Bilinear bi-parameter Calderón-Zygmund theory

Bilinear bi-parameter Calderon-Zygmund theory

Kangwei Li

BCAM-Basque Center for Applied Mathematics

18 May 2018

Joint work with Henri Martikainen and Emil Vuorinen

Linear bi-parameter Calderon-Zygmund operators

The classical multi-parameter singular integral theory was firstintroduced by Journe. Actually, he introduced it under the contextof vector-valued (operator-valued) Calderon-Zygmund theory.

Herewe reformulate the bi-parameter case as the following (tounderstand it we use ‘T1 ⊗ T2’ as a model operator): Letf = f1 ⊗ f2, and g = g1 ⊗ g2 with f1, g1 : Rn → C andf2, g2 : Rm → C . If spt f1 ∩ spt g1 = ∅ and spt f2 ∩ spt g2 = ∅, thenwe have the full kernel representation:⟨

T (f ), g⟩

∫Rn+m

K (x , y)f (y)g(x)dydx .

The kernel K : Rn+m × Rn+m \ (x , y) : x1 = y1 or x2 = y2 → Cis assumed to satisfy the following assumptions

K. Li(BCAM) Bilinear bi-parameter Calderon-Zygmund theory

The classical multi-parameter singular integral theory was firstintroduced by Journe. Actually, he introduced it under the contextof vector-valued (operator-valued) Calderon-Zygmund theory. Herewe reformulate the bi-parameter case as the following (tounderstand it we use ‘T1 ⊗ T2’ as a model operator):

Letf = f1 ⊗ f2, and g = g1 ⊗ g2 with f1, g1 : Rn → C andf2, g2 : Rm → C . If spt f1 ∩ spt g1 = ∅ and spt f2 ∩ spt g2 = ∅, thenwe have the full kernel representation:⟨

T (f ), g⟩

∫Rn+m

The classical multi-parameter singular integral theory was firstintroduced by Journe. Actually, he introduced it under the contextof vector-valued (operator-valued) Calderon-Zygmund theory. Herewe reformulate the bi-parameter case as the following (tounderstand it we use ‘T1 ⊗ T2’ as a model operator): Letf = f1 ⊗ f2, and g = g1 ⊗ g2 with f1, g1 : Rn → C andf2, g2 : Rm → C . If spt f1 ∩ spt g1 = ∅ and spt f2 ∩ spt g2 = ∅, thenwe have the full kernel representation:⟨

T (f ), g⟩

∫Rn+m

Full kernel assumptions

1 size condition |K (x , y)| ≤ C 1|x1−y1|n

1|x2−y2|m

2 regularity condition

|K (x , y)− K (x , (y ′1, y2))− K (x , (y1, y′2)) + K (x , y ′)|

≤ C|y1 − y ′1|δ

|x1 − y1|n+δ

|y2 − y ′2|δ

|x2 − y2|m+δ

|K (x , y)− K (x , (y ′1, y2))− K ((x1, x′2), y) + K ((x1, x

′2), (y ′1, y2))|

≤ C|y1 − y ′1|δ

|x1 − y1|n+δ

|x2 − x ′2|δ

|x2 − y2|m+δ

when |y1 − y ′1| ≤ |x1 − y1|/2 and |y2 − y ′2| ≤ |x2 − y2|/2, and|y1 − y ′1| ≤ |x1 − y1| and |x2 − x ′2| ≤ |x2 − y2|/2, respectively.We also assume the corresponding symmetrical case.

Full kernel assumptions

1 size condition |K (x , y)| ≤ C 1|x1−y1|n

1|x2−y2|m

2 regularity condition

|K (x , y)− K (x , (y ′1, y2))− K (x , (y1, y′2)) + K (x , y ′)|

≤ C|y1 − y ′1|δ

|x1 − y1|n+δ

|y2 − y ′2|δ

|x2 − y2|m+δ

|K (x , y)− K (x , (y ′1, y2))− K ((x1, x′2), y) + K ((x1, x

′2), (y ′1, y2))|

≤ C|y1 − y ′1|δ

|x1 − y1|n+δ

|x2 − x ′2|δ

|x2 − y2|m+δ

when |y1 − y ′1| ≤ |x1 − y1|/2 and |y2 − y ′2| ≤ |x2 − y2|/2, and|y1 − y ′1| ≤ |x1 − y1| and |x2 − x ′2| ≤ |x2 − y2|/2, respectively.We also assume the corresponding symmetrical case.

Kernel assumptions continued

We also need the following mixed regularity and size conditions

|K (x , y)− K (x , (y ′1, y2))| ≤ C|y1 − y ′1|δ

|x1 − y1|n+δ

|x2 − y2|m

when |y1 − y ′1| ≤ |x1 − y1|/2. We also assume the 3 symmetricalconditions.Now if we only have spt f1 ∩ spt g1 = ∅, then we only have partialkernel representation⟨

T (f ), g⟩

Kf2,g2(x1, y1)f1(x1)g1(y1)dy1dx1.

The kernel Kf2,g2 : Rn × Rn \ (x1, y1) : x1 = y1 → C satisfies theassumptions of Calderon-Zygmund kernel in Rn with someconstant C (f2, g2).

