Two-Weight Inequalities for Commutators with Calder on-Zygmund - PowerPoint PPT Presentation

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, in terms of the BMO norm of b .

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b .

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b . Recall: ◮ A p weights: � w � Q � w 1 − q � p − 1 [ w ] A p := sup Q Q

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b . Recall: ◮ A p weights: � w � Q � w 1 − q � p − 1 [ w ] A p := sup Q Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s)

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b . Recall: ◮ A p weights: � w � Q � w 1 − q � p − 1 [ w ] A p := sup Q Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s) ◮ M : L p ( w ) → L p ( w ) ⇔ w ∈ A p

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b . Recall: ◮ A p weights: � w � Q � w 1 − q � p − 1 [ w ] A p := sup Q Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s) ◮ H : L p ( w ) → L p ( w ) ⇔ w ∈ A p

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b . Recall: ◮ A p weights: � w � Q � w 1 − q � p − 1 [ w ] A p := sup Q Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s) ◮ H : L p ( w ) → L p ( w ) ⇔ w ∈ A p ◮ A 2 weights: � w � Q � w − 1 � Q [ w ] A 2 := sup Q

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b .

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b ??

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b ?? Recall: ◮ OK in the one-weight case µ = λ .

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b ?? Recall: ◮ OK in the one-weight case µ = λ . ◮ What if µ � = λ ?

Introduction GOAL: two-weight version of Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables , 1976 ◮ Characterize the norm of the commutator [ b , T ] : L p ( R n ; µ ) → L p ( R n ; λ ) , where T is a CZO, µ , λ are A p weights, in terms of the BMO norm of b ?? Recall: ◮ OK in the one-weight case µ = λ . ◮ What if µ � = λ ? Bloom!

Outline Introduction Bloom’s Result Main Results Upper Bound Lower Bound: Key Idea

Bloom (1985) 𝑐, 𝐼 : ¡𝑀 ' → ¡ 𝑀 ' 𝑐 ¡ ∈ 𝐶𝑁𝑃 bounded 1 𝑐 -./ ≔ ¡sup 𝑅 ¡ 7 𝑐 𝑦 − ¡ 𝑐 4 𝑒𝑦 4 4

Bloom (1985) 𝑐, 𝐼 : ¡𝑀 ' (𝑥) → ¡𝑀 ' (𝑥) 𝑐 ¡ ∈ 𝐶𝑁𝑃 bounded 1 𝑐 012 ≔ ¡sup 𝑅 ¡ : 𝑐 𝑦 − ¡ 𝑐 7 𝑒𝑦 7 7

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃 bounded 1 𝑐 123 ≔ sup 𝑅 ; 𝑐 𝑦 − 𝑐 8 𝑒𝑦 8 8

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃 bounded 1 𝑐 123 ≔ sup 𝑅 ; 𝑐 𝑦 − 𝑐 8 𝑒𝑦 8 8

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded 1 𝑐 234 ≔ sup 𝑅 < 𝑐 𝑦 − 𝑐 9 𝑒𝑦 9 9

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded @ ' A 𝜇 B@ ' A 𝜉 ≔ 𝜈 1 𝑐 234 ≔ sup 𝑅 < 𝑐 𝑦 − 𝑐 9 𝑒𝑦 9 9

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded A ' B 𝜇 CA ' B 𝜉 ≔ 𝜈 1 𝑐 234(5) ≔ sup 𝑅 = 𝑐 𝑦 − 𝑐 : 𝑒𝑦 : :

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded A ' B 𝜇 CA ' B 𝜉 ≔ 𝜈 1 𝑐 234(5) ≔ sup 𝜉(𝑅) = 𝑐 𝑦 − 𝑐 : 𝑒𝑦 : :

Bloom (1985) 𝑐, 𝐼 : ¡𝑀 ' (𝜈) → ¡𝑀 ' (𝜇) 𝑐 ¡ ∈ 𝐶𝑁𝑃(𝜉) bounded A ' B ¡𝜇 CA ' B 𝜉 ≔ 𝜈 1 𝑐 234(5) ≔ ¡sup 𝜉(𝑅) ¡ = 𝑐 𝑦 − ¡ 𝑐 : 𝑒𝑦 : :

Bloom (1985) 𝑐, 𝐼 : ¡𝑀 ' (𝜈) → ¡𝑀 ' (𝜇) 𝑐 ¡ ∈ 𝐶𝑁𝑃(𝜉) bounded

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded Ø Extend to all CZO’s 𝑈 on ℝ 4

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded Ø Extend to all CZO’s 𝑈 on ℝ 4 Ø Long-term: Extend to multiparameter setting

Bloom (1985) 𝑐, 𝐼 : 𝑀 ' (𝜈) → 𝑀 ' (𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) bounded Ø Extend to all CZO’s 𝑈 on ℝ 4 Ø Long-term: Extend to multiparameter setting Ø Dyadic approach

Outline Introduction Bloom’s Result Main Results Upper Bound Lower Bound: Key Idea

CRW: Upper Bound: � [ b , T ] : L p → L p � � � b � BMO Lower Bound: n � � [ b , R j ] : L p → L p � . � b � BMO � j =1

Main Results (H., Lacey, Wick): Upper Bound: � [ b , T ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Lower Bound: n � � [ b , R j ] : L p → L p � . � b � BMO � j =1

Main Results (H., Lacey, Wick): Upper Bound: � [ b , T ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Lower Bound: n � � [ b , R j ] : L p → L p � . � b � BMO � j =1 p λ − 1 1 ν := µ p � 1 � b � BMO ( ν ) := sup | b ( x ) − � b � Q | dx ν ( Q ) Q Q

