LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

26
LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR TUOMAS OIKARI ABSTRACT. Answering a key point left open in the recent work of Bongers, Guo, Li and Wick [2], we provide the lower bound kbk BMOγ (R 2 ) . k[b, Hγ ]k L p (R 2 )L p (R 2 ) , where Hγ is the parabolic Hilbert transform. 1. I NTRODUCTION The commutator of the parabolic Hilbert transform, [b, H γ ]f (x)= b(x)H γ f (x) - H γ (bf )(x), H γ f (x)= p.v. ˆ R f (x - γ (t)) dt t , where b L 1 loc (R 2 ; C)(t)=(t, t 2 ) and f : R 2 C, was recently studied in Bongers et al. [2], where they prove the following commutator estimates kbk test . k[b, H γ ]k L p (R 2 )L p (R 2 ) . kbk BMOγ (R 2 ) . (1.1) The upper bound involves the parabolic bmo norm kbk BMOγ (R 2 ) = sup Q∈Rγ Q |b -hbi Q |, where R γ is the collection of parabolic rectangles, i.e., rectangles R = I × J in the plane parallel with the coordinate axes such that (J )= (I ) 2 . The lower bound however in- volves the non-matching testing condition kbk test = sup Q∈Rγ Q b(x) - 1 μ ( I x,E Q ) ˆ I x,E Q b(x - γ (t)) dμ(t) dx, where μ(t)= dt t and E Q = x - γ (t): x Q, t [9(I ), 10(I )] , I x,E Q = t R : x - γ (t) E Q . Often, the necessity (the lower bound) is even more challenging than the corresponding sufficiency (the upper bound). In [2] the necessity was left open and we provide a proof here, thus completing the picture. Our main result is the following Theorem 1.2. 2010 Mathematics Subject Classification. 42B20. Key words and phrases. singular integrals, parabolic Hilbert transform, commutators, bounded mean os- cillation, parabolic bmo. T. Oikari was supported by the Academy of Finland project numbers 306901 and 314829, by the Finnish Centre of Excellence in Analysis and Dynamics Research project No. 307333, by the three-year research grant of the University of Helsinki No. 75160010 and by the Jenny and Antti Wihuri Foundation. 1 arXiv:2109.07939v3 [math.CA] 22 Oct 2021

Transcript of LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

Page 1: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

TUOMAS OIKARI

ABSTRACT. Answering a key point left open in the recent work of Bongers, Guo, Li andWick [2], we provide the lower bound

‖b‖BMOγ(R2) . ‖[b,Hγ ]‖Lp(R2)→Lp(R2),

where Hγ is the parabolic Hilbert transform.

1. INTRODUCTION

The commutator of the parabolic Hilbert transform,

[b,Hγ ]f(x) = b(x)Hγf(x)−Hγ(bf)(x), Hγf(x) = p.v.

ˆRf(x− γ(t))

dt

t,

where b ∈ L1loc(R2;C), γ(t) = (t, t2) and f : R2 → C, was recently studied in Bongers et

al. [2], where they prove the following commutator estimates

‖b‖test . ‖[b,Hγ ]‖Lp(R2)→Lp(R2) . ‖b‖BMOγ(R2).(1.1)

The upper bound involves the parabolic bmo norm

‖b‖BMOγ(R2) = supQ∈Rγ

Q|b− 〈b〉Q|,

where Rγ is the collection of parabolic rectangles, i.e., rectangles R = I × J in the planeparallel with the coordinate axes such that `(J) = `(I)2. The lower bound however in-volves the non-matching testing condition

‖b‖test = supQ∈Rγ

Q

∣∣b(x)− 1

µ(Ix,EQ

) ˆIx,EQ

b(x− γ(t)) dµ(t)∣∣dx,

where µ(t) = dtt and

EQ ={x− γ(t) : x ∈ Q, t ∈ [9`(I), 10`(I)]

}, Ix,EQ =

{t ∈ R : x− γ(t) ∈ EQ

}.

Often, the necessity (the lower bound) is even more challenging than the correspondingsufficiency (the upper bound). In [2] the necessity was left open and we provide a proofhere, thus completing the picture. Our main result is the following Theorem 1.2.

2010 Mathematics Subject Classification. 42B20.Key words and phrases. singular integrals, parabolic Hilbert transform, commutators, bounded mean os-

cillation, parabolic bmo.T. Oikari was supported by the Academy of Finland project numbers 306901 and 314829, by the Finnish

Centre of Excellence in Analysis and Dynamics Research project No. 307333, by the three-year researchgrant of the University of Helsinki No. 75160010 and by the Jenny and Antti Wihuri Foundation.

1

arX

iv:2

109.

0793

9v3

[m

ath.

CA

] 2

2 O

ct 2

021

Page 2: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

2 TUOMAS OIKARI

1.2. Theorem. Let b ∈ L1loc(R2;C) and p ∈ (1,∞). Then,

‖b‖BMOγ(R2) . ‖[b,Hγ ]‖Lp(R2)→Lp(R2).

Taken together the lower bound in Theorem 1.2 and the upper bound in (1.1) allow usto conclude the following.

1.3. Theorem. Let b ∈ L1loc(R2;C) and p ∈ (1,∞). Then,

‖b‖BMOγ(R2) ∼ ‖[b,Hγ ]‖Lp(R2)→Lp(R2).

We prove Theorem 1.2 with a non-trivial adaptation of the approximate weak factor-ization argument.

The approximate weak factorization (awf) argument for proving commutator lowerbounds for singular integral operators (SIOs) was recently developed and applied inHytönen [8] to complete the following picture. Let 1 < p, q <∞, b ∈ L1

loc(Rd) and T be anon-degenerate Calderón-Zygmund operator (CZO), then

‖[b, T ]‖Lp(Rd)→Lq(Rd) ∼

‖b‖BMO(Rd), q = p, [5] (1976),

‖b‖Cα,0(Rd), α = d(

1p −

1q

), q > p, [9] (1978),

‖b‖Ls(Rd),1q = 1

s + 1p , q < p, [8] (2018).

(1.4)

The commutator in (1.4) is defined by [b, T ]f = bTf −T (bf), and the listed references areCoifman, Rochberg, Weiss [5] and Janson [9]. The awf argument is strong in that it givesa unified approach to all of the three cases, in that it works for many singular integralswith kernels satisfying only minimum non-degeneracy assumptions, and in that it isflexible enough to grant e.g. multi-parameter and multilinear extensions. For the multi-parameter variants of the awf argument see Airta, Hytönen, Li, Martikainen, Oikari [1]and Oikari [15], where, respectfully, the commutators[

T2, [T1, b]],[b, T

]: Lp1(Rd1 ;Lp2(Rd2))→ Lq1(Rd1 ;Lq2(Rd2))(1.5)

were treated. On the line (1.5), 1 < p1, p2, q1, q2 < ∞, Ti is a one-parameter CZO on Rdi ,for i = 1, 2, and T is a bi-parameter CZO on Rd1+d2 . The adaptation of the awf argumentto the bi-parameter settings was not effortless and for both commutators on the line (1.5)the characterization of some cases is still open. For the multilinear extension see Oikari[16].

Another often-used argument, next to the awf argument, for proving commutatorlower bounds is through the median method. The median method can only handle real-valued functions b, however, the advantage is that it works for iterated commutators.For an account of the median method see [8].

Commutators have of course been studied outside the aforementioned research arti-cles and for some additional historically significant developments, we direct the readerto Nehari [14] (the case q = p with the Hilbert transform) and Uchiyama [18] (compact-ness of commutators and the case q = p with any Riesz transform). Lastly, we mentionthe notable recent developments of Lerner, Ombrosi, Rivera-Ríos [10] and Guo, Lian, Wu[6], before [8], that both recognized good non-degeneracy assumptions for commutatorlower bounds.

Commutator estimates, for example, imply factorization results for Hardy spaces, see[5], they have applications in PDEs through compensated compactness and div-curl lem-mas, and they have played a major role in investigations of the Jacobian problem, see

Page 3: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 3

Coifman, Lions, Meyer and Semmes [4], Lindberg [13], and [8]. It is crucial in theseapplications that we have both commutator upper and lower bounds.

In this article we almost solely focus on commutator lower bounds. For the conve-nience of the reader, we recall some of the timely developments in the theory of com-mutator upper bounds. The rough rule of thumb is that the upper bounds in the casesq 6= p are easy and the main work lies with the case q = p. Concerning the case q = p,a modern sparse domination proof of the linear (also, essentially the multilinear) casecan be found in [10]; for a proof of the multi-parameter cases through dyadic decom-position techniques we refer the reader to Holmes, Petermichl and Wick [7], and to Li,Martikainen and Vuorinen [11], [12].

The settings considered in all the aforementioned research articles, apart from [1]which treats an iterated commutator of product nature, are such that the dimension ofthe ambient space Rd is the same as that of the singular integral. Detaching from this,we consider singular integrals in the plane that are lower dimensional compared to thefunctions they hit, i.e., the kernel is localized to a curve. The challenge in adapting theawf argument to the parabolic setting lies with the fact that a priori a curve can onlyrecord one dimensional information, whereas the parabolic bmo involves a truly twodimensional quantity. This mismatch brings new elements to the awf argument and ne-cessitates a construction of a new kind of geometry compared to those present in theprevious cases.

