little hankel operators and product bmo - atlanta,...

OutlineNehari’s Theorem and its Extension

Upper BoundLower Bound & Paraproducts

References

Little Hankel Operators and Product BMO

Michael T. Lacey

Georgia Institute of Technology

March 15, 2005

M.T. Lacey Little Hankels



References

Coauthors

Sarah FergusonCUNY Staten Island

Erin TerwillegerU Conn




References

1 Nehari’s Theorem and its Extension

2 Upper Bound for Main Theorem

3 Lower Bound & Paraproducts




References

CommutatorWeak FactorizationBackground

Hankel and Toeplitz Operators

ϕ function on T. Mϕ ismultiplication by ϕ. Write thisin the basis {zn | n ∈ Z}.

We make some restrictions tothe doubly infinite matrix.




References



ϕ function on T. Mϕ ismultiplication by ϕ. Write thisin the basis {zn | n ∈ Z}.We make some restrictions tothe doubly infinite matrix.

ϕ(0)ϕ(1)ϕ(2)

ϕ(1)

ϕ(2)




References





Toeplitz




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Hankel




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Nehari’s Theorem (with C. Fefferman H1, BMO duality)

L2(R) = H2(R)⊕ H2−(R). P− is orthogonal projection of L2 onto

H2−. I = P+ + P−.

The Hankel operator with symbol b isHbϕ = P−bϕ as a map from H2 to H2

−.

Theorem (Nehari’s Theorem)

Hb is bounded iff there is a bounded function β such thatP+b = P+β.

Equivalently, P+b is in BMO.

Inner/Outer factorization is key to the proof.




References




H2−. I = P+ + P−. The Hankel operator with symbol b is

Hbϕ = P−bϕ as a map from H2 to H2−.


Hb is bounded iff there is a bounded function β such thatP+b = P+β.

Equivalently, P+b is in BMO.





References




H2−. I = P+ + P−. The Hankel operator with symbol b is

Hbϕ = P−bϕ as a map from H2 to H2−.


Hb is bounded iff there is a bounded function β such thatP+b = P+β. Equivalently, P+b is in BMO.





References


BMO

BMO = (H1)∗.

‖b‖BMO = supI|I |−1

∫I|b − bI | dx

bI =

∫Ib dx

We will need another formulation of definition later.




References


Main Theorem

H2(⊗n1C+) denotes the Hardy space of square integrable functions

analytic in each variable separately. Let P be the naturalprojection of L2(⊗n

1C+) onto H2(⊗n1C+). A Hankel operator with

symbol b is the linear operator from H2(⊗n1C+) to H2(⊗n

1C+)given by Hbϕ = Pbϕ.

Theorem

‖Hb‖ ' ‖P⊕b‖BMO(⊗n1C+)

where the right hand norm is the Chang Fefferman product BMO.




References


The Little Hankel Operator

H2(C+ ⊗ C+)

H2(C+ ⊗ C+)Hb




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Commutator

Define an operator from L2(Rn) to itself by

Cb := [· · · [ [Mb,H1],H2], · · · ,Hn],

in which Mb is the operator of pointwise multiplication by b, andHj denotes the Hilbert transform computed in the jth coordinate.

Corollary

‖Cb‖2 ' ‖b‖BMO(⊗n1R)

This is real BMO.




References


Weak Factorization of H1(⊗n1C+)

Theorem

A second equivalence gives us a weak factorization result, namely

H1(⊗n1C+) = H2(⊗n

1C+)⊗H2(⊗n1C+)

where the right hand side is the projective tensor product of H2.

‖h‖H2(⊗n1C+)b⊗H2(⊗n

1C+) = inf{∑

j

‖fj‖H2‖gj‖H2 | h =∑

j

fjgj

}




References


Essential Elements of the Proof

A ‘parameter’ is a free dilation in a coordinate.

