Weak Factorization and Hankel operators on Hardy spaces H 1.

Transcription

1 Weak Factorization and Hankel operators on Hardy spaces H 1. Aline Bonami Université d Orléans Bardonecchia, June 16, 2009

2 Let D be the unit disc (or B n C n ) the unit ball ), D its boundary. The Holomorphic Hardy space is H p := {F holo ; sup F (rξ) p dσ(ξ) < }. r<1 D

3 Let D be the unit disc (or B n C n ) the unit ball ), D its boundary. The Holomorphic Hardy space is H p := {F holo ; sup F (rξ) p dσ(ξ) < }. r<1 H p identifies with the space of its boundary values D H p = H p ( D) Hol,

4 Let D be the unit disc (or B n C n ) the unit ball ), D its boundary. The Holomorphic Hardy space is H p := {F holo ; sup F (rξ) p dσ(ξ) < }. r<1 H p identifies with the space of its boundary values D H p = H p ( D) Hol, H p ( D) = {f ; D Mf (ξ) p dσ(ξ) < }.

5 Let D be the unit disc (or B n C n ) the unit ball ), D its boundary. The Holomorphic Hardy space is H p := {F holo ; sup F (rξ) p dσ(ξ) < }. r<1 H p identifies with the space of its boundary values H p ( D) = {f ; D H p = H p ( D) Hol, Mf (ξ) := sup r<1 D D Mf (ξ) p dσ(ξ) < }. P S (rξ, η)f (η)dσ(η) with P S the Poisson (Szegö) kernel, which reproduces holomorphic functions.

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

7 Let Φ : [0, ) [0, ) be an increasing homeomorphism. We assume that φ is doubling, and, for some p < 1, Φ(st) Cs p Φ(t) for s < 1. Particular interest for Φ concave (and, in particular, sub-additive). For us: Φ(t) := t log(e+t). L Φ is the space of functions such that f L Φ := Φ( f )dσ <. The Luxembourg norm is f lux L Φ := inf { λ > 0 : D X ( ) } f (x) Φ dµ(x) 1. λ

8 H Φ := {F holo ; sup Φ ( F (rξ) ) dσ(ξ) <.} r<1 D The space H Φ identifies with the space of its boundary values H Φ ( D) := {f ; H Φ = H Φ ( D) Hol, D Φ (Mf (ξ)) dσ(ξ) < }. Studied by S. Janson and Viviani. The dual of H Φ is BMOA ρ, defined by 1 sup b b Q dσ <. Q ρ(σ(q))σ(q) Q ρ(t) := ρ Φ (t) = 1 tφ 1 (t).

9 H Φ := {F holo ; sup Φ ( F (rξ) ) dσ(ξ) <.} r<1 D The space H Φ identifies with the space of its boundary values H Φ ( D) := {f ; H Φ = H Φ ( D) Hol, D Φ (Mf (ξ)) dσ(ξ) < }. Studied by S. Janson and Viviani. The dual of H Φ is BMOA ρ, defined by 1 sup b b Q dσ <. Q ρ(σ(q))σ(q) Q ρ(t) := ρ Φ (t) = 1 tφ 1 (t). Main tool: the atomic decomposition.

10 Multiplication of functions in H 1 ( D) and BMO Theorem [B. Iwaniec Jones Zinsmeister]. The product b h, with b BMO and h H 1 can be given a meaning as a distribution, and e b h L 1 + H Φ, with Φ(t) = t/ log(e + t).

11 Multiplication of functions in H 1 ( D) and BMO Theorem [B. Iwaniec Jones Zinsmeister]. The product b h, with b BMO and h H 1 can be given a meaning as a distribution, and e b h L 1 + H Φ, with Φ(t) = t/ log(e + t). Main tool: Hölder Inequality. bh lux L Φ C h lux L 1 b lux exp L.