The main idea of the proof of Theorem 1.2 can however be recorded in a model situ-ation that involves only lines, in contrast to curves. Let us recall the directional Hilberttransforms:

Hσf(x) = p.v.

ˆRf(x− σt) dt

t, σ ∈ S1, f : R2 → C.

Let R denote the collection of all rectangles in the plane parallel to the coordinate axes.Then, the little bmo space is defined by the norm

‖b‖bmo(R2) = supR∈R

R|b− 〈b〉R|.

Our second result is the following.

1.6. Theorem. Let b ∈ L1loc(R2;C) and p ∈ (1,∞). Then,

‖b‖bmo(R2) ∼∑i=1,2

‖[b,Hei ]‖Lp(R2)→Lp(R2),

where e1 = (1, 0) and e2 = (0, 1).

Even though theorems 1.2 and 1.6 are independent, we recommend that the proof ofTheorem 1.6 is read first. As it is perhaps not clear that Theorem 1.6 is non-trivial, next,as a reminder, we record the bi-parameter result that follows immediately by applyingknown results.

1.7. Proposition. Let b ∈ L1loc(R2;C) and p ∈ (1,∞). Then,

‖b‖bmo(R2) ∼ max(

ess supx1∈R

∥∥[b(x1, ·), H]∥∥Lp(R)→Lp(R)

, ess supx2∈R

∥∥[b(·, x2), H]∥∥Lp(R)→Lp(R)

).

Page 4: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

4 TUOMAS OIKARI

Proof. Follows by Lemma 1.8 (see below) and the one-parameter result p = q recordedon the line (1.4). �

The following Lemma 1.8 was recorded at least in [7].

1.8. Lemma. Let b ∈ L1loc(Rd;C). Then,

‖b‖bmo(R2) ∼ max(

ess supx1∈R

‖b(x1, ·)‖BMO(R), ess supx2∈R

‖b(·, x2)‖BMO(R)

).

1.1. Basic notation. We denote L1loc(Rd;C) = L1

loc,´Rd =

´, and so on, mostly leaving

out the ambient space if this information is obvious.We denote averages with 〈f〉A =

fflA f = 1

|A|´A f, where |A| denotes the Lebesgue

measure of the set A. The indicator function of a set A is denoted by 1A.We denote z + A = {z + a : a ∈ A}, DilλA = {za : a ∈ A} for A ⊂ R2 and λ ∈ R.

For an interval I ⊂ R the centre-point is denoted cI and the concentric dilation is λI =

[cI − λ `(I)2 , cI + λ `(I)2 ], for λ > 0. We also denote −I = Dil−1 I.

A curve is a differentiable mapping γ = (γ1, . . . , γd) : I → Rd parametrized over someinterval I. We denote curve length with `(γ).

We denote A . B, if A ≤ CB for some constant C > 0 depending only on the dimen-sion of the underlying space, on integration exponents and on other absolute constantsappearing in the assumptions that we do not care about. Then A ∼ B, if A . B andB . A. Subscripts on constants (Ca,b,c,...) and quantifiers (.a,b,c,...) signify their depen-dence on those subscripts.

A large parameter A >> 1 will appear throughout the text and tracking it is essential.Then, we write .A if and only if the said estimate depends on the parameter A, and ifwe write X . CA · Y, then the implicit constant will never depend on the parameter A.

1.2. Acknowledgements. I thank Emil Vuorinen for reading through the manuscriptand for comments that led to improvements.

2. PROOF OF THEOREM 1.6

Proof of Theorem 1.6. We first show that the commutator norms are bounded above by‖b‖bmo. This follows immediately by the standard boundedness theory of commutatorsand lemma 1.8,

‖[b,He1 ]f‖Lp(R2) =∥∥∥∥∥[b(x1, x2), He1 ]f(x1, x2)

∥∥Lpx1 (R)

∥∥∥Lpx2 (R)

≤∥∥∥∥∥[b(x1, x2), H]

∥∥Lpx1 (R)→Lpx1 (R)

∥∥f(x1, x2)∥∥Lpx1 (R)

∥∥∥Lpx2 (R)

.∥∥∥∥∥b(x1, x2)

∥∥BMOx1 (R)

∥∥f(x1, x2)∥∥Lpx1 (R)

∥∥∥Lpx2 (R)

. ess supx2∈R

‖b(x1, x2)‖BMOx1 (R)

∥∥∥∥f(x1, x2)∥∥Lpx1 (R)

∥∥Lpx2 (R)

. ‖b‖bmo(R2)‖f‖Lp(R2).

The other commutator norms are estimated similarly. We turn to the lower bound.Fix a rectangle R0 = I × J and a constant A > 1 and define the three rectangles

R1 = R0 +A`(I)e1, R2 = R1 +A`(J)e2, R3 = R2 −A`(I)e1.

Page 5: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 5

Writing ψRi means that ψRi is a function supported on the set Ri. We begin with writingˆR0

|b− 〈b〉R0 | =ˆbf, f = (θ − 〈θ〉R0)1R0 , θ =

b− 〈b〉R0

|b− 〈b〉R0 |1{b6=〈b〉R0

},(2.1)

and

f =[hR0H

∗e1gR1 − gR1He1hR0

]+ fR1

=[hR0H

∗e1gR1 − gR1He1hR0

]+[hR1H

∗e2gR2 − gR2He2hR1

]+ fR2 ,

(2.2)

where

hR0 =f

H∗e1gR1

, gRi = 1Ri , fR1 = gR1He1hR0 , hR1 =fR1

H∗e2gR2

, fR2 = gR2He2hR1 .

The only possible problem in the above factorization of the function f is a division byzero in hRi , i = 1, 2, however, the estimates (2.6) and (2.5) below show the denominatorsto be strictly positive functions. Next, we will show that

|hR0 | .A 1R0 , |hR1 | .A 1R1 , ‖fR2‖∞ . A−1‖f‖∞.(2.3)

We reserve the following notation for the variables: x ∈ R0, y ∈ R1, z ∈ R2 and wedenote

I(x,+) = {t ∈ R : x+ te1 ∈ R1}, I(y,−) = {t ∈ R : y − te1 ∈ R0},J(y,+) = {t ∈ R : y + te2 ∈ R2}, J(z,−) = {t ∈ R : z − e2t ∈ R1}.

Notice that I(x,+), I(y,−) are intervals of length `(I) containing the point A`(I). Simi-larly, J(y,+), J(z,−) are intervals of length `(J) containing the point A`(J). Fix a pointz ∈ R2 and write

fR2(z) = He2hR1(z) =

ˆJ(z,−)

He1hR0(z − e2t)

H∗e2gR2(z − e2t)

dt

t

=

ˆJ(z,−)

1

H∗e2gR2(z − e2t)

ˆI(z−e2t,−)

f(z − e2t− e1s)

H∗e1gR1(z − e2t− e1s)

ds

s

dt

t

=

ˆJ(z,−)

ˆI(z−e2t,−)

f(z − e2t− e1s)

H∗e2gR2(z − e2t)H∗e1gR1(z − e2t− e1s)

ds

s

dt

t.

(2.4)

Let x ∈ R0 and y ∈ R1 be arbitrary. Then, there holds that

1

A+ 1≤ H∗e1gR1(x) =

ˆI(x,+)

dt

t≤ 1

A− 1(2.5)

and1

A+ 1≤ H∗e2gR2(y) =

ˆJ(y,+)

dt

t≤ 1

A− 1.(2.6)

From (2.6) and (2.5) it follows immediately that |hRi | .A 1Ri , for i = 0, 1, and hence forthe claims on the line (2.3) it remains to check that ‖fR2‖∞ . A−1‖f‖∞. For arbitraryt ∈ J(z,−) and s ∈ I(z − e2t,−), denoting

t′ = AH∗e2gR2(z − e2t)t, s′ = AH∗e1gR1(z − e2t− e1s)s,(2.7)

Page 6: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

6 TUOMAS OIKARI

there holds that

|t′ −A`(J)| . `(J), |s′ −A`(I)| . `(I).(2.8)

Let us briefly check the left estimate of (2.8). Assume e.g. that A`(J) ≤ t ≤ (A + 1)`(J),then by (2.6) we find that

|t′ −A`(J)| ≤ A

A− 1t−A`(J) ≤ A+ 1

A− 1A`(J)−A`(J) =

(A+ 1

A− 1A−A

)`(J)

=2A

A− 1`(J) . `(J),

whenever, say, A > 2. The other cases can be checked similarly. Now we come to thecrucial part of the argument. The double integral in (2.4) is exactly over the rectangle R0,i.e., for all z ∈ R2 there holds that

R0 = {z − e2t− e1s : t ∈ J(z,−) , s ∈ I(z − e2t,−)}(2.9)

and hence that ˆJ(z,−)

ˆI(z−e2t,−)

f(z − e2t− e1s) ds dt =

ˆR0

f = 0.(2.10)

By (2.10) we find that

(2.4) = A2

ˆJ(z,−)

ˆI(z−e2t,−)

f(z − e2t− e1s)

(1

t′ · s′− 1

A`(J) ·A`(I)

)ds dt.(2.11)

Now, applying the estimates on the line (2.8), that t′ ∼ A`(J) and s′ ∼ A`(I) (as A islarge, this is implied by (2.8)), triangle inequality, and the mean value theorem (appliedto x 7→ x−1 in the second passing), shows that∣∣∣ 1

t′ · s′− 1

A`(J) ·A`(I)

∣∣∣ ≤ 1

t′

∣∣∣ 1

s′− 1

A`(I)

∣∣∣+1

A`(I)

∣∣∣ 1t′− 1

A`(J)

∣∣∣.