The main inequality is ‖Hb‖ & ‖b‖BMO(⊗n1C+). The proof is

inductive on parameters:

The central enemy is the subtle way in which BMO(⊗n1C+) is

built up out of lower parameter BMOs. A new Journe’sLemma is needed to treat this aspect.

A non trivial (but not too dangerous) enemy concerns thedegenerate paraproducts that arise in higher parametersituations.




References


Background and Notation

L2(R) = H2(C+)⊕ H2(C+), with orthogonal projections P±.

This is extended to L2(⊗n1R) by taking products, thus P±

j is the

one dimensional projection P± applied in the jth coordinate.

P⊕ =n∏

j=1

P+j

Hp(⊗n1C+) is the Hardy space of functions F : ⊗n

j=1C+ −→ C+

that is analytic in each variable separately, and

‖F‖pHp(⊗n

1C+) = lim‖y‖↓0

∫Rn

|F (x + iy)|p dy .




References


Background and Notation

L2(R) = H2(C+)⊕ H2(C+), with orthogonal projections P±.This is extended to L2(⊗n

1R) by taking products, thus P±j is the

one dimensional projection P± applied in the jth coordinate.

P⊕ =n∏

j=1

P+j

Hp(⊗n1C+) is the Hardy space of functions F : ⊗n

j=1C+ −→ C+

that is analytic in each variable separately, and

‖F‖pHp(⊗n

1C+) = lim‖y‖↓0

∫Rn

|F (x + iy)|p dy .




References


The Y. Meyer Wavelet

w is a particular Schwartz function w(ξ) supported on [2/3, 8/3].The compact frequency implies that w is rapidly decreasing, andsupported on the whole real line.‘‘Schwarz tails.” Their control dictates most of the technicalitiesof the proof.




References


The Y. Meyer Wavelet, Cont’d

Let D denote the dyadic intervals on R. For an interval I ∈ D,define

wI (x) := |I |−1/2w(

x−c(I )|I |

),

where c(I ) denotes the center of I .

{wI}I∈D form an orthonormal basis on H2(C+).Littlewood-Paley inequalities,∥∥∥∑

I∈D〈f ,wI 〉wI

∥∥∥p'

∥∥∥( ∑I∈D

|〈f ,wI 〉|2|I | 1I

)1/2∥∥∥p, 1 < p <∞.




References




wI (x) := |I |−1/2w(

x−c(I )|I |

),

where c(I ) denotes the center of I .{wI}I∈D form an orthonormal basis on H2(C+).

Littlewood-Paley inequalities,∥∥∥∑I∈D

〈f ,wI 〉wI

∥∥∥p'

∥∥∥( ∑I∈D

|〈f ,wI 〉|2|I | 1I

)1/2∥∥∥p, 1 < p <∞.




References




wI (x) := |I |−1/2w(

x−c(I )|I |

),

where c(I ) denotes the center of I .{wI}I∈D form an orthonormal basis on H2(C+).Littlewood-Paley inequalities,∥∥∥∑

I∈D〈f ,wI 〉wI

∥∥∥p'

∥∥∥( ∑I∈D

|〈f ,wI 〉|2|I | 1I

)1/2∥∥∥p, 1 < p <∞.




References


BMO(⊗n1C+) and the Carleson Measure Condition

Let R = ⊗nj=1D be the dyadic rectangles. For a rectangle

R = ⊗nj=1Rj ∈ R, define

vR(x) :=n∏

j=1

wRj(xj).

We say f ∈ BMO(⊗n1C+) iff

supU

[|U|−1

∑R⊂U

|〈f , vR〉|2]1/2

<∞

where U is an open set in Rn of finite measure.




References


More on BMO in one and two dimensions

‖b‖BMO(R) = supJ

[|J|−1

∑I⊂J

|〈b,wI 〉|2]1/2

.




References




[|J|−1

∑I⊂J

|〈b,wI 〉|2]1/2

.

Seemingly, the most natural definition in two dimensions would be

‖b‖BMO(rect) = supJ is a rectangle

[|J|−1

∑I⊂J

|〈b,wI 〉|2]1/2

.