12 Multiplication of functions in H 1 ( D) and BMO Theorem [B. Iwaniec Jones Zinsmeister]. The product b h, with b BMO and h H 1 can be given a meaning as a distribution, and e b h L 1 + H Φ, with Φ(t) = t/ log(e + t). Main tool: Hölder Inequality. bh lux L Φ C h lux L 1 b lux exp L. Obtained as a consequence of the elementary inequality uv log(e + uv) u + ev 1.

13 Multiplication of functions in H 1 ( D) and BMO Theorem [B. Iwaniec Jones Zinsmeister]. The product b h, with b BMO and h H 1 can be given a meaning as a distribution, and e b h L 1 + H Φ, with Φ(t) = t/ log(e + t). Main tool: Hölder Inequality. bh lux L Φ C h lux L 1 b lux 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 L 1.

14 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) H Φ and T (b, h) L 1.

15 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) H Φ and T (b, h) L 1. Proof in the dyadic setting in [0, 1]: b h = j = j (P j b P j 1 b)(p k h P k 1 h) k P j 1 b Q j h + P j 1 h Q j b + Q j b Q j h. j One recognizes paraproducts.

16 Hankel operators Let B be the unit disc/unit ball. Then P : L 2 ( B) H 2 is the Szegö orthogonal projection. The Hankel operator h b, with symbol b H 2, is given by h b (f ) := P(b f ). Theorem[Nehari (d=1),coifman-rochberg-weiss]. h b bounded on H 2 b BMOA.

17 Hankel operators Let B be the unit disc/unit ball. Then P : L 2 ( B) H 2 is the Szegö orthogonal projection. The Hankel operator h b, with symbol b H 2, is given by h b (f ) := P(b f ). Theorem[Nehari (d=1),coifman-rochberg-weiss]. h b bounded on H 2 b BMOA. Theorem[Janson, Tolokonnikov (d=1), B.- Grellier-Sehba (d > 1)] h b bounded on H 1 b LMOA. That is, sup Q log 4/σ(Q) σ(q) Q b b Q dσ <,

18 Hankel operators Let B be the unit disc/unit ball. Then P : L 2 ( B) H 2 is the Szegö orthogonal projection. The Hankel operator h b, with symbol b H 2, is given by h b (f ) := P(b f ). Theorem[Nehari (d=1),coifman-rochberg-weiss]. h b bounded on H 2 b BMOA. Theorem[Janson, Tolokonnikov (d=1), B.- Grellier-Sehba (d > 1)] h b bounded on H 1 b LMOA. That is, sup Q b (H Φ) log 4/σ(Q) σ(q) Q b b Q dσ <, with Φ(t) = t log(e + t). For d = 1, Janson-Peetre-Semmes through commutators.

19 Characterizations of symbols of bounded Hankel operators in H 1. h b (f ), g = P(bf ), g = b, fg.

20 Characterizations of symbols of bounded Hankel operators in H 1. h b (f ), g = P(bf ), g = b, fg. Products of functions of BMOA and H 1 are in H Φ. The dual of H Φ is LMOA. Sufficient conditions are given by continuity properties of products.

21 Characterizations of symbols of bounded Hankel operators in H 1. h b (f ), g = P(bf ), g = b, fg. Products of functions of BMOA and H 1 are in H Φ. The dual of H Φ is LMOA. Sufficient conditions are given by continuity properties of products. Necessary conditions are given by (weak) factorization theorems.

22 Necessary conditions are given by factorization theorems. Want to estimate (log 4/σ(Q)) 2 σ(q) Q b b Q 2 dσ = b, a = b, Pa with a an atom (up to a constant), with zero mean, supported by Q with ( ) 1 log 4/σ(Q) a 2 b b (σ(q)) 1 Q 2 2 dσ. 2 Q

23 Necessary conditions are given by factorization theorems. Want to estimate (log 4/σ(Q)) 2 σ(q) Q b b Q 2 dσ = b, a = b, Pa with a an atom (up to a constant), with zero mean, supported by Q with ( ) 1 log 4/σ(Q) a 2 b b (σ(q)) 1 Q 2 2 dσ. 2 Q Done if we can write Pa = fg, so that b, a = b, Pa = h b f, g.