(A`(I))−2`(I)

A`(J)+

(A`(J))−2`(J)

A`(I).

1

A3`(I)`(J).

Plugging in the above estimates we continue from (2.11) and find that

|(2.11)| . ‖f‖∞ˆJ(z,−)

ˆI(z−e2t,−)

1

A`(I)`(J)ds dt ≤ A−1‖f‖∞

and hence we have established (2.3).Next, we repeat the above argument beginning from the function fR2 . We denote e3 =

−e1 and e0 = −e2 and write

fR2 =[hR2H

∗e3gR3 − gR3He3hR2

]+[hR3H

∗e0gR0 − gR0He0hR3

]+ fR0 ,(2.12)

where

hR2 =fR2

H∗e3gR3

, gRi = 1Ri , fR3 = gR3He3hR2 , hR3 =fR3

He0gR0

, fR0 = gR0He0hR3 .

Again, this decomposition is well-defined. By moving the adjoints we find thatˆfR2 =

ˆfR1 =

ˆf = 0,(2.13)

Page 7: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 7

e.g. the second identity follows asˆfR1 =

ˆgR1He1

( f

H∗e1gR1

)=

ˆH∗e1gR1

f

H∗e1gR1

=

ˆf.

Consequently, by similar arguments as above, we find that

|hRi | .A 1Ri , i = 2, 3, ‖fR0‖∞ . A−1‖fR2‖∞ . A−2‖f‖∞.

Then, we dualize as on the line (2.1) and factor according to the lines (2.2) and (2.12)to the extent that

ˆR0

|b− 〈b〉R0 | =ˆb

3∑i=1

[hRi−1H

∗eigRi − gRiHeihRi−1

]+

ˆb[hR3H

∗e0gR0 − gR0He0hR3

]+

ˆbfR0

= −ˆ 3∑

i=1

gRi [b,Hei ]hRi−1

−ˆgR0 [b,He0 ]hR3 +

ˆ(b− 〈b〉R0)fR0

≤3∑i=1

‖gRi‖Lp′∥∥[b,Hei ]

∥∥Lp→Lp‖hRi−1‖Lp

+ ‖gR0‖Lp′∥∥[b,He0 ]

∥∥Lp→Lp‖hR3‖Lp + ‖fR0‖∞

ˆR0

|b− 〈b〉R0 |

≤ CA( 3∑i=1

∥∥[b,Hei ]∥∥Lp→Lp +

∥∥[b,He0 ]∥∥Lp→Lp

)|R0|+ CA−1

ˆR0

|b− 〈b〉R0 |

≤ CA∑i=1,2

‖[b,Hei ]‖Lp(R2)→Lp(R2)|R0|+ CA−1

ˆR0

|b− 〈b〉R0 |,

where in the final estimate we note that Hσ = −H−σ, for any σ ∈ S, especially then,He3 = −He1 , He0 = −He2 so that ‖[b,He3 ]‖Lp→Lp = ‖[b,He1 ]‖Lp→Lp and ‖[b,He0 ]‖Lp→Lp =‖[b,He2 ]‖Lp→Lp . To conclude, using the assumption b ∈ L1

loc, we choose A large enoughand absorb the common term shared on both sides to the left-hand side, then divide with|R0|. �

3. PROOF OF THEOREM 1.3

Whereas Theorem 1.6 was in a sense proved on the go, now, due to the parabola, thesetup is more involved and we require a lengthier preparation. As the upper bound wasalready proved in [2], it remains to prove Theorem 1.2.

3.1. Geometry behind the factorization.

Page 8: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

8 TUOMAS OIKARI

3.1.1. Setup for analysis: Q, W and P . We fix a parabolic rectangle Q = I × J, i.e. `(J) =`(I)2. We work on a scale comparable to `(I) and hence define the auxiliary interval

IA = [`(I)A, `(I)(A+N)], A,N ≥ 1.

Then, we set

P = Q+ (2A+N)`(I)e1,

Q ={x+ γ(t) : x ∈ Q, t ∈ IA

}, P =

{z + γ(t) : z ∈ P, t ∈ −IA

},

W = Q ∩ P .

The following Figure 3.1 is a rough sketch of the sets Q, Q,W, P , P, when A ∼ 3, N ∼7, `(I) ∼ 2.

FIGURE 3.1: Setup for analysis

For our arguments to work we can take any fixedN ≥ 1, and we takeN = 1, however,considering a slightly larger N brings separation to the sets considered and streamlinesthe geometry.

The setup is symmetric with respect to a reflection across the line in Figure 3.1 thatsplits the set W vertically in half. Moreover, there holds that

|Q| ∼A |Q| ∼ |W | ∼ |P | ∼A |P |.(3.1)

The first estimate follows as Q contains a translate of Q and by considering the size ofthe set Q in the x1 and x2 directions. The second follows as W ⊂ Q and W contains atranslate of Q. That the second estimate is also independent of A is a fact that we do notneed, however, it is relatively clear from Lemma 3.10 below. The last two estimates aresymmetric with the first two.

We denote

lb, lt, rb, rt, l = left, r = right, t = top, b = bottom, c = centre(3.2)

and variables are reserved to be used as follows, x ∈ Q, y ∈W, z ∈ P. We also notate

I(a,±, B) = {t ∈ R : a± γ(t) ∈ B} ⊂ R, φ(a,±, B) = {a± γ(t) ∈ B : t ∈ R} ⊂ B.

The variable a ∈ R2 is the reference point, the sign ± ∈ {+,−} is either the plus or theminus sign and indicates direction, while the last variable is a set B ⊂ R2.

Page 9: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 9

3.3. Lemma. Let Q = I × J ∈ Rγ and x ∈ Q. Then, there holds that

limA→∞

|I(x,+,W )|`(I)

=1

2

with uniform convergence independent of the data x,Q.

Proof. Let x ∈ Q be arbitrary and let tx ∼ A`(I) be the smallest number such that x +γ(tx) ∈ ∂W. Let −sx ∼ A`(I) be the point such that x + γ(tx) = vlb + γ(sx), where vlb isthe left bottom vertex of P. Define the auxiliary point yx = x+ γ(tx) + `(I)e1. Then, thereholds that yx = vrb + γ(sx), where vrb is the right bottom vertex of P.

Next, we show that for each 0 < ε < 12 , there exists A large enough (independent of

`(I)) so that

π2(x+ γ(tx + (1

2− ε)`(I))) < π2(vrb + γ(sx −

`(I)

2)).(3.4)

To achieve (3.4), we choose A so large that

(1

2− ε)`(I) · 2(A+N)`(I) <

1

2`(I) · 2A`(I),(3.5)

clearly A as chosen on the line (3.5) is independent of `(I). Then, as

2A`(I) ≤ |γ′2(t)| ≤ 2(A+N)`(I), t ∈ −IA ∪ IAand π2(x+ γ(tx)) = π2(vrb + γ(sx)), (3.4) follows. By symmetry,

π2(x+ γ(tx +`(I)

2) > π2(vrb + γ(sx − (

1

2− ε)`(I))).(3.6)

Then, let 0 < −ux, vx < `(I) be the unique points such that x + γ(tx + vx) = vrb +γ(sx + ux). The line (3.4) shows that vx ≥ (1

2 − ε)`(I). Indeed, assume for contradictionthat vx < (1

2 − ε)`(I). Then by vx − ux = `(I) necessarily −ux > sx − `(I)2 and hence by

(3.4)

π2

(x+ γ(tx + vx)

)< π2

(vrb + γ(sx −

`(I)

2))< π2

(vrb + γ(sx + ux)

),

which contradicts x + γ(tx + vx) = vrb + γ(sx + ux). Similarly, from (3.6) it follows thatux ≤ −(1

2 − ε)`(I). Using vx − ux = `(I), it follows that

−ux, vx ∈ [(1

2− ε)`(I), (

1

2+ ε)`(I)].(3.7)

Notice that

x+ γ(tx + vx +`(I)

2A) 6∈ P .(3.8)

Indeed, (3.8) follows from the information

vrb + γ(sx + ux) = x+ γ(tx + vx), |γ′2(t)| > 2A`(I),

which implies that

x+ γ(tx + vx +`(I)

2A) 6∈ γ(sx + ux + h) + P, h ≥ 0,

along with the obvious fact that

x+ γ(tx + vx +`(I)

2A) 6∈ γ(sx + ux + h) + P, h < 0.

Page 10: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

10 TUOMAS OIKARI

From (3.7) and (3.8) we find that

(1

2− ε)`(I) ≤ |I(x,+,W )| ≤ (

1

2+ ε+

1

2A)`(I).(3.9)

Clearly (3.9) implies the claim. �

Lemma 3.3 immediately gives as a corollary the following Lemma 3.10.