Assumptions on C (f2, g2) and T (1) assumptions

Let uV be a function on Rm satisfying that spt uV ⊂ V , |uv | ≤ 1and

∫uV = 0, where V is a cube in Rm. We also need to assume

C (1V , 1V ) + C (1V , uV ) + C (uV , 1V ) ≤ C |V |.

When spt f2 ∩ spt g2 = ∅, everything is defined similarly. Thisfinishes the definition of the kernel assumption.

Next we introduce the natural T1 assumptions. Let S be T itselfor one of its 3 adjoint operators. We have to assumeS(1) ∈ BMOprod. Here we say b ∈ BMOD

prod , if

‖b‖BMOD

prod:= sup

|Ω|∑

K∈Dn,V∈Dm

K×V⊂Ω

|〈b, hK ⊗ hV 〉|2)1/2

We remark that the natural extension of the one-parameter BMO,namely, changing cubes to rectangles, is called bmo, which isstrictly contained in BMOprod.

C (1V , 1V ) + C (1V , uV ) + C (uV , 1V ) ≤ C |V |.

When spt f2 ∩ spt g2 = ∅, everything is defined similarly. Thisfinishes the definition of the kernel assumption.Next we introduce the natural T1 assumptions. Let S be T itselfor one of its 3 adjoint operators. We have to assumeS(1) ∈ BMOprod. Here we say b ∈ BMOD

prod , if

‖b‖BMOD

prod:= sup

|Ω|∑

K∈Dn,V∈Dm

K×V⊂Ω

|〈b, hK ⊗ hV 〉|2)1/2

C (1V , 1V ) + C (1V , uV ) + C (uV , 1V ) ≤ C |V |.

When spt f2 ∩ spt g2 = ∅, everything is defined similarly. Thisfinishes the definition of the kernel assumption.Next we introduce the natural T1 assumptions. Let S be T itselfor one of its 3 adjoint operators. We have to assumeS(1) ∈ BMOprod. Here we say b ∈ BMOD

prod , if

‖b‖BMOD

prod:= sup

|Ω|∑

K∈Dn,V∈Dm

K×V⊂Ω

|〈b, hK ⊗ hV 〉|2)1/2

Diagonal BMO assumptions: completion of T (1)

Finally, we also need the so-called diagonal BMO assumptions,which is the last condition related to T (1). We need∣∣⟨T (1K , uV ), 1K ⊗ 1V

⟩∣∣ ≤ C |K | · |V |,

and its 3 symmetrical conditions also hold. back

With all the assumptions listed in the above, one can prove that Tis bounded on Lp(w), with w ∈ Ap(Rn × Rm). How? Throughrepresentation theorem one can get a very easy proof.

⟩∣∣ ≤ C |K | · |V |,

With all the assumptions listed in the above, one can prove that Tis bounded on Lp(w), with w ∈ Ap(Rn × Rm).

How? Throughrepresentation theorem one can get a very easy proof.

⟩∣∣ ≤ C |K | · |V |,

With all the assumptions listed in the above, one can prove that Tis bounded on Lp(w), with w ∈ Ap(Rn × Rm). How?

Throughrepresentation theorem one can get a very easy proof.

⟩∣∣ ≤ C |K | · |V |,

With all the assumptions listed in the above, one can prove that Tis bounded on Lp(w), with w ∈ Ap(Rn × Rm). How? Throughrepresentation theorem one can get a very easy proof.

Representation theorem: The linear one parameter case

The motivation for representing Calderon-Zygmund operators byshift operators is due to the study of A2 theorem.

The first resultin this topic was given by Petermichl in 2000, where she showedthat for the Hilbert transform

Hf (x) =1

∫ ∞−∞

f (x − y)

can be written in the form

Hf = − 8

1Sω,r f

r, f ∈ Lp(R),

for certain specific dyadic operators Sω,r . Later on, in 2007, sheproved the sharp weighted estimate for the Hilbert transform.

The motivation for representing Calderon-Zygmund operators byshift operators is due to the study of A2 theorem. The first resultin this topic was given by Petermichl in 2000, where she showedthat for the Hilbert transform

Hf (x) =1

∫ ∞−∞

f (x − y)

Hf = − 8

1Sω,r f

r, f ∈ Lp(R),

for certain specific dyadic operators Sω,r .

Later on, in 2007, sheproved the sharp weighted estimate for the Hilbert transform.

The motivation for representing Calderon-Zygmund operators byshift operators is due to the study of A2 theorem. The first resultin this topic was given by Petermichl in 2000, where she showedthat for the Hilbert transform

Hf (x) =1

∫ ∞−∞

f (x − y)

Hf = − 8

1Sω,r f

r, f ∈ Lp(R),

for certain specific dyadic operators Sω,r . Later on, in 2007, sheproved the sharp weighted estimate for the Hilbert transform.

The linear one parameter case continued

The representation formula for general Calderon-Zygmundoperators was given by Hytonen in 2010. As a corollary, he provedthe A2 conjecture. The representation formula he obtained is thefollowing:

〈g ,Tf 〉 = cCKEω∞∑

i ,j=0maxi ,j>0

2−αmaxi ,j〈g ,S i ,jω f 〉

+ c(CK + ‖T‖WBT )Eω〈g , S0,0ω f 〉

+ Eω〈g , πωT1f 〉+ Eω〈g , (πωT∗1)∗f 〉.