Main Results (H., Lacey, Wick): Upper Bound: � [ b , T ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Lower Bound: n � � [ b , R j ] : L p ( µ ) → L p ( λ ) � . � b � BMO ( ν ) � j =1 p λ − 1 1 ν := µ p � 1 � b � BMO ( ν ) := sup | b ( x ) − � b � Q | dx ν ( Q ) Q Q

Outline Introduction Bloom’s Result Main Results Upper Bound Lower Bound: Key Idea

Upper Bound: Strategy � [ b , T ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν )

Upper Bound: Strategy � [ b , T ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift]

Upper Bound: Strategy � [ b , T ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν )

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift]

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids:

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D 0 0

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D 0 0

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D 0 0

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D 0 0

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D 0 0

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D ω 0

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D ω

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D ω ◮ | I | = 2 − k , k ∈ Z , ∀ I ∈ D ;

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D ω ◮ | I | = 2 − k , k ∈ Z , ∀ I ∈ D ; ◮ I ∩ J ∈ {∅ , I , J } , ∀ I , J ∈ D ;

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Dyadic Grids: D ω ◮ | I | = 2 − k , k ∈ Z , ∀ I ∈ D ; ◮ I ∩ J ∈ {∅ , I , J } , ∀ I , J ∈ D ; ◮ { I ∈ D : | I | = 2 − k } forms a partition of R .

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Haar Functions: I ∈ D � � 1 h I := � ✶ I − − ✶ I + | I | 𝑱 " 𝑱 # 𝑱

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Haar Functions: I ∈ D � � 1 h I := � ✶ I − − ✶ I + | I | 𝑱 " 𝑱 # 𝑱

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Haar Functions: { h I : I ∈ D} = onb for L 2 . 𝑱 " 𝑱 # 𝑱

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Haar Functions: � � f = f ( I ) h I I ∈D 𝑱 " 𝑱 # 𝑱

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Petermichl’s Dyadic Shift: � � � 1 � √ h I − − h I + X ω f := f ( I ) . 2 I ∈D ω

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Petermichl’s Dyadic Shift: � � � 1 � √ h I − − h I + X ω f := f ( I ) . 2 I ∈D ω 𝑱 " 𝑱 # 𝑱

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Petermichl’s Dyadic Shift: � � � 1 � √ h I − − h I + X ω f := f ( I ) . 2 I ∈D ω 𝑱 " 𝑱 # 𝑱 𝑱

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Petermichl’s Dyadic Shift: � � � 1 � √ h I − − h I + X ω f := f ( I ) . 2 I ∈D ω Petermichl (2000): Hf = c E ω ( X ω f )

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Petermichl’s Dyadic Shift: � � � 1 � √ h I − − h I + X ω f := f ( I ) . 2 I ∈D ω Petermichl (2000): Hf = c E ω ( X ω f ) ⇒ [ b , H ] f = c E ω ([ b , X ω ] f )

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] Petermichl’s Dyadic Shift: � � � 1 � √ h I − − h I + X ω f := f ( I ) . 2 I ∈D ω Petermichl (2000): Hf = c E ω ( X ω f ) ⇒ [ b , H ] f = c E ω ([ b , X ω ] f ) � [ b , X ω ] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν )

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] For general CZOs on R n :

Upper Bound: Strategy I. Use a Representation Theorem to reduce the problem to bounding [ b , Dyadic Shift] For general CZOs on R n : Hyt¨ onen Representation Theorem (2011).

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν )

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Paraproducts: � � f ( I ) ✶ I � Π ∗ � b ( I ) � b ( I ) � f � I h I Π b f := b f := | I | I I

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Paraproducts: � � f ( I ) ✶ I � Π ∗ � b ( I ) � b ( I ) � f � I h I Π b f := b f := | I | I I bf = Π b f + Π ∗ b f + Π f b

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Paraproducts: � � f ( I ) ✶ I � Π ∗ � b ( I ) � b ( I ) � f � I h I Π b f := b f := | I | I I bf = Π b f + Π ∗ b f + Π f b b ( X f ) − X ( bf ) [ b , X ] f =

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Paraproducts: � � f ( I ) ✶ I � Π ∗ b ( I ) � � b ( I ) � f � I h I Π b f := b f := | I | I I bf = Π b f + Π ∗ b f + Π f b b ( X f ) − X ( bf ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ = b ) f

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) Paraproducts: � � f ( I ) ✶ I � Π ∗ � b ( I ) � b ( I ) � f � I h I Π b f := b f := | I | I I bf = Π b f + Π ∗ b f + Π f b b ( X f ) − X ( bf ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ = b ) f + (Π X f b − X Π f b )

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ b ) f + (Π X f b − X Π f b )

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � �

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � � �� � �

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � � �� � � �

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � � �� � � � Known: X : L p ( w ) → L p ( w )

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � � �� � � � Known: X : L p ( w ) → L p ( w ) 𝑀 " (𝜈) 𝑀 " (𝜈)

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � � �� � � � Known: X : L p ( w ) → L p ( w ) Π ( 𝑀 " (𝜈) 𝑀 " (𝜇) 𝑀 " (𝜈)

Upper Bound: Strategy II. Bound: � [ b , Dyadic Shift] : L p ( µ ) → L p ( λ ) � � � b � BMO ( ν ) [ b , X ] f = (Π b X + Π ∗ b X − X Π b − X Π ∗ + (Π X f b − X Π f b ) b ) f � �� � � �� � � � Known: X : L p ( w ) → L p ( w ) Π ( 𝑀 " (𝜈) 𝑀 " (𝜇) Π ( 𝑀 " (𝜈)