24 The factorization Theorem [BIJZ]. BMOA H 1 = H Φ. Proof. Let F H Φ. Then ( ) MF dσ <. log(e + MF ) D

25 The factorization Theorem [BIJZ]. BMOA H 1 = H Φ. Proof. Let F H Φ. Then ( ) MF dσ <. log(e + MF ) D Assume that there exists G BMOA such that log(e + MF ) G. Then H := F /G is holomorphic with boundary values in L 1 ( D).

26 The factorization Theorem [BIJZ]. BMOA H 1 = H Φ. Proof. Let F H Φ. Then ( ) MF dσ <. log(e + MF ) D Assume that there exists G BMOA such that log(e + MF ) G. Then H := F /G is holomorphic with boundary values in L 1 ( D). Use Coifman-Rochberg Theorem on the maximal function of Hardy and Littlewood M HL, with u an integrable function: ( ) g := log e + M HL u BMO( D) Take for u the boundary values of the sub-harmonic function F p, so that MF C(M HL u) 1/p. Take G := g + ihg.

27 Weak factorization Coifman Rochberg Weiss in the unit ball: Every function in H 1 can be as a sum of products of H 2.

28 Weak factorization Coifman Rochberg Weiss in the unit ball: Every function in H 1 can be as a sum of products of H 2. Theorem [B. Grellier]. In the unit ball, there is a weak factorization of H Φ in products of functions in H 1 and BMOA. Key Point: log(1 w.z) is uniformly in BMOA.

29 Weak factorization Coifman Rochberg Weiss in the unit ball: Every function in H 1 can be as a sum of products of H 2. Theorem [B. Grellier]. In the unit ball, there is a weak factorization of H Φ in products of functions in H 1 and BMOA. Key Point: log(1 w.z) is uniformly in BMOA. Possible extensions to smooth bounded convex domains of finite type and to Hardy-Orlicz spaces.

30 The case of the bi-disc BMO(T 2 ) is the space of functions such that P Ω b 2 C Ω Ω where P Ω can be defined in terms of wavelets (project on the ones that are related to rectangles R Ω).

31 The case of the bi-disc BMO(T 2 ) is the space of functions such that P Ω b 2 C Ω Ω where P Ω 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 Theorem.The Hankel operator h b is bounded on H 2 (D 2 ) if and only if b is in BMOA.

32 The case of the bi-disc BMO(T 2 ) is the space of functions such that P Ω b 2 C Ω Ω where P Ω 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 Theorem.The Hankel operator h b is bounded on H 2 (D 2 ) if and only if b is in BMOA. What is the space LMOA for the bidisc?

33 Definition [Pott-Sehba].b is in LMOA if, for each rectangle I J containing Ω, ( ) ( ) 4 4 log log P Ω b 2 C Ω. I J Ω Theorem[Pott-Sehba].The Hankel operator h b is bounded on H 2 (D 2 ) if b is in LMOA.

34 Definition [Pott-Sehba].b is in LMOA if, for each rectangle I J containing Ω, ( ) ( ) 4 4 log log P Ω b 2 C Ω. I J Ω Theorem[Pott-Sehba].The Hankel operator h b is bounded on H 2 (D 2 ) if b is in LMOA.

35 Definition [Pott-Sehba].b is in LMOA if, for each rectangle I J containing Ω, ( ) ( ) 4 4 log log P Ω b 2 C Ω. I J Ω Theorem[Pott-Sehba].The Hankel operator h b is bounded on H 2 (D 2 ) if b is in LMOA. Theorem[B. Pott Sehba Wick].If the Hankel operator h b is bounded on H 2 (D 2 ), then b is in LMOA.