Model operators

1 Shifts

S i ,jw f =

∑K∈Dw

∑I (i)=J(j)=K

αIJK 〈f , hI 〉hJ , |αIJK | ≤|I |

12 |J|

Key idea∣∣∑

I (i)=J(j)=K αIJK 〈f , hI 〉hJ∣∣ ≤ 〈|f |〉K1K .

2 Paraproduct

πwb f =∑

K∈Dw

〈b, hK 〉〈f 〉KhK .

A direct corollary is we obtain T1 theorem. One more remark isthat, the representation formula has other applications as well, forinstance, Pott and Stoica use it to study the sharp UMDbounds;Ou et al. use it to study the commutators of tensorproduct type bi-parameter singular integrals.

Model operators

1 Shifts

S i ,jw f =

∑K∈Dw

∑I (i)=J(j)=K

12 |J|

Key idea∣∣∑

2 Paraproduct

πwb f =∑

K∈Dw

Model operators

1 Shifts

S i ,jw f =

∑K∈Dw

∑I (i)=J(j)=K

12 |J|

Key idea∣∣∑

2 Paraproduct

πwb f =∑

K∈Dw

Model operators

1 Shifts

S i ,jw f =

∑K∈Dw

∑I (i)=J(j)=K

12 |J|

Key idea∣∣∑

2 Paraproduct

πwb f =∑

K∈Dw

A direct corollary is we obtain T1 theorem.

One more remark isthat, the representation formula has other applications as well, forinstance, Pott and Stoica use it to study the sharp UMDbounds;Ou et al. use it to study the commutators of tensorproduct type bi-parameter singular integrals.

Model operators

1 Shifts

S i ,jw f =

∑K∈Dw

∑I (i)=J(j)=K

12 |J|

Key idea∣∣∑

2 Paraproduct

πwb f =∑

K∈Dw

A direct corollary is we obtain T1 theorem. One more remark isthat, the representation formula has other applications as well, forinstance, Pott and Stoica use it to study the sharp UMDbounds;

Ou et al. use it to study the commutators of tensorproduct type bi-parameter singular integrals.

Model operators

1 Shifts

S i ,jw f =

∑K∈Dw

∑I (i)=J(j)=K

12 |J|

Key idea∣∣∑

2 Paraproduct

πwb f =∑

K∈Dw

A quick view of how representation implies T (1)

Notation: ∆iK f =

∑I :I (i)=K ∆I f , where ∆I is the classical

martingale difference. For the proof, by duality we have∣∣⟨S i ,jw (f ), g

⟩∣∣ ≤ ∑K∈Dw

∑I (i)=J(j)=K

|αIJK ||〈f , hI 〉||〈g , hJ〉|

K∈Dw

∑I (i)=J(j)=K

|αIJK ||〈∆iK f , hI 〉||〈∆

jKg , hJ〉|

≤∑

K∈Dw

〈|∆iK f |〉K 〈|∆

jKg |〉K |K |

≤∫ ( ∑

K∈Dw

M(∆iK f )2

)1/2( ∑K∈Dw

M(∆jKg)2

≤ ‖( ∑

K∈Dw

M(∆iK f )2

)1/2‖Lp‖

( ∑K∈Dw

M(∆jKg)2

)1/2‖Lp′

≤ ‖f ‖Lp‖g‖Lp′ .

A quick view of how representation implies T (1)

Notation:

S(f ) =( ∑

K∈Dw

|〈f , hK 〉|21K|K |

For the paraproduct, again by H1 − BMO duality, we have

|〈πwb f , g〉| =∣∣∣⟨b, ∑

K∈Dw

〈f 〉K 〈g , hK 〉hK⟩∣∣∣ =: |〈b, φ〉|

≤ ‖b‖BMO‖S(φ)‖L1

≤ ‖b‖BMO

∫ ( ∑K∈Dw

|〈f 〉K |2|〈g , hK 〉|21K|K |

≤ ‖b‖BMO‖M(f )S(g)‖L1

≤ ‖b‖BMO‖M(f )‖Lp‖S(g)‖Lp′≤ ‖b‖BMO‖f ‖Lp‖g‖Lp′ .

Coifman–Meyer multiplier theorem

A classical (1978) result of Coifman and Meyer concerns theboundedness of one-parameter bilinear (or more generallymultilinear) multipliers of the form

Tm(f1, f2)(x) =

∫∫Rn×Rn

m(ξ1, ξ2)f1(ξ1)f2(ξ2)e2πix ·(ξ1+ξ2) dξ,

where f denotes the Fourier transform (here in Rn) and m satisfies

|∂α1ξ1∂α2ξ2m(ξ1, ξ2)| . (|ξ1|+ |ξ2|)−|α1|−|α2|.

Coifman and Meyer had only proved the boundedness when r > 1.

Coifman–Meyer multiplier theorem

A classical (1978) result of Coifman and Meyer concerns theboundedness of one-parameter bilinear (or more generallymultilinear) multipliers of the form

Tm(f1, f2)(x) =

∫∫Rn×Rn

m(ξ1, ξ2)f1(ξ1)f2(ξ2)e2πix ·(ξ1+ξ2) dξ,

where f denotes the Fourier transform (here in Rn) and m satisfies

|∂α1ξ1∂α2ξ2m(ξ1, ξ2)| . (|ξ1|+ |ξ2|)−|α1|−|α2|.