References




[|J|−1

∑I⊂J

|〈b,wI 〉|2]1/2

.

Important Mistake to Avoid

BMO(rect) is not Chang–Fefferman BMO.




References




[|J|−1

∑I⊂J

|〈b,wI 〉|2]1/2

.

Important Mistake to Avoid

BMO(rect) is not Chang–Fefferman BMO.

The Journe Lemma is a tool to bound the BMO by rectangularBMO.




References


S.-Y. Chang R. Fefferman Characterization ofBMO(⊗n

1C+)

Theorem

‖b‖H1(⊗n1C+)∗ ' sup

U

[|U|−1

∑R⊂U

|〈f , vR〉|2]1/2

The “test sets” U are arbitrary measurable subsets Rd of finitemeasure.




References


Definition of BMO−1(⊗n1C+)

For a collection of rectangles U ⊂ R, set the shadow of U , to be

sh(U) :=⋃

R∈UR.

We say that U has n − 1 parameters

iff there is a coordinate 1 ≤ k ≤ n, and a dyadic interval I , so thatfor all R ∈ U , we have Rk = I

Definition of BMO−1(⊗n1C+)

‖b‖BMO−1(⊗n1C+) = sup

U n − 1 parameters

[|sh(U)|−1

∑R∈U

|〈b, vR〉|2]1/2




References

Weak Factorization

Upper Bound on Hankel operator

‖Hb‖ = supf ,g∈H2(⊗n

1C+)‖f ‖2=1,‖g‖2=1

∫(Pbf )g dx

= supf ,g∈H2(⊗n

1C+)‖f ‖2=1,‖g‖2=1

∫P⊕bfg dx .

Since the product of H2 functions is in H1, we see that integralabove admits the upper bound of ‖P⊕b‖BMO(⊗n

1C+). This is theupper half of our Theorem.




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Weak Factorization

More on Weak Factorization

Previous Slide implies that the Hankel operator with symbol b isbounded if and only if the function P⊕b is in the dual ofH2(⊗n

1C+)⊗H2(⊗n1C+). Therefore, the weak factorization

statement is equivalent to our main theorem. That is, we have theequivalence

‖Hb‖ ' ‖b‖(H2(⊗n1C+)b⊗H2(⊗n

1C+))∗




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Weak Factorization

More on Weak Factorization

Previous Slide implies that the Hankel operator with symbol b isbounded if and only if the function P⊕b is in the dual ofH2(⊗n

1C+)⊗H2(⊗n1C+). Therefore, the weak factorization

statement is equivalent to our main theorem. That is, we have theequivalence

‖Hb‖ ' ‖b‖(H2(⊗n1C+)b⊗H2(⊗n

1C+))∗

This we need below.




References

BMO(n − 1) Lower BoundThe BMO Lower BoundApplication of Journe’s LemmaMain EstimateOrthogonality & DiagonalizationThe Positive Sums

The BMO−1(⊗n1C+) Lower Bound

An essential part of the argument is to, by induction, show that

‖Hb‖ & ‖P⊕b‖BMO−1(⊗n1C+)

This amounts to the assertion that

‖P⊕b‖(H2(⊗n1C+)b⊗H2(⊗n

1C+))∗ & ‖P⊕b‖BMO−1 ,




References


Take analytic b = b(x1, x2, . . . , xn) = b(x1, x′) of

‖b‖BMO(n−1) = 1.

Take a set U of rectangles n − 1 parameters. Assume that |I | = 1,|sh(U)| = 1, and for all R ∈ U we have R1 = I , and R = I × R ′.Claim: ∥∥∥∑

R∈U〈b, vR〉vR

∥∥∥H2(⊗n

1C+)b⊗H2(⊗n1C+)

. 1.

We have vR(x1, x′) = wI (x1)vR′(x ′), and∑

R∈U〈b, vR〉vR = wI (x1)

∑R∈U

〈b, vR〉vR′(x ′) := wI (x1)ψ′(x ′),

we can utilize factorization results in both x1 and x ′ to concludethe proof.