3.10. Lemma. Let x ∈ Q be arbitrary. Then,

|I(x,+,W )| ∼ `(I).

Also, there holds that

limA→∞

supQ∈Rγx,x′∈Q

|I(x,+,W )||I(x′,+,W )|

= 1.

Towards the next lemma define the reference rectangles

R(r) = [0, 2−(r−1) `(I)

A]× [0, 2−r`(I)2], 1 ≤ r <∞.

Then, we set

P lb(r) = vlb +R(r), P rt(r) = vrt + Dil−1R(r), P lb(r), P rt(r) ⊂ P

and define

∆r(Plb) = ∂P lb(r) \ ∂P , ∆r(P

rt) = ∂P rt(r) \ ∂P ,

P c = P \⋃r>1

(∆r(P

lb) ∪∆r(Prt)).

(3.11)

Notice that P c = P \(P lb(1) ∪ P rt(1)

). If z ∈ P c (c for centre) and y ∈ φ(z,+,W ), then∣∣I(y,−, P )

∣∣ ∼ `(I)A (relatively clear, also, see Lemma 3.12 below). The following Lemma

3.12 shows that the sets ∆r(Plb),∆r(P

rt) exactly quantify this same statement for pointssituated towards the vertices vlb, vrt of P.

3.12. Lemma. Let z ∈ ∆r(Plb) ∪∆r(P

rt) and y ∈ φ(z,+,W ). Then,

|I(y,−, P )| ∼ 2−r`(I)

A.

Let z ∈ P c and y ∈ φ(z,+,W ), then |I(y,−, P )| ∼ `(I)A .

Proof. Let z ∈ ∆r(Plb) for some r > 1. Then, either

π1(z − vlb) = 2−(r−1) `(I)

Aor π2(z − vlb) = 2−r`(I)2

holds. Assume first that π1(z − vlb) = 2−(r−1) `(I)A . As y ∈ φ(z,+,W ), there exists s ∼

−A`(I) so that y − γ(s) = z ∈ P. The claim will follow if we show the following: thereexists an absolute constant c > 0 so that

πi(y − γ(s+ h)) ∈ πi(P ), h ∈ (0, c2−(r−1) `(I)

A), i = 1, 2.(3.13)

Page 11: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 11

The case i = 1 is an immediate consequence of the following information

π1(y − γ(s+ h)) = z1 − h, z1 − π1(vlb) = 2−(r−1) `(I)

Ah ∈ (0, c2−(r−1) `(I)

A),

as long as we choose c small enough. For the case i = 2, we note that (s, s+ h) ⊂ −2IA isan interval of length h and hence for some absolute constant c1 > 0 there holds that

`(π2(γ(s, s+ h))) ≤ c1hA`(I) ≤ c1c2−(r−1) `(I)

AA`(I) ≤ 1

2`(I)2,(3.14)

as long as we choose c small enough. The inequality (3.14) implies that π2(y−γ(s+h)) ≤π2(vlt).Also clearly π2(y−γ(s+h)) ≥ π2(vlb). Together these show that π2(y−γ(s+h)) ∈π2(P ) and so we have also checked the case i = 2 on the line (3.13).

The case π2(z − vlb) = 2−r`(I)2 and then z ∈ ∆r(Prt) are handled in very much the

same way and we leave the details to the reader. �

3.1.2. Auxiliary functions. Recall, that a fixed parabolic rectangle Q ∈ Rγ and the pa-rameters A,N determine the sets W (= W (Q)), P (= P (Q)). During the factorization wewill in total make use of four auxiliary functions, the first two are particularly simple,gQ = 1Q, gP = 1P . The other two functions are supported on the set W, more precisely,we find two collections of functions, the first one being {gW }Q∈Rγ , and we show that thefollowing three conditions are met.

(i) There holds that

1W cgW = 1W c , W c ={y ∈W : ∃z ∈ P c : y ∈ φ(z,+,W )

}.(3.15)

(ii) There holds that

gW (y) ∼ |I(y,−, P )| A`(I)

,(3.16)

where the implicit constants do not depend on y,Q.(iii) There holds that

limA→∞

supQ∈Rγz∈P

t,t′∈I(z,+,W )

gW (z + γ(t))

gW (z + γ(t′))= 1.(3.17)

The reader who feels comfortable with the existence of such a family {gW }Q∈Rγ mayimmediately skip to Section 3.2.

Next we explicitly define the functions gW . Let ηQ ≥ 0 be the smallest constant so thatthe following is a partition,

W =⋃

s∈[−ηQ,∞)

W (s), W (s) ={y ∈W : |I(y,−, P )| = 2−s

`(I)

A

}.

Lemma 3.12 implies that supQ∈Rγ ηQ < ∞, a fact worth noting, which, however, we donot need anywhere. Then, we define

ϕ : W × R+ → R+, ϕ(y,M) =∑

−ηQ≤s<M1W (s)(y) +

∑s≥M

1W (s)(y)2−(s−M).

Page 12: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

12 TUOMAS OIKARI

By the following Lemma 3.18 we choose M large enough and define

gW (·) = ϕ(·,M).

3.18. Lemma. There exists M ∈ R+ so that ϕ(·,M) satisfies the points (i), (ii) and (iii).

Proof. If y ∈ W c, then by Lemma 3.12, |I(y,−, P )| ∼ `(I)A and hence y ∈ W (sy) for

some sy ≥ −ηQ. Clearly supQ∈Rγ supy∈W c sy <∞. Consequently, for a choice of M largeenough, W c ⊂ ∪s∈[ηQ,M)W (s) and the point (i) follows from the definition of ϕ(·,M).

By definition

|I(y,−, P )| A`(I)

∼M ϕ(y,M),

hence ϕ(·,M) satisfies the point (ii) with any choice of M ≥ 1.Lastly, we check the point (iii). Fix z ∈ P . If φ(z,+,W ) ⊂ ∪s∈[ηQ,M)W (s), then ϕ(z +

γ(t),M) = 1 for all t ∈ I(z,+,W ) and the claim is clear. If φ(z,+,W ) 6⊂ ∪s∈[ηQ,M)W (s),

then ϕ(φ(z,+,W ),M) = ϕ(φ(z,+,

[∪s∈[M,∞) W (s)

]),M), i.e. the function ϕ(·,M) al-

ready attains all possible values on the set φ(z,+,

[∪s∈[M,∞) W (s)

]). Then, as

ϕ(z + γ(t),M) = |I(z + γ(t),−, P )| A`(I)

2M , t ∈ I(z,+,∪s∈[M,∞)W (s)

),

the claim follows from Lemma 3.19 below. �

3.19. Lemma. There holds that

limA→∞

supQ∈Rγz∈P

t,t′∈I(z,+,W )

|I(z + γ(t),−, P )||I(z + γ(t′),−, P )|

= 1.(3.20)

Proof. Fix z ∈ P, denote fz,t(s) = z + s(1,−t) and note that

2A`(I) ≤ |γ′2(t)| ≤ 2(A+N)`(I), t ∈ I(z,+,W ) ⊂ IA.Hence, for each t ∈ IA, there exists th ∈ [2A`(I), 2(A+N)`(I)] so that |I(z+γ(t),−, P )| =|f−1z,th

(P )|.First, assume that the lines fz,th , fz,t′h exit the rectangle P through the bottom and top

edges. Then, there holds that |f−1z,th

(P )|th = `(I)2, and hence

A

A+N≤ |I(z + γ(t),−, P )||I(z + γ(t′),−, P )|

=t′hth≤ A+N

A(3.21)

from which the claim follows, with this configuration of the data.Then, let t > t′ and assume that the lines fz,th , fz,t′h exit the rectangle P from the right

edge ∂rP (then, as A is large, they exit P through the top edge) and respectfully letet, et′ ∈ ∂rP be these points. By t > t′, it follows that th > t′h and π2(et′) > π2(et), andhence that

|f−1z,t′h

(P )| > |f−1z,th

(P )|, π2(vrt − et′) > π2(vrt − et).(3.22)

Using the estimates on the line (3.22) and |f−1z,sh

(P )|sh = π2(vrt − es), for s ∈ {t, t′}, wefind that

1 >|f−1z,th

(P )||f−1z,t′h

(P )|=π2(vrt − et)π2(vrt − et′)

t′hth>t′hth>

A

A+N.(3.23)

Page 13: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 13

From (3.23) we find that

A

A+N≤|f−1z,th

(P )||f−1z,t′h

(P )|≤ A+N

A

and the claim follows with this configuration of the data.Lastly, we consider the case t > t′ when fz,th and fz,t′h , respectively, exit through the

bottom edge and through the right edge. This case follows from the two cases above bywriting

|f−1z,th

(P )||f−1z,t′h

(P )|=|f−1z,th

(P )||f−1z,t′′h

(P )|·|f−1z,t′′h

(P )|

|f−1z,t′h

(P )|

for t′h < t′′h < th such that fz,t′′h passes through the vertex vrb. This last case along withthe first two were representative of all possible cases, and as all the estimates were inde-pendent of the rectangle Q, the proof is concluded. �

The other collection of functions we use is {uW }Q∈Rγ , where uW (y) = (gW ◦Ξ)(y) andΞ is the following reflection

Ξ(x) =(wd − (x1 − wd), x2

), wd = π1

({x = (x1, x2) ∈W : x2 = inf

y∈Wy2}).(3.24)

The reflection Ξ is exactly across the line depicted in Figure 3.1 and the function uW isthe symmetric version of gW with respect to this reflection.

