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

Weak Factorization and Hankel operators on Hardy spaces H 1 . Aline Bonami Universit´ e d’Orl´ eans Bardonecchia, June 16, 2009

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 ( r ξ ) | p d σ ( ξ ) < ∞} . r < 1 ∂ D

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 ( r ξ ) | p d σ ( ξ ) < ∞} . r < 1 ∂ D H p identifies with the space of its boundary values H p = H p ( ∂ D ) ∩ Hol ,

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 ( r ξ ) | p d σ ( ξ ) < ∞} . r < 1 ∂ D H p identifies with the space of its boundary values H p = H p ( ∂ D ) ∩ Hol , � H p ( ∂ D ) = { f |M f ( ξ ) | p d σ ( ξ ) < ∞} . ; ∂ D

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 ( r ξ ) | p d σ ( ξ ) < ∞} . r < 1 ∂ D H p identifies with the space of its boundary values H p = H p ( ∂ D ) ∩ Hol , � H p ( ∂ D ) = { f |M f ( ξ ) | p d σ ( ξ ) < ∞} . ; ∂ D � � � � � M f ( ξ ) := sup P S ( r ξ, η ) f ( η ) d σ ( η ) � � r < 1 � � ∂ D with P S the Poisson (Szeg¨ o) 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 ) ≤ Cs p Φ( t ) for s < 1 . Particular interest for Φ concave (and, in particular, sub-additive). t For us: Φ( 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 ) ≤ Cs p Φ( t ) for s < 1 . Particular interest for Φ concave (and, in particular, sub-additive). t For us: Φ( t ) := log( e + t ) . L Φ is the space of functions such that � � f � L Φ := Φ( | f | ) d σ < ∞ . D The Luxembourg “norm” is � � � | f ( x ) | � � � f � lux L Φ := inf λ > 0 : Φ d µ ( x ) ≤ 1 . λ X

� H Φ := { F holo ; sup Φ ( | F ( r ξ ) | ) d σ ( ξ ) < ∞ . } r < 1 ∂ D The space H Φ identifies with the space of its boundary values H Φ = H Φ ( ∂ D ) ∩ Hol , � H Φ ( ∂ D ) := { f ; Φ ( M f ( ξ )) d σ ( ξ ) < ∞} . ∂ D Studied by S. Janson and Viviani. The dual of H Φ is B M O A ρ , defined by 1 � sup | b − b Q | d σ < ∞ . ρ ( σ ( Q )) σ ( Q ) Q Q 1 ρ ( t ) := ρ Φ ( t ) = t Φ − 1 ( t ) .

� H Φ := { F holo ; sup Φ ( | F ( r ξ ) | ) d σ ( ξ ) < ∞ . } r < 1 ∂ D The space H Φ identifies with the space of its boundary values H Φ = H Φ ( ∂ D ) ∩ Hol , � H Φ ( ∂ D ) := { f ; Φ ( M f ( ξ )) d σ ( ξ ) < ∞} . ∂ D Studied by S. Janson and Viviani. The dual of H Φ is B M O A ρ , defined by 1 � sup | b − b Q | d σ < ∞ . ρ ( σ ( Q )) σ ( Q ) Q Q 1 ρ ( t ) := ρ Φ ( t ) = t Φ − 1 ( t ) . Main tool: the atomic decomposition.

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

Multiplication of functions in H 1 ( ∂ D ) and B M O Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with O and h ∈ H 1 can be given a meaning as a distribution, b ∈ B M and e b × h ∈ L 1 + H Φ , with Φ( t ) = t / log( e + t ) . Main tool: H¨ older Inequality. � bh � lux L Φ ≤ C � h � lux L 1 � b � lux exp L .

Multiplication of functions in H 1 ( ∂ D ) and B M O Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with O and h ∈ H 1 can be given a meaning as a distribution, b ∈ B M and e b × h ∈ L 1 + H Φ , with Φ( t ) = t / log( e + t ) . Main tool: H¨ older 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 + e v − 1 .