References



‖b‖BMO(n−1) = 1.Take a set U of rectangles n − 1 parameters. Assume that |I | = 1,|sh(U)| = 1, and for all R ∈ U we have R1 = I , and R = I × R ′.

Claim: ∥∥∥∑R∈U

〈b, vR〉vR

∥∥∥H2(⊗n

1C+)b⊗H2(⊗n1C+)

. 1.



∑R∈U

〈b, vR〉vR′(x ′) := wI (x1)ψ′(x ′),





References



‖b‖BMO(n−1) = 1.Take a set U of rectangles n − 1 parameters. Assume that |I | = 1,|sh(U)| = 1, and for all R ∈ U we have R1 = I , and R = I × R ′.Claim: ∥∥∥∑

R∈U〈b, vR〉vR

∥∥∥H2(⊗n

1C+)b⊗H2(⊗n1C+)

. 1.



∑R∈U

〈b, vR〉vR′(x ′) := wI (x1)ψ′(x ′),





References


Increasing the Lower Bound

The remainder of the proof is

Concerned with increasing the BMO−1 bound.

Exploits a new form of Journe’s Lemma.

Heavily harmonic analytic, using e.g. orthogonality, LittlewoodPaley, John Nirenberg, etc.

More function theoretic/operator theoretic proofs would bewelcome.

The current proof should provide some hints as what such aproof would look like.




References


First Steps in Lower Bound

The bound ‖Hb‖ & ‖P⊕b‖BMO(⊗n1C+) can be established under the

assumptions

‖P⊕b‖BMO(⊗n1C+) = 1, ‖P⊕b‖BMO−1(⊗n

1C+) ≤ δ−1,

for a free choice of absolute δ−1 > 0.

Assume that there is collection of rectangles U with |sh(U)| = 1,for which

α =∑R∈U

〈b, vR〉vR

has L2 norm one. Then, we show ‖Hbα‖ & 1.




References


First Steps in Lower Bound

The bound ‖Hb‖ & ‖P⊕b‖BMO(⊗n1C+) can be established under the

assumptions

‖P⊕b‖BMO(⊗n1C+) = 1, ‖P⊕b‖BMO−1(⊗n

1C+) ≤ δ−1,

for a free choice of absolute δ−1 > 0.Assume that there is collection of rectangles U with |sh(U)| = 1,for which

α =∑R∈U

〈b, vR〉vR

has L2 norm one. Then, we show ‖Hbα‖ & 1.




References


Application of Journe Lemma

The Function Theoretic Application of Journe Lemma

For any δjourne > 0, We can select V ⊃ sh(U), and a mapemb : U → [1,∞) so that

|V | ≤ 1 + δjourne, emb(R)R ⊂ V ,∥∥∥∑R∈U

emb(R)−2n〈b, vR〉vR

∥∥∥BMO(⊗n

1C+)≤ Kδjourne

δ−1

The power −2n is of no consequence to us. What is important isthe presence of the BMO−1 norm as an upper bound on theBMO(⊗n

1C+) norm.




References


Geometric Formulation of Journe Lemma

Given a a collection of rectangles U of dyadic rectangles, whoseshadow has finite area, suppose that V ⊃ sh(U), and define

emb(R) := sup{µ ≥ 1 | µR1 × R2 × · · · × Rn ⊂ V }.

For an arbitrary subset U ′ ⊂ U , let

F (I , j ,U ′) :=⋃{I × R ′ | I × R ′ ∈ U ′, 2j−1 ≤ emb(I × R ′) < 2j}.

Notice that these sets are the shadows of collections of rectangleswith n − 1 parameters.




References


Geometric Formulation of Journe Lemma, Cont’d

Journe Lemma

For all δ, ε > 0, we can select V ⊃ sh(U) with|V | ≤ (1 + δ)|sh(U)|, for which we have the estimate

∞∑j=1

∑I∈D

2−(n+ε)j |F (I , j ,U ′)| . |sh(U ′)|, U ′ ⊂ U .