3.2. Approximate weak factorization.

3.2.1. The first two iterations. In this section we prove the following Proposition 3.25.

3.25. Proposition. Let f ∈ L1loc be supported on a parabolic rectangle Q = I × J. Then, for all

A large enough (independently of Q), the function f can be written as

f =[hQH

∗γgW − gWHγhQ

]+[hWHγgP − gPH∗γhW

]+ fP ,(3.26)

where

hQ =f

H∗γgW, hW =

gWHγhQHγgP

, fP = gPH∗γ

( gWHγgP

( f

H∗γgW

)),(3.27)

and there holds that

|hQ| .A |f |, |hW | .A ‖f‖∞1W .(3.28)

Moreover, suppose that´Q f = 0 and let ε > 0. Then, for all A large enough (independently

of Q), there holds that

|fP | . ε‖f‖∞1P .(3.29)

The identities on the lines (3.26) and (3.27) are simply algebraic, as long as the functionshQ, hW are well-defined, which we will see below, and hence, it is enough to prove theestimates on the lines (3.28) and (3.29).

Page 14: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

14 TUOMAS OIKARI

Proof of the estimates (3.28). We begin with a better estimate than is actually needed for(3.28), which will be reused in the proof of the estimate (3.29). Let ε > ε′ > 0 andconsider the set

Ic(x,+,W ) = {t ∈ I(x,+,W ) : gW (x+ γ(t)) = 1}.

We choose A (independently of the data) so large that that

|Ic(x,+,W )| ≥ (1− ε′)|I(x,+,W )|.(3.30)

Let us give a short argument for (3.30). Let x ∈ Q be arbitrary and let tx be the smallestnumber with y := x + γ(tx) ∈ ∂W. Let c > 0 be a constant and define r = c `(I)A . Definethe point y′ := x + γ(tx + r) and let R denote the rectangle with opposite vertices y, y′.Then, by |γ′2(t)| ∼ A`(I), the rectangle R has dimensions

R = I × J, `(I) = c`(I)

A, `(J) ∼ c`(I)2.

Then, let sy be such that y − γ(sy) = vlb and notice that the points vlb and z′ := y′ − γ(sy)are the opposite vertices of the rectangle G = −γ(sy) +R ⊂ P and that φ(y′,−, P )∩G =vrt(G), where vrt(G) is the right-top vertex of G. Since the rectangle G has the samedimensions as R, it follows with a choice of the constant c large enough that φ(y′−, P ) ∩P c 6= ∅ which implies that gW (y′) = 1 by property (i) of the function gW . The sameargument also works if we begin with "let tx be the largest number with y := x+ γ(tx) ∈∂W." It follows that the two sections of the curve φ(x,+,W ) where gW 6= 1 both havelengths . `(I)

A . By Lemma 3.10 we have |I(x,+,W )| ∼ `(I) and hence we conclude that

|I(x,+,W ) \ Ic(x,+,W )| . `(I)

A,

which implies (3.30).Then, by I(x,+,W ) ⊂ IA and (3.30) we find

(1− ε′)|I(x,+,W )|(A+N)`(I)

≤ |Ic(x,+,W )|

(A+N)`(I)≤ H∗γgW (x)

=

ˆI(x,+,W )

gW (x+ γ(s))ds

s≤ |I(x,+,W )|

A`(I).

(3.31)

It follows from Lemma 3.10 that with a choice of A large enough for some absolute con-stants κ1, κ2 (independently of the data) there holds that

0 < κ1 ≤ supQ∈Rγx∈Q

|I(x,+,W )|`(I)

≤ κ2 <∞.(3.32)

Moreover, by Lemma 3.10

limA→∞

supQ∈Rγx,x′∈Q

|I(x,+,W )||I(x′,+,W )|

= 1.(3.33)

Page 15: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 15

Now choose arbitrary x′ ∈ Q and define κQ,A = |I(x′,+,W )|`(I) . Let x ∈ Q be arbitrary, then,

for a choice of A large enough combining both (3.32) and (3.33) it follows that

κQ,A(1− ε′) ≤ |I(x,+,W )|`(I)

≤ κQ,A(1 + ε′), 0 < κ1 ≤ κQ,A ≤ κ2 <∞.(3.34)

Then, choose A so large that (3.31) and (3.34) imply

(1− ε)(A+N)

κQ,A ≤ H∗γgW (x) ≤ (1 + ε)

AκQ,A, 0 < κ1 ≤ κQ,A ≤ κ2 <∞.(3.35)

By I(y,−, P ) ⊂ −2IA we find

|HγgP (y)| =ˆI(y,−,P )

ds

|s|∼ |I(y,−, P )|

A`(I).(3.36)

It follows from (3.35) and (3.36), respectively, that hQ, hW are both well-defined and

|hQ| ∼ A|f |, |hW (y)| ∼ A`(I)

|I(y,−, P )||I(y,−, P )| A

`(I)|HγhQ(y)| = A2|HγhQ(y)|,

for the above estimate of hW , recall (3.16). Estimating |HγhQ(y)| a little further we findthat

|HγhQ(y)| =∣∣∣ ˆ

I(y,−,Q)hQ(y − γ(s))

ds

s

∣∣∣ . ‖hQ‖∞ ˆI(y,−,Q)

ds

s

. A‖f‖∞|I(y,−, Q)|A`(I)

. A−1‖f‖∞,

where in the last estimate we used Lemma 3.12. Hence, we find that |hW | . A‖f‖∞1W .A‖f‖∞1W . �

Now we come to the crucial part of the argument. Let z ∈ P be arbitrary. Then, thereholds that (see Figure 3.2)

Q =⋃

y∈φ(z,+,W )

φ(y,−, Q).(3.37)

There also holds that⋃y∈φ(z,+,W )

φ(y,−, Q) ={z + γ(t)− γ(s) : s ∈ I(z + γ(t),−, Q), t ∈ I(z,+,W )

},(3.38)

easily checked from definitions. Then, we notate

Izf =

ˆI(z,+,W )

ˆI(z+γ(t),−,Q)

f(z + γ(t)− γ(s))ds

s

dt

t,(3.39)

and (3.37), (3.38) show the double integral in (3.39) to be over the rectangle Q. Next, werecognize the density ϑz : Q→ R+ satisfying

Izf =

ˆQfϑz, ∀f ∈ L1

loc.(3.40)

Consider the mapping

hz : F → Q, F ={

(t, s) : s ∈ I(z + γ(t),−, Q), t ∈ I(z,+,W )},

hz(t, s) = z + γ(t)− γ(s).

Page 16: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

16 TUOMAS OIKARI

FIGURE 3.2: Why (3.37) holds; a parabola {z + γ(t) : t ≤ 0} begins from apoint z ∈ P, exits P, travels and penetrates intoW ; then, another parabola{y − γ(t) : t ≥ 0} begins from a point y ∈ φ(z,+,W ), exits the set W,travels and penetrates into the rectangle Q.

For a change of variables, we need to check that hz is bijective and differentiable. Dif-ferentiability is obvious and by (3.37) and (3.38) we have surjectivity. For injectivity, itis enough to show the following: let y, y′ ∈ φ(z,+,W ) be distinct, then φ(y,−, Q) ∩φ(y′,−, Q) = ∅. Notice that φ(y,−, Q) and φ(y′,−, Q) are both contained in differenttranslates of φ(0,−,R− × R−), where R− = (−∞, 0] (and R+ = −R−). The other fact weuse is(a+ φ(0,−,R− × R−)

)∩(b+ φ(0,−,R− × R−)

)= ∅, a− b ∈ R− × R+ ∪ R+ × R−.

Now injectivity follows by noting that y − y′ ∈ R− × R+ ∪ R+ × R−.Denote ρ(t, s) = (ts)−1. Then, a change of variables tells us that

Izf =

ˆFf ◦ hz(t, s)

d(t, s)

ρ(t, s)=

ˆQf(x)

|det Jh−1z

(x)|ρ ◦ h−1

z (x)dx,

and hence it remains to evaluate the density. There holds that

det Jh−1z

(hz(t, s)) =1

det Jhz(t, s), det Jhz(t, s) = det

[1 −12t −2s

]= 2(t− s),

where for arbitrary x ∈ Q we denote

x = hz(tx, sx) = z + γ(tx)− γ(sx), −tx ∼ sx ∼ A`(I)(3.41)

for the unique choice of such tx, sx. Then, we find that

ϑz(x) =|det Jh−1

z(x)|

ρ ◦ h−1z (x)

=1

2|tx − sx|1

sxtx.(3.42)

Page 17: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 17

Proof of the estimate (3.29). We begin with gathering three two sided estimates: (3.43),(3.45) and (3.47). Let ε′ > 0. Let x ∈ Q. Repeating the contents of the line (3.35), aftera choice of A sufficiently large, we find that

(1− ε′)(A+N)

κQ,A ≤ H∗γgW (x) ≤ (1 + ε′)

(A− 1)κQ,A, 0 < κ1 ≤ κQ,A ≤ κ2 <∞.(3.43)