Coifman and Meyer had only proved the boundedness when r > 1.

Bilinear CZ theory and representation theorem

Grafakos and Torres (2002) refined the results of Coifman andMeyer by systematically formulating the bilinear (multilinear)Calderon-Zygmund theory, showing that these operators are in factbilinear singular integrals.

Using a bilinear T1 theorem they getthat

Tm : Lp(Rn)× Lq(Rn)→ Lr (Rn)

for all 1 < p, q ≤ ∞, 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .Recently, the representation formula for bilinear Calderon-Zygmundoperators was obtained by L.-Martikainen-Ou-Vuorinen. Therelated model operators are natural extensions of the linear shifts(non-cancellative haar functions appear) and paraproducts.

Grafakos and Torres (2002) refined the results of Coifman andMeyer by systematically formulating the bilinear (multilinear)Calderon-Zygmund theory, showing that these operators are in factbilinear singular integrals. Using a bilinear T1 theorem they getthat

for all 1 < p, q ≤ ∞, 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .

Recently, the representation formula for bilinear Calderon-Zygmundoperators was obtained by L.-Martikainen-Ou-Vuorinen. Therelated model operators are natural extensions of the linear shifts(non-cancellative haar functions appear) and paraproducts.

Grafakos and Torres (2002) refined the results of Coifman andMeyer by systematically formulating the bilinear (multilinear)Calderon-Zygmund theory, showing that these operators are in factbilinear singular integrals. Using a bilinear T1 theorem they getthat

for all 1 < p, q ≤ ∞, 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .Recently, the representation formula for bilinear Calderon-Zygmundoperators was obtained by L.-Martikainen-Ou-Vuorinen. Therelated model operators are natural extensions of the linear shifts(non-cancellative haar functions appear) and paraproducts.

Bi-parameter version of Coifman–Meyer multiplier theorem

Let m ∈ L∞(Rn+m × Rn+m) satisfy the product type estimate

|∂α1ξ1∂α2ξ2∂β1η1∂β2η2m(ξ, η)| . (|ξ1|+ |η1|)−|α1|−|β1|(|ξ2|+ |η2|)−|α2|−|β2|.

For Schwartz functions f1, f2 : Rn+m → C and x ∈ Rn+m define

Tm(f1, f2)(x) =

∫∫Rn+m

m(ξ, η)f1(ξ)f2(η)e2πix ·(ξ+η) dξ dη.

This is a much harder class of operators as there simply are muchmore symbols satisfying the above product estimate, as opposed tothe stronger estimate

|∂α1ξ ∂

α2η m(ξ, η)| . (|ξ|+ |η|)−|α1|−|α2|

required by the usual Coifman–Meyer theorem.

Tm(f1, f2)(x) =

∫∫Rn+m

|∂α1ξ ∂

α2η m(ξ, η)| . (|ξ|+ |η|)−|α1|−|α2|

Tm(f1, f2)(x) =

∫∫Rn+m

|∂α1ξ ∂

α2η m(ξ, η)| . (|ξ|+ |η|)−|α1|−|α2|

When you start studying this operator you encounter at least thefollowing:

1 There is no general theory of bilinear bi-parameter singularintegrals available, so the Grafakos–Torres route is notavailable.

2 The boundedness of Tm in the Banach range,

for all 1 < p, q, r <∞ satisfying 1/p + 1/q = 1/r , is not verydifficult.

3 But there no more is a general principle how to get thedifficult range r ≤ 1 (i.e. the quasi-Banach range) from this.Indeed, the bi-parameter setup removes many of the key tools:you cannot anymore expect end point estimates likeL1 × L1 → L1/2,∞, and you don’t have tools like theCalderon–Zygmund decomposition or sparse dominationavailable due to the bi-parameter setup!

Muscalu–Pipher–Tao–Thiele theorem

Nevertheless, the main result of a 2004 Acta paper byMuscalu–Pipher–Tao–Thiele is that

Tm : Lp(Rn+m)× Lq(Rn+m)→ Lr (Rn+m)

for all 1 < p, q ≤ ∞, 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .

As an application, the following Leibniz rule type estimate isobtained

‖Dα1 D

β2 (fg)‖r . ‖Dα

1 Dβ2 f ‖p1‖g‖q1 + ‖f ‖p2‖Dα

1 Dβ2 g‖q2

+ ‖Dα1 f ‖p3‖D

β2 g‖q3 + ‖Dβ

2 f ‖p4‖Dα1 g‖q4 ,

where 1/pi + 1/qi = 1/r , 1 < pi , qi ≤ ∞, andmax((1 + α)−1, (1 + β)−1) < r <∞.

Muscalu–Pipher–Tao–Thiele theorem

Nevertheless, the main result of a 2004 Acta paper byMuscalu–Pipher–Tao–Thiele is that

Tm : Lp(Rn+m)× Lq(Rn+m)→ Lr (Rn+m)

for all 1 < p, q ≤ ∞, 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .As an application, the following Leibniz rule type estimate isobtained

‖Dα1 D

β2 (fg)‖r . ‖Dα

1 Dβ2 f ‖p1‖g‖q1 + ‖f ‖p2‖Dα

1 Dβ2 g‖q2

+ ‖Dα1 f ‖p3‖D

β2 g‖q3 + ‖Dβ

2 f ‖p4‖Dα1 g‖q4 ,

where 1/pi + 1/qi = 1/r , 1 < pi , qi ≤ ∞, andmax((1 + α)−1, (1 + β)−1) < r <∞.