The implied constants in these estimates depend only ondimensions and the choices of ε, δ.

This estimate is implicit in the work of Pipher.




References


Images to Keep in Mind

sh(U)

R

V

µR

It is important to expand the rectangle in all directions, in order tocontrol Schwartz tails.




References


Images to Keep in Mind, Part 2

The Carleson Example showsBMO−1(⊗n

1C+) is essentially smaller thanthe BMO(⊗n

1C+) norm. These exampleslive on a “thinly woven fabric” The“embeddedness” quantity pulls out thesethin sheets.




References


Images to Keep in Mind, Part 2




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Elementary Calculations

‖Hαα‖ = ‖P|α|2‖2 & 1.

β :=∑R⊂V

R 6⊂sh(U)

〈b, vR〉vR

‖β‖2 . δ1/2journe, BMO(⊗n

1C+) norm 1, hence small in all Lp.

‖Hβα‖2 . δ1/4journe.




References


The Main Estimate & Paraproducts

γ :=∑R 6⊂V

〈b, vR〉vR , ‖Hγα‖2 ≤ Kδjourneδ−1

Cancellation is essential to this estimate

Pv ′RvR = 0 if |R ′k | > 8|Rk | for any k.

Hγα =∑

|R′k |≤8|Rk |, all k

〈b, vR′〉〈b, vR〉vR′vR .

This is an example of a (sum of) paraproduct.




References


A Picture for the Main Estimate

R

R ′

This choice of R ′ is not allowed.

R ′

Nor is this one.

R ′

R ′ is smaller in one dimension

R ′

Smaller dimension, bigger the gain.

R ′

R ′ is smaller in two dimensions

R ′

R ′ is smaller in three dimensions




References


Orthogonality, Part 1

Given a subset J ⊂ {1, 2, . . . , n}, write R ′ ≺J R iff for indices j ∈ Jwe have 8|R ′

j | < |Rj |, whereas for indices in j ∈ {1, 2, . . . , n} − J

we have 8−1|R ′j | ≤ |Rj | ≤ 8|R ′

j |. This encodes Orthogonality.

Set

X (J) := {(R ′,R) ∈ W × U | R ′ ≺J R}

X(J) :=∑

(R′,R)∈X (J)

〈b, vR′〉vR′〈b, vR〉vR .

The remainder of the proof is devoted to the assertion that

‖X(J)‖2 ≤ Kδjourneδ−1, J ⊂ {1, 2, . . . , n}.




References


Orthogonality, Part 1

Given a subset J ⊂ {1, 2, . . . , n}, write R ′ ≺J R iff for indices j ∈ Jwe have 8|R ′

j | < |Rj |, whereas for indices in j ∈ {1, 2, . . . , n} − J

we have 8−1|R ′j | ≤ |Rj | ≤ 8|R ′

j |. This encodes Orthogonality.Set

X (J) := {(R ′,R) ∈ W × U | R ′ ≺J R}

X(J) :=∑

(R′,R)∈X (J)


The remainder of the proof is devoted to the assertion that

‖X(J)‖2 ≤ Kδjourneδ−1, J ⊂ {1, 2, . . . , n}.




References


Diagonalization, Part 1

Diagonalization parameters are d1, d2, . . . to gain geometric decayin estimates.

By the Journe Lemma, we can decompose U intocollections Ud1 , for d1 ∈ N, for which we have2d1 ≤ emb(R) < 2d1+1 for R ∈ Ud1 , and∥∥∥ ∑

R∈Ud1

〈b, vR〉vR

∥∥∥BMO(⊗n

1C+)≤ Kδjourne

22nd1δ−1.