The property (iii) (the line (3.17)) shows that with a choice of A large enough and anarbitrary t′′ ∈ I(z,+,W ), after defining Cz,Q,A = gW (z + γ(t′′)), there holds that

limA→∞

supQ∈Rγz∈P

t∈I(z,+,W )

gW (z + γ(t))

Cz,Q,A= 1.(3.44)

By (3.44) and the property (ii) (the line (3.16)), for A sufficiently large, there holds that

(1− ε′)Cz,Q,A ≤ |gW (z + γ(t))| ≤ (1 + ε′)Cz,Q,A, Cz,Q,A ∼ |I(z + γ(tz),−, P )| A`(I)

,

(3.45)

where we fix some arbitrary choice of tz ∈ I(z,+,W ).There holds that

|I(z + γ(t),−, P )|(A+N)`(I)

≤∣∣HγgP (z + γ(t))

∣∣ =

ˆI(z+γ(t),−,P )

ds

s≤ |I(z + γ(t),−, P )|

(A− 1)`(I).(3.46)

By (3.20) we find that with a choice of A large enough (3.46) implies

(1− ε′)|I(z + γ(tz),−, P )|(A+N)`(I)

≤∣∣HγgP (z + γ(t))

∣∣ ≤ (1 + ε′)|I(z + γ(tz),−, P )|(A− 1)`(I)

,(3.47)

where both sides of the estimate now depend only on the fixed choice tz. Denote

Cz,A(t, s) :=( gWHγgP

)(z + γ(t))

( 1

H∗γgW

)(z + γ(t)− γ(s)).

Together (3.43), (3.45) and (3.47) show that

|Cz,A(t, s)| ≤ (1 + ε′)Cz,Q,A(A+N)`(I)

(1− ε′)|I(z + γ(tz),−, P )|A+N

(1− ε′)κQ,A

=1 + ε′

(1− ε′)2

Cz,Q,A`(I)

A|I(z + γ(tz),−, P )|A(A+N)2

κQ,A.

(3.48)

Similarly, we find that

|Cz,A(t, s)| ≥ (1− ε′)Cz,Q,A(A− 1)`(I)

(1 + ε′)|I(z + γ(tz),−, P )|A− 1

(1 + ε′)κQ,A

=1− ε′

(1 + ε′)2

Cz,Q,A`(I)

A|I(z + γ(tz),−, P )|A(A− 1)2

κQ,A.

(3.49)

Let ε > 0. As Cz,Q,A ∼ |I(z+ γ(tz),−, P )| A`(I) and 0 < κ1 ≤ κQ,A ≤ κ2 <∞, we find witha choice of A large enough from the estimates (3.48) and (3.49) that

(1− ε) ≤|Cz,A(t, s)|

A3C1z,Q,A ≤ (1 + ε),(3.50)

Page 18: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

18 TUOMAS OIKARI

whereC1z,Q,A is some constant uniformly bounded from above and below (independently

of the data z,Q,A). Denote Cz = Cz,A(tz, sz) for some fixed (tz, sz) ∈ F so that especially(3.50) is valid with Cz in place of Cz,A(t, s). Consequently,

sup(t,s)∈F

|Cz,A(t, s)− Cz|A3

. ε.(3.51)

Then, we write out the error term to the extent that

fP (z) = H∗γ

( gWHγgP

( f

H∗γgW

))(z)

=

ˆI(z,+,W )

( gWHγgP

)(z + γ(t))Hγ

( f

H∗γgW

)(z + γ(t))

dt

t

=

ˆI(z,+,W )

ˆI(z+γ(t),−,Q)

Cz,A(t, s)f(z + γ(t)− γ(s))ds

s

dt

t= CzIzf + I∆,zf,

where Izf was defined on the line (3.39) and

I∆,zf =

ˆI(z,+,W )

ˆI(z+γ(t),−,Q)

(Cz,A(t, s)− Cz)f(z + γ(t)− γ(s))ds

s

dt

t.(3.52)

We apply the estimates

|I(z,+,W )| . `(I), |I(z + γ(t),−, Q)| . `(I)

A(3.53)

and (3.51) to find

|I∆,zf | ≤ ‖f‖∞ˆI(z,+,W )

ˆI(z+γ(t),−,Q)

|Cz,A(t, s)− Cz|ds

|s|dt

|t|

. ‖f‖∞ sup(t,s)∈F

|Cz,A(t, s)− Cz|A3

. ε‖f‖∞.

This estimate is of the desired form. Then, we analyse the term Izf. By the lines (3.40),(3.42) and using the zero-mean of the function f, we write

Izf =

ˆI(z,+,W )

ˆI(z+γ(t),−,Q)

f(hz(t, s)) (1− ψz(hz(s, t)))ds

s

dt

t,

where

ψz(x) =ϑz(cQ)

ϑz(x)=

|tx − sx|sxtx|tcQ − scQ |scQtcQ

.

As

|sa −A`(I)|, | − ta −A`(I)| . `(I), a ∈ {x, cQ}we find that ψz(x) → 1, as A → ∞, and independently of the data x,Q. Then, choosingA so large that |1− ψz| ≤ ε and again using the estimates on the line (3.53), we find that

Cz|Izf | . A3

ˆI(z,+,W )

ˆI(z+γ(t),−,Q)

|f(hz(t, s))||1− ψz(hz(s, t))|ds

|s|dt

|t|

. A3‖f‖∞`(I)`(I)

A

ε

A2`(I)2= ε‖f‖∞.

Page 19: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 19

3.2.2. The last two iterations. Next, we repeat the contents of the previous Section 3.2.1,but this time beginning from the rectangle P instead of Q. For the above arguments topass through a second time we need to respect the symmetry present in the first iterationof the argument. The only nonsymmetric object with respect to the reflection Ξ in thestatement of Proposition 3.25 is gW . Hence, this time, we simply use the function uW =gW ◦ Ξ in place of gW .

3.54. Proposition. Let f ∈ L1loc be supported on a parabolic rectangle P. Then, for all A large

enough (independently of P ), the function f can be written as

f =[oPH

∗γuW − uWHγoP

]+[oWHγgQ − gQH∗γoW

]+ fQ,(3.55)

where

oP =f

H∗γuW, oW =

uWHγoPHγgQ

, fQ = gQH∗γ

( uWHγgQ

( f

H∗γuW

)),(3.56)

and the following estimates hold

|oP | .A |f |, |oW | .A ‖f‖∞1W .(3.57)

Moreover, suppose that´P f = 0 and let ε > 0. Then, for all A large enough (independently

of P ), there holds that

|fQ| . ε‖f‖∞1Q.(3.58)

3.3. Closing the argument.

Proof of Theorem 1.2. WriteˆQ|b− 〈b〉Q| =

ˆbf, f = (σ − 〈σ〉Q)1Q, σ = sgn(b− 〈b〉Q), sgn(ϕ) =

ϕ

|ϕ|1ϕ 6=0.

According to the line (3.26) of Proposition 3.25 we factorize the function f asˆbf =

ˆb[hQH

∗γgW − gWHγhQ

]+

ˆb[hWHγgP − gPH∗γhW

]+

ˆbfP .

Then, by |f | ≤ 2, and the estimates on the line (3.28), the first term above with bracketsis controlled as

∣∣∣ ˆ b[hQH

∗γgW − gWHγhQ

] ∣∣∣ =∣∣∣ ˆ gW [b,Hγ ]hQ

∣∣∣ ≤ ‖[b,Hγ ]‖Lp→Lp‖gW ‖Lp′‖hQ‖Lp

.A ‖[b,Hγ ]‖Lp→Lp |W |1p′ |Q|

1p ∼ ‖[b,Hγ ]‖Lp→Lp |Q|,

(3.59)

where in the last estimate we used the estimates (3.1). The second term with brackets issimilarly estimated to the same upper bound. Proceeding, accordingly to the line (3.55)of Proposition 3.54, we factorize the function fP asˆ

bfP =

ˆb[oPH

∗γuW − uWHγoP

]+

ˆb[oWHγgQ − gQH∗γoW

]+

ˆb(fP )Q.

Page 20: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

20 TUOMAS OIKARI

There holds that ˆQ

(fP )Q =

ˆPfP =

ˆQf = 0,(3.60)

all of which are easy to check by moving the adjoints, see e.g. the similar argument for

(2.13). Then, by (3.29) and (3.58) there holds that |(fP )Q| . ε2 . 1. Hence, the estimateson the lines (3.57) and (3.58) allow us to conclude, similarly as the estimate (3.59), that∣∣∣ ˆ b

[oPH

∗γuW − uWHγoP

]∣∣∣+∣∣∣ˆ b

[oWHγgQ − gQH∗γoW

]∣∣∣ .A ‖[b,Hγ ]‖Lp→Lp |Q|.

By (3.60) we find∣∣∣ ˆ b(fP )Q

∣∣∣ =∣∣∣ ˆ

Q

(b− 〈b〉Q

)(fP )Q

∣∣∣ ≤ ‖(fP )Q‖∞ˆQ|b− 〈b〉Q| . ε2

ˆQ|b− 〈b〉Q|.