Our goal

Our goal was to develop a completely general theory of bilinearbi-parameter singular integrals so that:

We establish the boundedness of the singular integrals on thefull range, and get the above Muscalu–Pipher–Tao–Thieletheorem, and other more recent developments, as a specialcase of our general theory of singular integrals.

We can prove weighted bi-parameter estimates.

We present the exact dyadic structure of these operators via arepresentation theorem.

The representation is the key to all the boundedness results we willprove (quasi–Banach, weighted, mixed-norm), but has otherapplications as well, for example: commutators.

Our goal

Representation theorem in bi-parameter case

The exact dyadic structure given by the representation theorems isimportant, especially so in the multi-parameter world, where manyother tools (good lambda, Carleson embedding etc.) are missing.

Martikainen proved a dyadic representation of bi-parametersingular integrals in 2011. A multi-parameter version is by Ou.With the representation formula, Holmes, Petermichl and Wickwere able to significantly simply the proof of weighted boundednessof bi-parameter singular integrals (which was proved by R.Fefferman first with Ap/2 and finally improved by himself to Ap),moreover, it is also crucial for the boundedness for commutators(actually, a two weight Bloom type inequality was proved there).

Martikainen proved a dyadic representation of bi-parametersingular integrals in 2011. A multi-parameter version is by Ou.

With the representation formula, Holmes, Petermichl and Wickwere able to significantly simply the proof of weighted boundednessof bi-parameter singular integrals (which was proved by R.Fefferman first with Ap/2 and finally improved by himself to Ap),moreover, it is also crucial for the boundedness for commutators(actually, a two weight Bloom type inequality was proved there).

Martikainen proved a dyadic representation of bi-parametersingular integrals in 2011. A multi-parameter version is by Ou.With the representation formula, Holmes, Petermichl and Wickwere able to significantly simply the proof of weighted boundednessof bi-parameter singular integrals

(which was proved by R.Fefferman first with Ap/2 and finally improved by himself to Ap),moreover, it is also crucial for the boundedness for commutators(actually, a two weight Bloom type inequality was proved there).

Martikainen proved a dyadic representation of bi-parametersingular integrals in 2011. A multi-parameter version is by Ou.With the representation formula, Holmes, Petermichl and Wickwere able to significantly simply the proof of weighted boundednessof bi-parameter singular integrals (which was proved by R.Fefferman first with Ap/2 and finally improved by himself to Ap),moreover, it is also crucial for the boundedness for commutators(actually, a two weight Bloom type inequality was proved there).

General bilinear bi-parameter singular integrals

A model of a bilinear CZO T in Rn is

T (f1, f2)(x) := T (f1 ⊗ f2)(x , x), x ∈ Rn,

where T is a usual linear CZO in R2n.

A model of a bilinear bi-parameter CZO in Rn × Rm is

(T1 ⊗ T2)(f1 ⊗ f2, g1 ⊗ g2)(x) := T1(f1, g1)(x1)T2(f2, g2)(x2),

where f1, g1 : Rn → C, f2, g2 : Rm → C, x = (x1, x2) ∈ Rn+m, T1 isa bilinear CZO in Rn and T2 is a bilinear CZO in Rm.

A model of a bilinear CZO T in Rn is

T (f1, f2)(x) := T (f1 ⊗ f2)(x , x), x ∈ Rn,

where T is a usual linear CZO in R2n.

A model of a bilinear bi-parameter CZO in Rn × Rm is

(T1 ⊗ T2)(f1 ⊗ f2, g1 ⊗ g2)(x) := T1(f1, g1)(x1)T2(f2, g2)(x2),

where f1, g1 : Rn → C, f2, g2 : Rm → C, x = (x1, x2) ∈ Rn+m, T1 isa bilinear CZO in Rn and T2 is a bilinear CZO in Rm.

The general definition is derived from the above considerations.Therefore, a bilinear bi-parameter singular integral is a bilinearoperator T hitting two functions f1, f2 : Rn+m → C

so that giventhree tensor product form functions fi = f 1

i ⊗ f 2i we have that

〈T (f1, f2), f3〉 can be written using different kernel representationsdepending on whether

1 spt f 1i ∩ spt f 1

j = ∅ for some i , j and spt f 2i ′ ∩ spt f 2

j ′ = ∅ forsome i ′, j ′; or

j = ∅ for some i , j ; or

3 spt f 2i ′ ∩ spt f 2

j ′ = ∅ for some i ′, j ′.

In case 1 we have the so called full kernel representation, and inthe remaining cases we have some partial kernel representations inRn (case 2) or Rm (case 3). The exact definition is rather long(but similar with the linear case introduced in the very beginning),so we skip it here.

The general definition is derived from the above considerations.Therefore, a bilinear bi-parameter singular integral is a bilinearoperator T hitting two functions f1, f2 : Rn+m → C so that giventhree tensor product form functions fi = f 1

j ′ = ∅ for some i ′, j ′.

