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

Two-Weight Inequalities for Commutators with Calder´

• n-Zygmund Operators

Irina Holmes

Joint work with B. D. Wick, M. Lacey,

• R. Rahm, S. Spencer,
• S. Petermichl

Michigan State University

Midwestern Workshop on Asymptotic Analysis IUPUI, October 6–8, 2017.

Outline

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

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b.

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Hilbert transform - R:

Hf (x) := 1 π p. v.

• R

f (y) x − y dy

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Riesz transforms - Rn:

Rjf (x) := Γ ((n + 1)/2) π(n+1)/2

• p. v.
• Rn f (y)

xj − yj |x − y|n+1 dy, j = 1, . . . , n.

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Calder´

• n-Zygmund Operators - Rn:

Tf (x) :=

• Rn K(x, y)f (y) dy

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Commutators:

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Commutators:

[b, T]f := b(Tf ) − T(bf )

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Bounded Mean Oscillation:

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Bounded Mean Oscillation:

bBMO := sup

Q

1 |Q|

• Q

|b(x) − bQ | dx

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Bounded Mean Oscillation:

bBMO := sup

Q

1 |Q|

• Q

|b(x) − bQ | dx

◮ bQ := 1 |Q|

• Q b(x) dx.

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Bounded Mean Oscillation:

bBMO := sup

Q

1 |Q|

• Q

|b(x) − bQ | dx

◮ bQ := 1 |Q|

• Q b(x) dx.

◮ H1(Rn) – BMO(Rn) Duality (Fefferman, 1971)

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b.

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b.

Upper Bound:

[b, T] : Lp → Lp bBMO

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b.

Upper Bound:

[b, T] : Lp → Lp bBMO

Lower Bound:

bBMO

n

• j=1

[b, Rj] : Lp → Lp.

Introduction

Starting point: Coifman, Rochberg and Weiss, Factorization theorems for Hardy spaces in several variables, 1976

◮ Characterize the norm of the commutator

[b, T] : Lp(Rn) → Lp(Rn), 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] : Lp(Rn) → Lp(Rn), 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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn.

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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|pw(x) dx

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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|pw(x) dx

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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|p dw

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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|p dw

◮ One-weight Inequalities: T : Lp(w) → Lp(w)

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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|p dw

◮ One-weight Inequalities: T : Lp(w) → Lp(w) – mostly

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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|p dw

◮ One-weight Inequalities: T : Lp(w) → Lp(w) – mostly ◮ Two-weight Inequalities: T : Lp(µ) → Lp(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] : Lp(Rn) → Lp(Rn), where T is a CZO, in terms of the BMO norm of b. Recall:

◮ Weight: non-negative, locally integrable function w on Rn. ◮ Lp(w):

• |f (x)|p dw

◮ One-weight Inequalities: T : Lp(w) → Lp(w) – mostly ◮ Two-weight Inequalities: T : Lp(µ) → Lp(λ) – much harder!

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] : Lp(Rn) → Lp(Rn), 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] :Lp(Rn; µ) → Lp(Rn; 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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap 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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap weights, in terms of the BMO norm of b. Recall:

◮ Ap weights:

[w]Ap := sup

Q

wQ w1−qp−1

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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap weights, in terms of the BMO norm of b. Recall:

◮ Ap weights:

[w]Ap := sup

Q

wQ w1−qp−1

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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap weights, in terms of the BMO norm of b. Recall:

◮ Ap weights:

[w]Ap := sup

Q

wQ w1−qp−1

Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s) ◮ M : Lp(w) → Lp(w) ⇔ w ∈ Ap

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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap weights, in terms of the BMO norm of b. Recall:

◮ Ap weights:

[w]Ap := sup

Q

wQ w1−qp−1

Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s) ◮ H : Lp(w) → Lp(w) ⇔ w ∈ Ap

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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap weights, in terms of the BMO norm of b. Recall:

◮ Ap weights:

[w]Ap := sup

Q

wQ w1−qp−1

Q ◮ Muckenhoupt, Hunt, Wheeden (1970’s) ◮ H : Lp(w) → Lp(w) ⇔ w ∈ Ap ◮ A2 weights:

[w]A2 := sup

Q

wQ w−1Q

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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap 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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap 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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap 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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap 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] :Lp(Rn; µ) → Lp(Rn; λ), where T is a CZO, µ, λ are Ap 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)

bounded 𝑐, 𝐼 : ¡𝑀' → ¡ 𝑀' 𝑐 ¡ ∈ 𝐶𝑁𝑃 𝑐 -./ ≔ ¡sup

4

1 𝑅 ¡ 7 𝑐 𝑦 − ¡ 𝑐 4 𝑒𝑦

4

Bloom (1985)

bounded 𝑐, 𝐼 : ¡𝑀'(𝑥) → ¡𝑀'(𝑥) 𝑐 ¡ ∈ 𝐶𝑁𝑃 𝑐 012 ≔ ¡sup

7

1 𝑅 ¡ : 𝑐 𝑦 − ¡ 𝑐 7 𝑒𝑦

7

Bloom (1985)

bounded 𝑐, 𝐼 : 𝑀'(𝜈) → 𝑀'(𝜇) 𝑐 ∈ 𝐶𝑁𝑃 𝑐 123 ≔ sup

8

1 𝑅 ; 𝑐 𝑦 − 𝑐 8 𝑒𝑦

8

Bloom (1985)

bounded 𝑐, 𝐼 : 𝑀'(𝜈) → 𝑀'(𝜇) 𝑐 ∈ 𝐶𝑁𝑃 𝑐 123 ≔ sup

8

1 𝑅 ; 𝑐 𝑦 − 𝑐 8 𝑒𝑦

8

Bloom (1985)

bounded 𝑐, 𝐼 : 𝑀'(𝜈) → 𝑀'(𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) 𝑐 234 ≔ sup

9

1 𝑅 < 𝑐 𝑦 − 𝑐 9 𝑒𝑦

9

Bloom (1985)

bounded 𝑐, 𝐼 : 𝑀'(𝜈) → 𝑀'(𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) 𝑐 234 ≔ sup

9

1 𝑅 < 𝑐 𝑦 − 𝑐 9 𝑒𝑦

9

𝜉 ≔ 𝜈

@ ' A 𝜇B@ ' A

Bloom (1985)

bounded 𝑐, 𝐼 : 𝑀'(𝜈) → 𝑀'(𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) 𝑐 234(5) ≔ sup

:

1 𝑅 = 𝑐 𝑦 − 𝑐 : 𝑒𝑦

:

𝜉 ≔ 𝜈

A ' B 𝜇CA ' B

Bloom (1985)

bounded 𝑐, 𝐼 : 𝑀'(𝜈) → 𝑀'(𝜇) 𝑐 ∈ 𝐶𝑁𝑃(𝜉) 𝑐 234(5) ≔ sup

:

1 𝜉(𝑅) = 𝑐 𝑦 − 𝑐 : 𝑒𝑦

:

𝜉 ≔ 𝜈

A ' B 𝜇CA ' B

Bloom (1985)

bounded 𝑐, 𝐼 : ¡𝑀'(𝜈) → ¡𝑀'(𝜇) 𝑐 ¡ ∈ 𝐶𝑁𝑃(𝜉) 𝑐 234(5) ≔ ¡sup

:

1 𝜉(𝑅) ¡ = 𝑐 𝑦 − ¡ 𝑐 : 𝑒𝑦

:

𝜉 ≔ 𝜈

A ' B ¡𝜇CA ' B

Bloom (1985)

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

Bloom (1985)

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

Ø Extend to all CZO’s 𝑈 on ℝ4

Bloom (1985)

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

Ø Extend to all CZO’s 𝑈 on ℝ4 Ø Long-term: Extend to multiparameter setting

Bloom (1985)

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

Ø Extend to all CZO’s 𝑈 on ℝ4 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] : Lp → Lp bBMO

Lower Bound:

bBMO

n

• j=1

[b, Rj] : Lp → Lp.

Main Results (H., Lacey, Wick):

Upper Bound:

[b, T] : Lp(µ) → Lp(λ) bBMO(ν)

Lower Bound:

bBMO

n

• j=1

[b, Rj] : Lp → Lp.