Putting all of the above together, we conclude that for some absolute constants CA, C(independent of Q) there holds thatˆ

Q|b− 〈b〉Q| ≤ CA‖[b,Hγ ]‖Lp→Lp |Q|+ Cε2

ˆQ|b− 〈b〉Q|.(3.61)

As b ∈ L1loc, the common term shared on both sides of the estimate (3.61) is finite. Hence,

as ε can be made arbitrarily small, by choosing A sufficiently large, by absorbing thecommon term to the left-hand side we find from (3.61) thatˆ

Q|b− 〈b〉Q| .A ‖[b,Hγ ]‖Lp→Lp |Q|.

A division by |Q| closes the argument. �

4. EXTENSIONS, OPEN PROBLEMS

In this section we present some extensions of Theorem 1.3 and Theorem 1.6 along withsome open problems. We are more interested in the parabolic case and hence only recordthe following Theorem 4.1 as an extension to Theorem 1.6.

4.1. Theorem. Let b ∈ L1loc(Rd;C) and p ∈ (1,∞). Then,

‖b‖bmo(Rd) ∼d∑i=1

‖[b,Hei ]‖Lp(Rd)→Lp(Rd).

It is clear how to adapt the proof of Theorem 1.6 to prove Theorem 4.1. Then, we moveto discuss the immediately available extensions to Theorem 1.3.

4.1. Monomial curves. A function γ : R → Rd is said to be a monomial curve if it is ofthe form

γ(t) =

{(ε1|t|β1 , . . . , εn|t|βd), t > 0

(δ1|t|β1 , . . . , δn|t|βd), t ≤ 0,

where βi > 0, εi, δi ∈ {−1, 1}, and there exists at least one index j so that εj 6= δj . Letβ, ε, δ denote these parameter tuples. Associated to a monomial curve γ, and hence tothe parameter tuple β = (β1, . . . , βd), is the related bmo space; let Rβ = Rγ denote the

Page 21: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 21

collection of all rectangles Q = I1 × · · · × Id parallel to the coordinate axes such that

`(I1)1β1 = `(I2)

1β2 = · · · = `(Id)

1βd , and define the space BMOβ(Rd) by the norm

‖b‖BMOβ(Rd) = supQ∈Rβ

Q|b− 〈b〉Q|.

Notice that ‖b‖BMOβ(Rd) depends only on the vector β and not on ε, δ ∈ {−1, 1}d. Withthis notation and γ(t) = (t, t2) we have BMOγ = BMO(1,2) .

For Theorem 1.3 the extension is the following.

4.2. Theorem. Let b ∈ L1loc(R2;C), let p ∈ (1,∞) and let γ : R → R2 be a monomial curve

with the associated parameter tuples β = (β1, β2), ε = (ε1, ε2) and δ = (δ1, δ2). Let εi = δi forexactly one index i ∈ {1, 2}. Then,

‖b‖BMOβ(R2) ∼ ‖[b,Hγ ]‖Lp(R2)→Lp(R2).

Proof. The upper bound was proved in [2] and holds for any d ∈ N and any monomialcurve γ : R→ Rd. Here we only need that εi 6= δi for at least a single index i ∈ {1, 2}.

The lower bound follows by similar arguments as Theorem 1.2 did. We need to makesure that a suitable geometry exists that allows for the approximate weak factorization(e.g. propositions 3.25 and 3.54) which in turn allows the abstract awf argument (e.g. theproof of Theorem 1.2 in Section 3.3) to pass through.

In the parabolic case the geometry was formed by the three sets Q,W (Q), P (Q) forQ ∈ Rγ , see Figure 3.1. Next, we describe the correct geometry for any monomial curveγ : R→ R2, with exactly one index εi 6= δi, and a rectangle Q = I1 × I2 ∈ Rβ . Set

J = {i ∈ {1, 2} : εi 6= δi}, J = {1, 2} \ J ,

`(Q) = `(I1)1β1 = `(I2)

1β2 , IA = [A`(Q), (A+N)`(Q)]

and

P (Q) = P1 × P2, Pi =

{((2A+N)`(Q)

)βiei + Ii, i ∈ J ,Ii, i ∈ J

and

Q = {x+ γ(t) : x ∈ Q, t ∈ IA}, P (Q) = {z + γ(t) : z ∈ P (Q), t ∈ −IA},

W (Q) = Q ∩ P (Q).

With the above setup it is easy to verify the following key points of the argument

|Q| ∼ |W (Q)| ∼ |P (Q)|, Q =⋃

y∈φ(z,+,W (Q))

φ(y,−, Q).

The geometries of the two cases are similar, with the case ε1 = −δ1 and ε2 = δ2 beingalmost identical to the case of the parabola.

When d ≥ 3 the argument for the lower bound in Theorem 4.2 does not pass throughas such and a description of an essentially different geometry is needed. A further goalwill be to relax the assumptions concerning the sign tuples ε, δ. We leave this to a futurework.

Page 22: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

22 TUOMAS OIKARI

4.3. Problem. Extend the lower bound in Theorem 4.2 to the largest possible class of mono-mial curves and to the case d ≥ 3.

4.2. Off-diagonal. Recall the following characterization of the homogeneous Hölder space

‖b‖Cα,0(Rd) = supx 6=y∈Rd

|b(x)− b(y)||x− y|α

∼ supQ`(Q)−α

Q|b− 〈b〉Q|,(4.4)

where the supremum is taken over all cubes Q ⊂ Rd and α > 0. The reader unfamiliarwith (4.4) can essentially read the proof from the proof of Proposition 4.8 below. Consid-ering the right-hand side of (4.4) we set the following definition.

4.5. Definition. Let b ∈ L1loc(Rd;C), let α > 0 and let β = (β1, . . . , βd) be a tuple of

exponents (possibly associated to a monomial curve). Then, we define the norm

‖b‖Cα,0β (Rd)

= supQ∈Rβ

|Q|−α Q|b− 〈b〉Q|.(4.6)

Notice the difference in normalizations before the integrals on the lines (4.4) and (4.6).Next, we will connect the norm of Definition 4.5 with a pointwise definition.

4.7. Definition. Let γ : R→ R2 and for each x ∈ R2 let us denote

Xγ(x) = φγ(x,+,R2) ∪ φγ(x,−,R2), φγ(a,±, B) = {a± γ(t) ∈ B : t ∈ R}.Then, we define the norm

‖b‖Cα,0γ (R2)

= supx∈R2

supy∈Xγ(x)

|b(x)− b(y)|∏2i=1 |xi − yi|α

.

4.8. Proposition. Let γ : R → R2 be a monomial curve with the associated parameter tupleβ = (β1, β2) and εi 6= δi for exactly one index i ∈ {1, 2}. Let b ∈ L1

loc(Rd;C). Then, there holdsthat

‖b‖Cα,0γ (R2)

∼ ‖b‖Cα,0β (R2)

.

Proof. Fix a rectangle Q = I × J ∈ Rβ and for distinct x, y ∈ Q pick a point z(x, y) ∈Xγ(x) ∩ Xγ(y) ∩ 3Q. That such a point exists follows from εi 6= δi for exactly one indexi ∈ {1, 2}. Then, we estimate

Q|b− 〈b〉Q| .

Q

Q|b(x)− b(z(x, y))| dx dy

. ‖b‖Cα,0γ (R2)

Q

Q

2∏i=1

|xi − z(x, y)i|α dx dy

. ‖b‖Cα,0γ (R2)

`(I)α`(J)α = ‖b‖Cα,0γ (R2)

|Q|α.

Let x ∈ R2 and y ∈ Xγ(x). Let Q = I × J ∈ Rβ be the rectangle with the points x, y asvertices. We let Qk(x) = Ik(x) × Jk(x) ∈ Rβ be the rectangle such that x ∈ Qk(x) ⊂ Q

and `(Ik(x)) = 2−kI. Similarly for the point y. Then, we write

b(x)− b(y) =

∞∑k=0

(〈b〉Qk+1(x) − 〈b〉Qk(x)

)−∞∑k=0

(〈b〉Qk+1(y) − 〈b〉Qk(y)

),

Page 23: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 23

where we use the Lebesgue differentiation theorem with Rβ rectangles. A comment onthis. For the standard argument to pass through, we need to know that for each pointx ∈ R2 there exists a sequence Qk(x) → x with diam(Qk(x)) → 0 (obviously true) andthat the related maximal function MRβf(x) = supx∈Q∈Rβ

fflQ |f | is of weak type (1, 1).

To see the weak type (1, 1), dominate MRβ ≤∑N

i=1MDiβby a finite number of dyadic

operators over some anisotropic dyadic grids Diβ. Such grids are constructed at least in[3], and these grids have all the important properties that we would expect of a dyadicgrid, hence, a standard argument shows that MDiβ is of weak type (1, 1). With this detailin the clear, we estimate the second of the martingale differences as

∞∑k=0

∣∣〈b〉Qk+1(y) − 〈b〉Qk(y)

∣∣ ≤ ∞∑k=0

Qk+1(y)

|b− 〈b〉Qk(y)| .∞∑k=0

‖b‖Cα,0β (R2)

|Qk(y)|α

. ‖b‖Cα,0β (R2)

|Q|α = ‖b‖Cα,0β (R2)

2∏i=1

|xi − yi|α.