In case 1 we have the so called full kernel representation, and inthe remaining cases we have some partial kernel representations inRn (case 2) or Rm (case 3).

The exact definition is rather long(but similar with the linear case introduced in the very beginning),so we skip it here.

j ′ = ∅ for some i ′, j ′.

Model operators: shifts

For k = (k1, k2, k3), k1, k2, k3 ≥ 0, v = (v1, v2, v3), v1, v2, v3 ≥ 0,K ∈ Dn and V ∈ Dm define

AV ,vK ,k(f1, f2) =

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

∑J1,J2,J3∈Dm

J(v1)1 =J

(v2)2 =J

(v3)3 =V

aK ,V ,(Ii ),(Jj )

× 〈f1, hI1 ⊗ hJ1〉〈f2, hI2 ⊗ hJ2〉h0I3 ⊗ h0

We also demand that the scalars aK ,V ,(Ii ),(Jj ) satisfy the estimate

|aK ,V ,(Ii ),(Jj )| ≤|I1|1/2|I2|1/2|I3|1/2

|K |2|J1|1/2|J2|1/2|J3|1/2

|V |2.

A shift of complexity (k, v) of a particular form (thenon-cancellative Haar functions are in certain positions) is

Svk (f1, f2) =

∑K∈Dn

∑V∈Dm

AV ,vK ,k(f1, f2).

An operator of the above form, but having the non-cancellativeHaar functions h0

I and h0J in some of the other slots, is also a shift.

AV ,vK ,k(f1, f2) =

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

∑J1,J2,J3∈Dm

J(v1)1 =J

(v2)2 =J

(v3)3 =V

aK ,V ,(Ii ),(Jj )

× 〈f1, hI1 ⊗ hJ1〉〈f2, hI2 ⊗ hJ2〉h0I3 ⊗ h0

|aK ,V ,(Ii ),(Jj )| ≤|I1|1/2|I2|1/2|I3|1/2

|K |2|J1|1/2|J2|1/2|J3|1/2

|V |2.

Svk (f1, f2) =

∑K∈Dn

∑V∈Dm

AV ,vK ,k(f1, f2).

AV ,vK ,k(f1, f2) =

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

∑J1,J2,J3∈Dm

J(v1)1 =J

(v2)2 =J

(v3)3 =V

aK ,V ,(Ii ),(Jj )

× 〈f1, hI1 ⊗ hJ1〉〈f2, hI2 ⊗ hJ2〉h0I3 ⊗ h0

|aK ,V ,(Ii ),(Jj )| ≤|I1|1/2|I2|1/2|I3|1/2

|K |2|J1|1/2|J2|1/2|J3|1/2

|V |2.

Svk (f1, f2) =

∑K∈Dn

∑V∈Dm

AV ,vK ,k(f1, f2).

AV ,vK ,k(f1, f2) =

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

∑J1,J2,J3∈Dm

J(v1)1 =J

(v2)2 =J

(v3)3 =V

aK ,V ,(Ii ),(Jj )

× 〈f1, hI1 ⊗ hJ1〉〈f2, hI2 ⊗ hJ2〉h0I3 ⊗ h0

|aK ,V ,(Ii ),(Jj )| ≤|I1|1/2|I2|1/2|I3|1/2

|K |2|J1|1/2|J2|1/2|J3|1/2

|V |2.

Svk (f1, f2) =

∑K∈Dn

∑V∈Dm

AV ,vK ,k(f1, f2).

Model operators: partial paraproducts

Let k = (k1, k2, k3), k1, k2, k3 ≥ 0. For each K , I1, I2, I3 ∈ Dn weare given a function bK ,I1,I2,I3 : Rm → C such that

‖bK ,I1,I2,I3‖BMO(Rm) ≤|I1|1/2|I2|1/2|I3|1/2

|K |2. memorandum

A partial paraproduct of complexity k of a particular form is

Pk(f1, f2) =∑K∈Dn

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

h0I3⊗πbK ,I1,I2,I3

(〈f1, hI1〉1, 〈f2, hI2〉1),

where πb denotes a bilinear paraproduct in Rm:

πb(g1, g2) :=∑

V∈Dm

〈b, hV 〉〈g1〉V 〈g2〉V hV .

Again, an operator of the above form, but having thenon-cancellative Haar function h0

I or the cancellative Haar functionhV in some other slot, is also a partial paraproduct. Of course, theparaproduct component can also be in Rn.

‖bK ,I1,I2,I3‖BMO(Rm) ≤|I1|1/2|I2|1/2|I3|1/2

|K |2. memorandum

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

(〈f1, hI1〉1, 〈f2, hI2〉1),

πb(g1, g2) :=∑

V∈Dm

〈b, hV 〉〈g1〉V 〈g2〉V hV .

‖bK ,I1,I2,I3‖BMO(Rm) ≤|I1|1/2|I2|1/2|I3|1/2

|K |2. memorandum

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

(〈f1, hI1〉1, 〈f2, hI2〉1),

πb(g1, g2) :=∑

V∈Dm

〈b, hV 〉〈g1〉V 〈g2〉V hV .