For integers d2 ≥ d1, set

X (J, d2) := {(R ′,R) ∈ X (J) | R ∈ Ud1 , R ′ ⊂ 2d2+3R, R ′ 6⊂ 2d2R}

X(J, d2) :=∑

(R′,R)∈X (J,d2)





References



By the Journe Lemma, we can decompose U into collections Ud1 ,for d1 ∈ N, for which we have 2d1 ≤ emb(R) < 2d1+1 for R ∈ Ud1 ,and ∥∥∥ ∑

R∈Ud1

〈b, vR〉vR

∥∥∥BMO(⊗n

1C+)≤ Kδjourne

22nd1δ−1.

For integers d2 ≥ d1, set

X (J, d2) := {(R ′,R) ∈ X (J) | R ∈ Ud1 , R ′ ⊂ 2d2+3R, R ′ 6⊂ 2d2R}

X(J, d2) :=∑

(R′,R)∈X (J,d2)





References



For integers−→d3 = (

−→d31, . . . ,

−→d3 |J|) ∈ N|J|, set

X (J, d2,−→d3) := {(R ′,R) ∈ X (J, d2) | 2

−→d3 j |R ′

j | = |Rj |, j ∈ J},

X(J, d2,−→d3) :=

∑(R′,R)∈X (J,d2,

−→d3)

〈b, vR′〉vR′〈b, vR〉vR

The estimate is:

‖X(J, d2,−→d3)‖2 ≤ Kδjourne

2−d2+‖−→d3‖δ−1.




References


Orthogonality Part 2

For R ′ ≺J R and R ′ ≺J R, if for some coordinate j ∈ J we have8|Rj | < |R ′

j |, then vR′vR and veR′veR are orthogonal.




References



For R ′ ≺J R and R ′ ≺J R, if for some coordinate j ∈ J we have8|Rj | < |R ′

j |, then vR′vR and veR′veR are orthogonal.

For −→v ∈ Z|J|,

X (J, d2,−→d3,

−→v ) := {(R ′,R) ∈ X (J, d2,−→d3) | |R ′

j | = 2−→v j , j ∈ J},

X(J, d2,−→d3,

−→v ) :=∑

(R′,R)∈X (J,d2,−→d3 ,−→v )





References




X (J, d2,−→d3,

−→v ) := {(R ′,R) ∈ X (J, d2,−→d3) | |R ′

j | = 2−→v j , j ∈ J},

X(J, d2,−→d3,

−→v ) :=∑

(R′,R)∈X (J,d2,−→d3 ,−→v )


‖X(J, d2,−→d3)‖2

2 .∑

−→v ∈Z|J|

‖X(J, d2,−→d3,

−→v )‖22




References




X (J, d2,−→d3,

−→v ) := {(R ′,R) ∈ X (J, d2,−→d3) | |R ′

j | = 2−→v j , j ∈ J},

X(J, d2,−→d3,

−→v ) :=∑

(R′,R)∈X (J,d2,−→d3 ,−→v )


‖X(J, d2,−→d3)‖2

2 .∑

−→v ∈Z|J|

‖X(J, d2,−→d3,

−→v )‖22

Next we replace the product vR′vR above.




References


The Positive Sums

For (R ′,R) ∈ X (J, d2,−→d3), we have

|vR′vR(x)| . 2−Nd2− 12‖−→d3‖1

1

|R ′|(M 1R′)N .

Here, N is arbitrary, due to Schwartz tails.

It is essential to observe: For all R ′, there are only O(2nd2) possible

values of R for which (R ′,R) ∈ X(J, d2,−→d3). This moderate

number can be absorbed into the estimate above.We are finished with diagonalization & orthogonality. The finalform of the estimate is:∑−→v ∈Z|J|

∥∥∥ ∑(R,R′)∈X (J,d2,

−→d3 ,−→v )

|〈b, vR′〉〈b, vR〉||R ′|

1R′

∥∥∥2

2. 22nd2− 1

4‖−→d3‖1 .




References


The Positive Sums


|vR′vR(x)| . 2−Nd2− 12‖−→d3‖1

1

|R ′|(M 1R′)N .

