ;

• ∂D

Φ (Mf (ξ)) dσ(ξ) < ∞}. Studied by S. Janson and Viviani. The dual of HΦ is B M O Aρ, defined by sup

Q

1 ρ(σ(Q))σ(Q)

• Q

|b − bQ|dσ < ∞. ρ(t) := ρΦ(t) = 1 tΦ−1(t).

HΦ := {F holo ; sup

r<1

• ∂D

Φ (|F(rξ)|) dσ(ξ) < ∞.} The space HΦ identifies with the space of its boundary values HΦ = HΦ(∂D) ∩ Hol, HΦ(∂D) := {f ;

• ∂D

Φ (Mf (ξ)) dσ(ξ) < ∞}. Studied by S. Janson and Viviani. The dual of HΦ is B M O Aρ, defined by sup

Q

1 ρ(σ(Q))σ(Q)

• Q

|b − bQ|dσ < ∞. ρ(t) := ρΦ(t) = 1 tΦ−1(t). Main tool: the atomic decomposition.

Multiplication of functions in H1(∂D) and B M O

Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with b ∈ B M O and h ∈ H1 can be given a meaning as a distribution, and e b × h ∈ L1 + HΦ, with Φ(t) = t/ log(e + t).

Multiplication of functions in H1(∂D) and B M O

Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with b ∈ B M O and h ∈ H1 can be given a meaning as a distribution, and e b × h ∈ L1 + HΦ, with Φ(t) = t/ log(e + t). Main tool: H¨

• lder Inequality.

bhlux

LΦ ≤ Chlux L1 blux exp L.

Multiplication of functions in H1(∂D) and B M O

Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with b ∈ B M O and h ∈ H1 can be given a meaning as a distribution, and e b × h ∈ L1 + HΦ, with Φ(t) = t/ log(e + t). Main tool: H¨

• lder Inequality.

bhlux

LΦ ≤ Chlux L1 blux exp L.

Obtained as a consequence of the elementary inequality uv log(e + uv) ≤ u + ev − 1.

Multiplication of functions in H1(∂D) and B M O

Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with b ∈ B M O and h ∈ H1 can be given a meaning as a distribution, and e b × h ∈ L1 + HΦ, with Φ(t) = t/ log(e + t). Main tool: H¨

• lder Inequality.

bhlux

LΦ ≤ Chlux L1 blux exp L.

Obtained as a consequence of the elementary inequality uv log(e + uv) ≤ u + ev − 1. Moreover, for holomorphic functions, one can erase the term in L1.

• Remark. One can answer a question in [BIJZ]: find two

continuous bilinear operators S and T, such that b×h = S(b, h)+T(b, h) with S(b, h) ∈ HΦ and T(b, h) ∈ L1.

• Remark. One can answer a question in [BIJZ]: find two

continuous bilinear operators S and T, such that b×h = S(b, h)+T(b, h) with S(b, h) ∈ HΦ and T(b, h) ∈ L1. Proof in the dyadic setting in [0, 1]: b × h =

• j
• k

(Pjb − Pj−1b)(Pkh − Pk−1h) =

• j

Pj−1b Qjh +

• j

Pj−1h Qjb +

• Qjb Qjh.

One recognizes paraproducts.

Hankel operators

Let B be the unit disc/unit ball. Then P : L2(∂B) → H2 is the Szeg¨

• orthogonal projection.

The Hankel operator hb, with symbol b ∈ H2, is given by hb(f ) := P(b f ). Theorem[Nehari (d=1),Coifman-Rochberg-Weiss]. hb bounded on H2 ⇔ b ∈ B M O A.

Hankel operators

Let B be the unit disc/unit ball. Then P : L2(∂B) → H2 is the Szeg¨

• orthogonal projection.

The Hankel operator hb, with symbol b ∈ H2, is given by hb(f ) := P(b f ). Theorem[Nehari (d=1),Coifman-Rochberg-Weiss]. hb bounded on H2 ⇔ b ∈ B M O A. Theorem[Janson, Tolokonnikov (d=1), B.- Grellier-Sehba (d > 1)] hb bounded on H1 ⇔ b ∈ LMOA. That is, sup

Q

log 4/σ(Q) σ(Q)

• Q

|b − bQ|dσ < ∞,

Hankel operators

Let B be the unit disc/unit ball. Then P : L2(∂B) → H2 is the Szeg¨

• orthogonal projection.

The Hankel operator hb, with symbol b ∈ H2, is given by hb(f ) := P(b f ). Theorem[Nehari (d=1),Coifman-Rochberg-Weiss]. hb bounded on H2 ⇔ b ∈ B M O A. Theorem[Janson, Tolokonnikov (d=1), B.- Grellier-Sehba (d > 1)] hb bounded on H1 ⇔ b ∈ LMOA. That is, sup

Q

log 4/σ(Q) σ(Q)

• Q

|b − bQ|dσ < ∞, b ∈

• HΦ∗

with Φ(t) = t log(e + t). For d = 1, Janson-Peetre-Semmes through commutators.

Characterizations of symbols of bounded Hankel

• perators in H1.

hb(f ), g = P(bf ), g = b, fg.

Characterizations of symbols of bounded Hankel

• perators in H1.

hb(f ), g = P(bf ), g = b, fg. Products of functions of B M O A and H1 are in HΦ. The dual of HΦ is LMOA. Sufficient conditions are given by continuity properties of products.