Main Results (H., Lacey, Wick):

Upper Bound:

[b, T] : Lp(µ) → Lp(λ) bBMO(ν)

Lower Bound:

bBMO

n

• j=1

[b, Rj] : Lp → Lp. ν := µ

1 p λ− 1 p

bBMO(ν) := sup

Q

1 ν(Q)

• Q

|b(x) − bQ | dx

Main Results (H., Lacey, Wick):

Upper Bound:

[b, T] : Lp(µ) → Lp(λ) bBMO(ν)

Lower Bound:

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). ν := µ

1 p λ− 1 p

bBMO(ν) := sup

Q

1 ν(Q)

• Q

|b(x) − bQ | dx

Outline

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

Upper Bound: Strategy

[b, T] : Lp(µ) → Lp(λ) bBMO(ν)

Upper Bound: Strategy

[b, T] : Lp(µ) → Lp(λ) bBMO(ν)

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift]

Upper Bound: Strategy

[b, T] : Lp(µ) → Lp(λ) bBMO(ν)

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift]

• II. Bound:

[b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(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: D0

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: D0

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: D0

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: D0

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: D0

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: Dω

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: Dω

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Dyadic Grids: Dω

◮ |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: Dω

◮ |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: Dω

◮ |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 hI := 1

• |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 hI := 1

• |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: {hI : I ∈ D} = onb for L2. 𝑱 𝑱" 𝑱#

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Haar Functions: f =

• I∈D
• f (I)hI

𝑱 𝑱" 𝑱#

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Petermichl’s Dyadic Shift: Xωf := 1 √ 2

• I∈Dω
• f (I)
• hI− − hI+
• .

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Petermichl’s Dyadic Shift: Xωf := 1 √ 2

• I∈Dω
• f (I)
• hI− − hI+
• .

𝑱 𝑱" 𝑱#

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Petermichl’s Dyadic Shift: Xωf := 1 √ 2

• I∈Dω
• f (I)
• hI− − hI+
• .

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: Xωf := 1 √ 2

• I∈Dω
• f (I)
• hI− − hI+
• .

Petermichl (2000): Hf = cEω (Xωf )

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Petermichl’s Dyadic Shift: Xωf := 1 √ 2

• I∈Dω
• f (I)
• hI− − hI+
• .

Petermichl (2000): Hf = cEω (Xωf ) ⇒ [b, H]f = cEω ([b, Xω]f )

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] Petermichl’s Dyadic Shift: Xωf := 1 √ 2

• I∈Dω
• f (I)
• hI− − hI+
• .

Petermichl (2000): Hf = cEω (Xωf ) ⇒ [b, H]f = cEω ([b, Xω]f ) [b, Xω] : Lp(µ) → Lp(λ) bBMO(ν)

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] For general CZOs on Rn:

Upper Bound: Strategy

• I. Use a Representation Theorem to reduce the problem to

bounding [b, Dyadic Shift] For general CZOs on Rn: Hyt¨

• nen Representation Theorem

(2011).

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

Paraproducts: Πbf :=

• I
• b(I) f I hI

Π∗

bf :=

• I
• b(I)

f (I)✶I |I|

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

Paraproducts: Πbf :=

• I
• b(I) f I hI

Π∗

bf :=

• I
• b(I)

f (I)✶I |I| bf = Πbf + Π∗

bf + Πf b

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

Paraproducts: Πbf :=

• I
• b(I) f I hI

Π∗

bf :=

• I
• b(I)

f (I)✶I |I| bf = Πbf + Π∗

bf + Πf b

[b, X]f = b(Xf ) − X(bf )

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

Paraproducts: Πbf :=

• I
• b(I) f I hI

Π∗

bf :=

• I
• b(I)

f (I)✶I |I| bf = Πbf + Π∗

bf + Πf b

[b, X]f = b(Xf ) − X(bf ) = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

Paraproducts: Πbf :=

• I
• b(I) f I hI

Π∗

bf :=

• I
• b(I)

f (I)✶I |I| bf = Πbf + Π∗

bf + Πf b

[b, X]f = b(Xf ) − X(bf ) = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

+ (ΠXf b − XΠf b)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f + (ΠXf b − XΠf b)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)
• Known: X : Lp(w) → Lp(w)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)
• Known: X : Lp(w) → Lp(w)

𝑀"(𝜈) 𝑀"(𝜈)

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)
• Known: X : Lp(w) → Lp(w)

𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π(

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)
• Known: X : Lp(w) → Lp(w)

𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

Upper Bound: Strategy

• II. Bound: [b, Dyadic Shift] : Lp(µ) → Lp(λ) bBMO(ν)

[b, X]f = (ΠbX + Π∗

bX − XΠb − XΠ∗ b)f

• + (ΠXf b − XΠf b)
• Known: X : Lp(w) → Lp(w)

𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

Upper Bound: Strategy 𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

Upper Bound: Strategy 𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

◮ Reduce to one-weight maximal and square function estimates!