‖bK ,I1,I2,I3‖BMO(Rm) ≤|I1|1/2|I2|1/2|I3|1/2

|K |2. memorandum

∑I1,I2,I3∈Dn

I(k1)1 =I

(k2)2 =I

(k3)3 =K

(〈f1, hI1〉1, 〈f2, hI2〉1),

πb(g1, g2) :=∑

V∈Dm

〈b, hV 〉〈g1〉V 〈g2〉V hV .

Model operators: full paraproducts

Given a function b : Rn+m → C a full paraproduct Πb of aparticular form is

Πb(f1, f2) =∑K∈Dn

V∈Dm

〈b, hK × hV 〉〈f1〉K×V 〈f2〉K×V hK ⊗ hV .

Again, an operator of the above form, but having the cancellativeHaar functions hK or hV in some other slots, is also a fullparaproduct.

Recall the product BMO norm

‖b‖BMOD

prod:= sup

|Ω|∑

K∈Dn,V∈Dm

K×V⊂Ω

|〈b, hK × hV 〉|2)1/2

where the supremum is taken over all the bounded open setsΩ ⊂ Rn+m.

V∈Dm

‖b‖BMOD

prod:= sup

|Ω|∑

K∈Dn,V∈Dm

K×V⊂Ω

|〈b, hK × hV 〉|2)1/2

V∈Dm

‖b‖BMOD

prod:= sup

|Ω|∑

K∈Dn,V∈Dm

K×V⊂Ω

|〈b, hK × hV 〉|2)1/2

Results: the representation theorem

Theorem (L.-Martikainen-Vuorinen, 2018)

Suppose T is a bilinear bi-parameter Calderon–Zygmund operatorsatisfying certain natural T1 type conditions.

Then for boundedand compactly supported functions fi : Rn+m → C we can write〈T (f1, f2), f3〉 as the sum

CTEω,ω′∑

k=(k1,k2,k3)∈Z3+

v=(v1,v2,v3)∈Z3+

⟨Uvk,u,Dn

ω ,Dmω′

(f1, f2), f3⟩,

where CT . 1, αk,v = 2−αmax ki/22−αmax vj/2, the summation overu is finite, and Uv

k,u,Dnω ,Dm

ω′is always either a shift of complexity

(k , v), a partial paraproduct of complexity k or v (this requiresv = 0 or k = 0) or a full paraproduct (this requires k = v = 0)associated with some product BMO function b satisfying‖b‖BMOprod(Rn+m)

≤ 1.

Suppose T is a bilinear bi-parameter Calderon–Zygmund operatorsatisfying certain natural T1 type conditions. Then for boundedand compactly supported functions fi : Rn+m → C we can write〈T (f1, f2), f3〉 as the sum

CTEω,ω′∑

k=(k1,k2,k3)∈Z3+

v=(v1,v2,v3)∈Z3+

⟨Uvk,u,Dn

ω ,Dmω′

(f1, f2), f3⟩,

k,u,Dnω ,Dm

≤ 1.

Suppose T is a bilinear bi-parameter Calderon–Zygmund operatorsatisfying certain natural T1 type conditions. Then for boundedand compactly supported functions fi : Rn+m → C we can write〈T (f1, f2), f3〉 as the sum

CTEω,ω′∑

k=(k1,k2,k3)∈Z3+

v=(v1,v2,v3)∈Z3+

⟨Uvk,u,Dn

ω ,Dmω′

(f1, f2), f3⟩,

k,u,Dnω ,Dm

≤ 1.

Results: bounds for CZO that are free of full paraproducts

A bilinear bi-parameter CZO is free of full paraproducts if for allthe nine adjoints and partial adjoints S of T (i.e.,S ∈ T ,T 1∗,T 2∗,T 1∗

1 ,T 1∗2 ,T 1∗

2 ,T 2∗2 ,T 1∗,2∗

1,2 ,T 1∗,2∗2,1 ), and for all

cubes I ⊂ Rn, J ⊂ Rm there holds

〈S(1, 1), hI ⊗ hJ〉 = 0.

Suppose T is bilinear bi-parameter Calderon–Zygmund operatorsatisfying the assumptions of the representation theorem, and thatT is free of full paraproducts. Then we have

‖T (f1, f2)‖Lr (v3) . ‖f1‖Lp(w1)‖f2‖Lq(w2)

for all 1 < p, q <∞ and 1/2 < r <∞ satisfying 1/p + 1/q = 1/r ,and for all bi-parameter weights w1 ∈ Ap(Rn × Rm),

w2 ∈ Aq(Rn × Rm) with v3 := wr/p1 w

r/q2 .

A bilinear bi-parameter CZO is free of full paraproducts if for allthe nine adjoints and partial adjoints S of T (i.e.,S ∈ T ,T 1∗,T 2∗,T 1∗

1 ,T 1∗2 ,T 1∗

2 ,T 2∗2 ,T 1∗,2∗

1,2 ,T 1∗,2∗2,1 ), and for all

cubes I ⊂ Rn, J ⊂ Rm there holds

〈S(1, 1), hI ⊗ hJ〉 = 0.

Suppose T is bilinear bi-parameter Calderon–Zygmund operatorsatisfying the assumptions of the representation theorem, and thatT is free of full paraproducts. Then we have

‖T (f1, f2)‖Lr (v3) . ‖f1‖Lp(w1)‖f2‖Lq(w2)

for all 1 < p, q <∞ and 1/2 < r <∞ satisfying 1/p + 1/q = 1/r ,and for all bi-parameter weights w1 ∈ Ap(Rn × Rm),

w2 ∈ Aq(Rn × Rm) with v3 := wr/p1 w

r/q2 .