Here, N is arbitrary, due to Schwartz tails.It is essential to observe: For all R ′, there are only O(2nd2) possible


number can be absorbed into the estimate above.

We are finished with diagonalization & orthogonality. The finalform of the estimate is:∑−→v ∈Z|J|

∥∥∥ ∑(R,R′)∈X (J,d2,

−→d3 ,−→v )

|〈b, vR′〉〈b, vR〉||R ′|

1R′

∥∥∥2

2. 22nd2− 1

4‖−→d3‖1 .




References


The Positive Sums


|vR′vR(x)| . 2−Nd2− 12‖−→d3‖1

1

|R ′|(M 1R′)N .

Here, N is arbitrary, due to Schwartz tails.It is essential to observe: For all R ′, there are only O(2nd2) possible


number can be absorbed into the estimate above.We are finished with diagonalization & orthogonality. The finalform of the estimate is:∑−→v ∈Z|J|

∥∥∥ ∑(R,R′)∈X (J,d2,

−→d3 ,−→v )

|〈b, vR′〉〈b, vR〉||R ′|

1R′

∥∥∥2

2. 22nd2− 1

4‖−→d3‖1 .




References


The Essential Difficulty Left to Face

The side lengths of R ′ are for those coordinates in J completelyspecified by −→v ∈ Z|J|. The remaining side lengths of R ′ are thenpermitted to vary. Thus, the ways that two possible choices of

R ′,R ′′ ∈ X (J, d2,−→d3,

−→v ) can intersect are as general as theintersections of two dyadic rectangles of dimension n − |J|.The John Nirenberg, Littlewood Paley Inequalities are the keyingredient to invoke.




References


The Essential Difficulty Left to Face

Figure: Intersecting 3 dimensional RectanglesM.T. Lacey Little Hankels



References


John Nirenberg Estimates

John Nirenberg estimates bound high Lp norms in terms of e.g. L1

norms.∥∥∥∥[ ∑

(R′,R)∈X (J,d2,−→d3)

|〈b, vR′〉|2

|R ′|1R′

]1/2∥∥∥∥p

. 24nd2 , 1 < p <∞.

This is immediate from the fact that b is in BMO and that therectangles in X (J, d2,

−→d3,

−→v ) are contained in{M 1sh(U) > c2−nd2}.A similar fact is∥∥∥∥

[ ∑R′∈X (J,d2,

−→d3)

|〈b, vR〉|2

|R|1R′

]1/2∥∥∥∥p

. δ−124nd2 , 1 < p <∞.




References


John Nirenberg Estimates, Cont’d

What is important here is that the estimate is independent of−→d3,

and is uniform in −→v ∈ Z|J|. Due to the John Nirenberg inequality,this will follow from this estimate.∑

R′⊂V

(R′,R)∈X (J,d2,−→d3)

|〈b, vR〉|2 . 24nd2+‖−→d3‖δ2−1|V |, V ⊂ Rn.

As this estimate is uniform in the choice of V , it provides a boundfor a Carleson measure, to which the John Nirenberg inequalityapplies.




References

[1] Sun-Yung A. Chang, Carleson measure on the bi-disc, Ann. of Math. (2)109 (1979), no. 3, 613–620.

[2] Sun-Yung A. Chang and Robert Fefferman, A continuous version of dualityof H1 with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1,179–201.MR 82a:32009

[3] Sarah H. Ferguson and Michael T. Lacey, A characterization of productBMO by commutators, Acta Math. 189 (2002), no. 2, 143–160.1 961 195

[4] Jean-Lin Journe, A covering lemma for product spaces, Proc. Amer. Math.Soc. 96 (1986), no. 4, 593–598.MR 87g:42028

[5] Michael T. Lacey and Erin Terwileger, Little Hankel Operators and ProductBMO.

[6] Jill Pipher, Journe’s covering lemma and its extension to higher dimensions,Duke Math. J. 53 (1986), no. 3, 683–690.MR 88a:42019


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