Multiplication of functions in H 1 ( ∂ D ) and B M O Theorem [B. Iwaniec Jones Zinsmeister]. The product b × h, with O and h ∈ H 1 can be given a meaning as a distribution, b ∈ B M and e b × h ∈ L 1 + H Φ , with Φ( t ) = t / log( e + t ) . Main tool: H¨ older 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 + e v − 1 . Moreover, for holomorphic functions, one can erase the term in L 1 .

Remark. One can answer a question in [BIJZ]: find two continuous bilinear operators S and T , such that with S ( b , h ) ∈ H Φ and T ( b , h ) ∈ L 1 . b × h = S ( b , h )+ T ( b , h )

Remark. One can answer a question in [BIJZ]: find two continuous bilinear operators S and T , such that with S ( b , h ) ∈ H Φ and T ( b , h ) ∈ L 1 . b × h = S ( b , h )+ T ( b , h ) Proof in the dyadic setting in [0 , 1]: � � b × h = ( P j b − P j − 1 b )( P k h − P k − 1 h ) j k � � � = P j − 1 b Q j h + P j − 1 h Q j b + Q j b Q j h . j j One recognizes paraproducts.

Hankel operators Let B be the unit disc/unit ball. Then P : L 2 ( ∂ B ) �→ H 2 is the Szeg¨ o 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 ∈ B M O A .

Hankel operators Let B be the unit disc/unit ball. Then P : L 2 ( ∂ B ) �→ H 2 is the Szeg¨ o 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 ∈ B M O A . Theorem[Janson, Tolokonnikov (d=1), B.- Grellier-Sehba h b bounded on H 1 ⇔ b ∈ LMOA . ( d > 1) ] That is, log 4 /σ ( Q ) � sup | b − b Q | d σ < ∞ , σ ( Q ) Q Q

Hankel operators Let B be the unit disc/unit ball. Then P : L 2 ( ∂ B ) �→ H 2 is the Szeg¨ o 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 ∈ B M O A . Theorem[Janson, Tolokonnikov (d=1), B.- Grellier-Sehba h b bounded on H 1 ⇔ b ∈ LMOA . ( d > 1) ] That is, log 4 /σ ( Q ) � sup | b − b Q | d σ < ∞ , σ ( Q ) Q Q t � H Φ � ∗ b ∈ Φ( t ) = log( e + t ) . with For d = 1, Janson-Peetre-Semmes through commutators.

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

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

Characterizations of symbols of bounded Hankel operators in H 1 . � h b ( f ) , g � = � P ( bf ) , g � = � b , fg � . A and H 1 are in H Φ . The dual of H Φ Products of functions of B M O 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 � | b − b Q | 2 d σ = � b , a � σ ( Q ) Q = � b , Pa � with a an atom (up to a constant), with zero mean, supported by Q with � 1 � a � 2 ≤ log 4 /σ ( Q ) �� 2 | b − b Q | 2 d σ . 1 ( σ ( Q )) 2 Q

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

The factorization A × H 1 = H Φ . Theorem [BIJZ]. B M O Proof. Let F ∈ H Φ . Then � � � M F d σ < ∞ . log( e + M F ) ∂ D

The factorization A × H 1 = H Φ . Theorem [BIJZ]. B M O Proof. Let F ∈ H Φ . Then � � � M F d σ < ∞ . log( e + M F ) ∂ D Assume that there exists G ∈ B M O A such that log( e + M F ) ≤ | G | . Then H := F / G is holomorphic with boundary values in L 1 ( ∂ D ).

The factorization A × H 1 = H Φ . Theorem [BIJZ]. B M O Proof. Let F ∈ H Φ . Then � � � M F d σ < ∞ . log( e + M F ) ∂ D Assume that there exists G ∈ B M O A such that log( e + M F ) ≤ | 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: � � e + M HL u g := log ∈ B M O ( ∂ D ) Take for u the boundary values of the sub-harmonic function | F | p , so that M F ≤ C ( M HL u ) 1 / p . Take G := g + i H g .

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