Theorem (continued)

In the unweighted case we have the following general mixed-normestimates

‖T (f1, f2)‖Lr1 (Rn;Lr2 (Rm)) . ‖f1‖Lp1 (Rn;Lp2 (Rm))‖f2‖Lq1 (Rn;Lq2 (Rm))

for all 1 < pi , qi ≤ ∞ and 1/2 < ri < 1 with 1/pi + 1/qi = 1/ri ,except that if r2 < 1 we have to assume ∞ 6∈ p1, q1.

The Muscalu–Pipher–Tao–Thiele bi-parameter multipliers areparticular operators satisfying the above assumptions: they areconvolution form and free of partial paraproducts. Therefore, we

1 Recover the theorem of Muscalu–Pipher–Tao–Thiele, and infact prove a weighted version of their result, and

2 Recover the mixed-norm estimates of the multipliers provedby Benea–Muscalu (which they proved using their recenthelicoidal method).

Theorem (continued)

The Muscalu–Pipher–Tao–Thiele bi-parameter multipliers areparticular operators satisfying the above assumptions: they areconvolution form and free of partial paraproducts.

Therefore, we

Theorem (continued)

Results: bounds for all CZO

Suppose T is bilinear bi-parameter Calderon–Zygmund operatorsatisfying the assumptions of the representation theorem. Then wehave

‖T (f1, f2)‖Lr (Rn+m) . ‖f1‖Lp(Rn+m)‖f2‖Lq(Rn+m)

for all 1 < p, q ≤ ∞ and 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .

Moreover, we have the mixed-norm estimates

for all 1 < pi , qi ≤ ∞ with 1/pi + 1/qi = 1/ri , i = 1, 2,1/2 < r1 <∞, 1 ≤ r2 <∞.

This proves that all T have completely satisfactory unweightedbounds in the non-mixed norm case – a substantialgeneralisation of the Muscalu–Pipher–Tao–Thiele theorem.The mixed-norm bounds are not as general as above.

for all 1 < p, q ≤ ∞ and 1/2 < r <∞ satisfying 1/p + 1/q = 1/r .Moreover, we have the mixed-norm estimates

This proves that all T have completely satisfactory unweightedbounds in the non-mixed norm case – a substantialgeneralisation of the Muscalu–Pipher–Tao–Thiele theorem.

The mixed-norm bounds are not as general as above.

Results: commutators

To study commutators, bmo space is involved, which is also provedto be necessary for some special cases. Recall thatb ∈ bmo(Rn × Rm), if

‖b‖bmo := supR

∫R|b − 〈b〉R | <∞,

where the supremum is taken over all the rectangles in Rn+m.

Wealso have some results for commutators, which are listed in below:

1 Banach range: commutators and iterated commutators arebounded

2 Quasi Banach range: commutators are bounded if T is free ofparaproduct, i.e., for

S ∈ T ,T 1∗,T 2∗,T 1∗1 ,T 1∗

2 ,T 1∗2 ,T 2∗

2 ,T 1∗,2∗1,2 ,T 1∗,2∗

〈S(1⊗ g1, 1⊗ g2), hI ⊗ hJ〉 = 〈S(f1 ⊗ 1, f2 ⊗ 1), hI ⊗ hJ〉 = 0.

Weighted boundedness ? Open.

‖b‖bmo := supR

∫R|b − 〈b〉R | <∞,

where the supremum is taken over all the rectangles in Rn+m. Wealso have some results for commutators, which are listed in below:

S ∈ T ,T 1∗,T 2∗,T 1∗1 ,T 1∗

2 ,T 1∗2 ,T 2∗

2 ,T 1∗,2∗1,2 ,T 1∗,2∗

〈S(1⊗ g1, 1⊗ g2), hI ⊗ hJ〉 = 〈S(f1 ⊗ 1, f2 ⊗ 1), hI ⊗ hJ〉 = 0.

Weighted boundedness ? Open.

‖b‖bmo := supR

∫R|b − 〈b〉R | <∞,

S ∈ T ,T 1∗,T 2∗,T 1∗1 ,T 1∗

2 ,T 1∗2 ,T 2∗

2 ,T 1∗,2∗1,2 ,T 1∗,2∗

〈S(1⊗ g1, 1⊗ g2), hI ⊗ hJ〉 = 〈S(f1 ⊗ 1, f2 ⊗ 1), hI ⊗ hJ〉 = 0.

Weighted boundedness ?

‖b‖bmo := supR

∫R|b − 〈b〉R | <∞,

S ∈ T ,T 1∗,T 2∗,T 1∗1 ,T 1∗

2 ,T 1∗2 ,T 2∗

2 ,T 1∗,2∗1,2 ,T 1∗,2∗

〈S(1⊗ g1, 1⊗ g2), hI ⊗ hJ〉 = 〈S(f1 ⊗ 1, f2 ⊗ 1), hI ⊗ hJ〉 = 0.

Weighted boundedness ? Open.K. Li(BCAM) Bilinear bi-parameter Calderon-Zygmund theory

Thank you for your attention!

Bilinear bi-parameter Calderón-Zygmund theory

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