The first martingale difference estimates identically and we conclude. �

4.9. Theorem. Let b ∈ L1loc, let 1 < p < q < ∞ and define α = 1

p −1q . Let γ : R → R2 be

a monomial curve with the associated parameter tuples β, ε, δ and εi 6= δi for exactly one indexi ∈ {1, 2}. Then, there holds that

‖b‖Cα,0β (R2)

. ‖[b,Hγ ]‖Lp(R2)→Lq(R2).

Proof. Follows by the same proof as Theorem 1.2 in Section 3.3 with the only differencebeing the replacement of ‖[b,Hγ ]‖Lp→Lp with ‖[b,Hγ ]‖Lp→Lq which gives the correct nor-malization. �

4.10. Remark. If Problem 4.3 has a positive solution by the awf argument, then this auto-matically leads to the generalization of Theorem 4.9 to d ≥ 3 and any monomial curve astherein.

One is tempted to attempt to prove the upper bound ‖[b,Hγ ]‖Lp(R2)→Lq(R2) . ‖b‖Cα,0β (R2)

as follows. By Proposition 4.8 and

|[b,Hγ ]f(x)| =∣∣∣p.v.ˆ (b(x)− b(x− γ(t)))f(x− γ(t))

dt

t

∣∣∣≤ ‖b‖

Cα,0γ (R2)

ˆ|f(x− γ(t))|

2∏i=1

|γi(t)|αdt

|t|,

it would be enough to bound the following fractional integral

Iαγ f(x) =

ˆf(x− γ(t))

2∏i=1

|γi(t)|αdt

|t|=

ˆf(x− γ(t))

dt

|t|1−α|β|,

where |β| =∑2

i=1 |βi|. It is not clear whether this operator is bounded Lp → Lq or not.In the other direction, a standard scaling argument shows that if Iαγ is bounded Lp → Lq

then necessarily α = 1p −

1q .

Page 24: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

24 TUOMAS OIKARI

For completeness, we give this scaling argument in Rd. For ~λ ∈ Rd and x ∈ Rd denote~λx = (λ1x1, . . . , λdxd) and Dil~λ f(x) = f(~λx). For β ∈ Rd and λ ∈ R denote λβ =

(λβ1 , . . . , λβd). Dilation structure for Iαγ is contained in the following identity,

Dilλβ Iαγ f(x) =

ˆf(λβx− γ(t))|t|α|β| dt

t

=

ˆDilλβ f(x− γ(t))λα|β||t|α|β| dt

t= λα|β|Iαγ Dilλβ f(x).

Then, assuming ‖Iαγ ‖Lp(Rd)→Lq(Rd) <∞, and by ‖Dil~λ f‖Ls(Rd) =∏di=1 λ

− 1s

i ‖f‖Ls(Rd), wefind that

λ− |β|

q ‖Iαγ f‖Lq(Rd) = ‖Dilλβ Iαγ f‖Lq(Rd) = λα|β|‖Iαγ Dilλβ f‖Lq(Rd)

. λα|β|‖Dilλβ f‖Lp(Rd) = λα|β|λ− |β|

p ‖f‖Lp(Rd).

Choosing f so that both sides of the above estimate are positive and varying λ showsthat necessarily α = 1

p −1q .

With the above discussion at hand we are led to two problems.

4.11. Problem. Let 1 < p < q < ∞, α = 1p −

1q and let γ be a monomial curve. Does there

hold that ‖[b,Hγ ]‖Lp→Lq . ‖b‖Cα,0β?

4.12. Problem. Let 1 < p < q < ∞ and α = 1p −

1q . For what curves γ, is Iαγ bounded

Lp → Lq?

The second off-diagonal case is when q < p.

4.13. Theorem. Let b ∈ L1loc(R2;C) and let γ : R→ R2 be a monomial curve such that εi = δi

for exactly one index i ∈ {1, 2}. Let 1 < q < p < ∞ and define the exponent r > 1 by therelation 1

q = 1r + 1

p . Then, there holds that

‖b‖Lr(R2) ∼ ‖[b,Hγ ]‖Lp(R2)→Lq(R2).

Proof. It was shown in Stein, Wainger [17] thatHγ : Lp → Lp, 1 < p <∞ is a bounded op-erator. By the boundedness of Hγ , that the commutator is unchanged modulo constantsand Hölder’s inequality we find that

‖[b,Hγ ]‖Lp→Lq = ‖[b− c,Hγ ]‖Lp→Lq ≤ ‖(b− c)Hγ‖Lp→Lq + ‖Hγ(b− c)‖Lp→Lq≤ ‖b− c‖Lr‖Hγ‖Lp→Lp + ‖Hγ‖Lq→Lq‖f 7→ (b− c)f‖Lq→Lq≤ ‖b− c‖Lr‖Hγ‖Lp→Lp + ‖Hγ‖Lq→Lq‖b− c‖Lr. ‖b− c‖Lr ,

for any constant c ∈ C. This upper bound is valid for any d and any monomial curve.For the lower bound we only discuss the case of the parabola and it is clear how to

adapt the proof to any monomial curve γ : R → R2 with the sign tuples agreeing inexactly one entry. With the factorization results at hand, the lower bound follows withthe same argument as the case q < p in [8]. The important points to consider whenadapting the proof are the following:

(i) If K ⊂ Rd is compact, then K ⊂ Q for some Q ∈ Rγ .

Page 25: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR 25

(ii) Any parabolic rectangle Q ∈ Rγ allows parabolic stopping time arguments withconstants independent of Q.

(iii) Each Q ∈ Rγ allows a factorization of functions, as in propositions 3.25 and 3.54.(iv) There exists λ > 0 so that for each Q ∈ Rγ , there holds that

|S| ∼ |Q|, Q ⊂ λS, S ∈ {W (Q), P (Q)},

where the implicit constants do not depend on Q and the dilation λS of any con-nected bounded set S is

λS ={y : yi = cπi(S) + λ(xi − cπi(S)), x ∈ S, i = 1, . . . , d

}.

Since S is connected and bounded πi(S) is a finite interval and the meaning of thecentre point cπi(S) is clear.

We leave the details to the interested reader. �

4.14. Remark. Theorem 4.13 extends to the case d ≥ 3 as long as we can carry out the awfargument in that setting, similarly as with Remark 4.10.

The author has no competing interests to declare.

REFERENCES

[1] E. Airta, T. Hytönen, K. Li, H. Martikainen, and T. Oikari, Off-Diagonal Estimates for Bi-Commutators,International Mathematics Research Notices (2021). ↑2, 3

[2] T. Bongers, Z. Guo, J. Li, and B. Wick, Commutators of hilbert transforms along monomial curves, StudiaMathematica (2021). ↑1, 7, 21

[3] L. Cladek and Y. Ou, Sparse domination of hilbert transforms along curves, Mathematical Research Letters25 (201704). ↑23

[4] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and hardy spaces, J. Math.Pures Appl. 9 (1993), 247–286. ↑3

[5] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann.of Math. (2) 103 (1976), no. 3, 611–635. ↑2

[6] W. Guo, J. Lian, and H. Wu, The unified theory for the necessity of bounded commutators and applications,The Journal of Geometric Analysis 30 (201709). ↑2

[7] I. Holmes, S. Petermichl, and B. D. Wick, Weighted little bmo and two-weight inequalities for Journé commu-tators, Anal. PDE 11 (2018), no. 7, 1693–1740. ↑3, 4

[8] T. P. Hytönen, The Lp-to-Lq boundedness of commutators with applications to the Jacobian operatorr, J. Math.Pures Appl. (2021). ↑2, 3, 24

[9] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), 263–270. ↑2[10] A. K. Lerner, S. Ombrosi, and I. P. Rivera-Ríos, Commutators of singular integrals revisited, Bull. Lond.

Math. Soc. 51 (2019), no. 1, 107–119. ↑2, 3[11] K. Li, H. Martikainen, and E. Vuorinen, Bloom type upper bounds in the product bmo setting, The Journal

of Geometric Analysis 30 (2018), 3181–3203. ↑3[12] , Bloom-Type Inequality for Bi-Parameter Singular Integrals: Efficient Proof and Iterated Commutators,

International Mathematics Research Notices 2021 (201904), no. 11, 8153–8187. ↑3[13] S. Lindberg, On the hardy space theory of compensated compactness quantities, Archive for Rational Mechan-

ics and Analysis 224 (2017), 709–742. ↑3[14] Z. Nehari, On bounded bilinear forms, Annals of Mathematics 65 (1957), 153–162. ↑2[15] T. Oikari, Off-diagonal estimates for commutators of bi-parameter singular integrals, prerint arXiv (2020),

available at https://arxiv.org/abs/2010.01538. ↑2[16] , Off-diagonal estimates for bilinear commutators, prerint arXiv (2021), available at https://

arxiv.org/abs/2102.13535. ↑2[17] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bulletin of the American

Mathematical Society 84 (1978), no. 6, 1239 –1295. ↑24

Page 26: LOWER BOUND OF THE PARABOLIC HILBERT COMMUTATOR

26 TUOMAS OIKARI

[18] A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Mathematical Journal 30 (1978),no. 1, 163 –171. ↑2

(T.O.) DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF HELSINKI, P.O.B. 68, FI-00014UNIVERSITY OF HELSINKI, FINLAND

Email address: [email protected]