Upper Bound: Strategy 𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

◮ Reduce to one-weight maximal and square function estimates! ◮ Key idea for this:

Upper Bound: Strategy 𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

◮ Reduce to one-weight maximal and square function estimates! ◮ Key idea for this: a weighted dyadic form of H1 – BMO

duality

Upper Bound: Strategy 𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

◮ Reduce to one-weight maximal and square function estimates! ◮ Key idea for this: a weighted dyadic form of H1 – BMO

duality (very nice for A2 weights in particular)

Upper Bound: Strategy 𝑀"(𝜈) 𝑀"(𝜈) 𝑀"(𝜇) Π( Π(

◮ Reduce to one-weight maximal and square function estimates! ◮ Key idea for this: a weighted dyadic form of H1 – BMO

duality (very nice for A2 weights in particular)

◮ ν = µ1/pλ−1/p ∈ A2 !!!

Outline

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

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ).

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW.

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW. Key fact: equivalent definitions of Bloom BMO:

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW. Key fact: equivalent definitions of Bloom BMO:

𝑐 "#$(&) ≔ sup

• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 - 𝑒𝑦

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW. Key fact: equivalent definitions of Bloom BMO:

𝑐 "#$(&) ≔ sup

• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 - 𝑒𝑦

• 𝑐 "#$ (&) ≅ sup
• 1

𝜈(𝑅) 1 𝑐 𝑦 − 𝑐 -

7

• 𝑒𝜇

9 7 :

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW. Key fact: equivalent definitions of Bloom BMO:

𝑐 "#$(&) ≔ sup

• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 - 𝑒𝑦

• 𝑐 "#$ (&) ≅ sup
• 1

𝜈(𝑅) 1 𝑐 𝑦 − 𝑐 -

7

• 𝑒𝜇

9 7 :

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW. Key fact: equivalent definitions of Bloom BMO:

𝑐 "#$(&) ≔ sup

• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 - 𝑒𝑦

• 𝑐 "#$5(&) ≔ sup
• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 -

6

• 𝑒𝜉78

8 6 9

≅ 𝑐 "#$ (&) ≅ sup

• 1

𝜈(𝑅) 1 𝑐 𝑦 − 𝑐 -

<

• 𝑒𝜇

8 < 9

Lower Bound

bBMO(ν)

n

• j=1

[b, Rj] : Lp(µ) → Lp(λ). Follows the same strategy in CRW. Key fact: equivalent definitions of Bloom BMO:

𝑐 "#$(&) ≔ sup

• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 - 𝑒𝑦

• 𝑐 "#$5(&) ≔ sup
• 1

𝜉(𝑅) 1 𝑐 𝑦 − 𝑐 -

6

• 𝑒𝜉78

8 6 9

≅ 𝑐 "#$ (&) ≅ sup

• 1

𝜈(𝑅) 1 𝑐 𝑦 − 𝑐 -

<

• 𝑒𝜇

8 < 9

Muckenhoupt & Wheeden (‘75)

• S. Bloom: A commutator theorem and weighted BMO -
• Trans. Amer. Math. Soc. 292 (1985), no. 1
• R. R. Coifman, R. Rochberg, G. Weiss: Factorization theorems

for Hardy spaces in several variables, Ann. of Math. 103 (1976), no. 3

• T. Hyt¨
• nen: The sharp weighted bound for general

Calder´

• n-Zygmund operators, Ann. of Math. 175 (2012), no.

3.

• B. Muckenhoupt, R. L. Wheeden: Weighted bounded mean
• scillation and the Hilbert transform, Studia Math. 54

(1975/76), no. 3

• S. Petermichl:

Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Ser I. Math. 330 (2000), no. 6