Characterizations of symbols of bounded Hankel

• perators in H1.

hb(f ), g = P(bf ), g = b, fg. Products of functions of B M O A and H1 are in HΦ. The dual of HΦ is LMOA. Sufficient conditions are given by continuity properties of products. Necessary conditions are given by (weak) factorization theorems.

Necessary conditions are given by factorization theorems.

Want to estimate (log 4/σ(Q))2 σ(Q)

• Q

|b − bQ|2dσ = b, a = b, Pa with a an atom (up to a constant), with zero mean, supported by Q with a2 ≤ log 4/σ(Q) (σ(Q))

1 2

• Q

|b − bQ|2dσ 1

2

.

Necessary conditions are given by factorization theorems.

Want to estimate (log 4/σ(Q))2 σ(Q)

• Q

|b − bQ|2dσ = b, a = b, Pa with a an atom (up to a constant), with zero mean, supported by Q with a2 ≤ log 4/σ(Q) (σ(Q))

1 2

• Q

|b − bQ|2dσ 1

2

. Done if we can write Pa = fg, so that b, a = b, Pa = hbf , g.

The factorization

Theorem [BIJZ]. B M O A × H1 = HΦ.

• Proof. Let F ∈ HΦ. Then
• ∂D
• MF

log(e + MF)

• dσ < ∞.

The factorization

Theorem [BIJZ]. B M O A × H1 = HΦ.

• Proof. Let F ∈ HΦ. Then
• ∂D
• MF

log(e + MF)

• dσ < ∞.

Assume that there exists G ∈ B M O A such that log(e + MF) ≤ |G|. Then H := F/G is holomorphic with boundary values in L1(∂D).

The factorization

Theorem [BIJZ]. B M O A × H1 = HΦ.

• Proof. Let F ∈ HΦ. Then
• ∂D
• MF

log(e + MF)

• dσ < ∞.

Assume that there exists G ∈ B M O A such that log(e + MF) ≤ |G|. Then H := F/G is holomorphic with boundary values in L1(∂D). Use Coifman-Rochberg Theorem on the maximal function of Hardy and Littlewood MHL, with u an integrable function: g := log

• e + MHLu
• ∈ B

M O(∂D) Take for u the boundary values of the sub-harmonic function |F|p, so that MF ≤ C(MHLu)1/p. Take G := g + iHg.

Weak factorization

Coifman Rochberg Weiss in the unit ball: Every function in H1 can be as a sum of products of H2.

Weak factorization

Coifman Rochberg Weiss in the unit ball: Every function in H1 can be as a sum of products of H2. Theorem [B. Grellier]. In the unit ball, there is a weak factorization of HΦ in products of functions in H1 and B M O A. Key Point: log(1 − w.z) is uniformly in B M O A.

Weak factorization

Coifman Rochberg Weiss in the unit ball: Every function in H1 can be as a sum of products of H2. Theorem [B. Grellier]. In the unit ball, there is a weak factorization of HΦ in products of functions in H1 and B M O A. Key Point: log(1 − w.z) is uniformly in B M O A. Possible extensions to smooth bounded convex domains of finite type and to Hardy-Orlicz spaces.

The case of the bi-disc

B M O(T2) is the space of functions such that

• Ω

|PΩb|2 ≤ C|Ω| where PΩ can be defined in terms of wavelets (project on the ones that are related to rectangles R ⊂ Ω).

The case of the bi-disc

B M O(T2) is the space of functions such that

• Ω

|PΩb|2 ≤ C|Ω| where PΩ can be defined in terms of wavelets (project on the ones that are related to rectangles R ⊂ Ω). Can also be defined in terms of Carleson measures. The analogue of Nehari’s Theorem has been proved by Lacey and Ferguson in 2002. Theorem.The Hankel operator hb is bounded on H2(D2) if and

• nly if b is in B

M O A.

The case of the bi-disc

B M O(T2) is the space of functions such that

• Ω

|PΩb|2 ≤ C|Ω| where PΩ can be defined in terms of wavelets (project on the ones that are related to rectangles R ⊂ Ω). Can also be defined in terms of Carleson measures. The analogue of Nehari’s Theorem has been proved by Lacey and Ferguson in 2002. Theorem.The Hankel operator hb is bounded on H2(D2) if and

• nly if b is in B

M O A. What is the space L M O A for the bidisc?

Definition [Pott-Sehba].b is in L M O A if, for each rectangle I × J containing Ω, log 4 |I|

• × log

4 |J|

Ω

|PΩb|2 ≤ C|Ω|. Theorem[Pott-Sehba].The Hankel operator hb is bounded on H2(D2) if b is in L M O A.

Definition [Pott-Sehba].b is in L M O A if, for each rectangle I × J containing Ω, log 4 |I|

• × log

4 |J|

Ω

|PΩb|2 ≤ C|Ω|. Theorem[Pott-Sehba].The Hankel operator hb is bounded on H2(D2) if b is in L M O A.

Definition [Pott-Sehba].b is in L M O A if, for each rectangle I × J containing Ω, log 4 |I|

• × log

4 |J|

Ω

|PΩb|2 ≤ C|Ω|. Theorem[Pott-Sehba].The Hankel operator hb is bounded on H2(D2) if b is in L M O A. Theorem[B. Pott Sehba Wick].If the Hankel operator hb is bounded on H2(D2), then b is in L M O A.