Bloom type upper bounds in the product BMO setting
aa r X i v : . [ m a t h . C A ] A p r BLOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING
KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINENA
BSTRACT . We prove some Bloom type estimates in the product BMO setting. Morespecifically, for a bounded singular integral T n in R n and a bounded singular integral T m in R m we prove that k [ T n , [ b, T m ]] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) , where p ∈ (1 , ∞ ) , µ, λ ∈ A p and ν := µ /p λ − /p is the Bloom weight. Here T n is T n acting on the first variable, T m is T m acting on the second variable, A p stands for thebi-parameter weights of R n × R m and BMO prod ( ν ) is a weighted product BMO space.
1. I
NTRODUCTION
Let µ and λ be two general Radon measures in R n . A two-weight problem asks fora characterisation of the boundedness T : L p ( µ ) → L p ( λ ) , where T is, for instance, asingular integral operator (SIO). Singular integral operators take the form T f ( x ) = ˆ R n K ( x, y ) f ( y ) d y, where different assumptions on the kernel K lead to important classes of linear transfor-mations arising across pure and applied analysis. There exists a two-weight characteri-sation for the Hilbert transform T = H , where K ( x, y ) = 1 / ( x − y ) , by Lacey [25] andLacey, Sawyer, Uriarte-Tuero and Shen [26] (see also Hytönen [20]).In a Bloom type variant of the two-weight question we require that µ and λ are Muck-enhoupt A p weights and that the problem involves a function b that can be taken to bein some appropriate weighted BMO space
BMO( ν ) formed using the Bloom weight ν := µ /p λ − /p ∈ A . The presence of the function b lands us naturally to the commutator set-ting. Coifman–Rochberg–Weiss [3] showed that the commutator [ b, T ] f := bT f − T ( bf ) satisfies k b k BMO . k [ b, T ] k L p → L p . k b k BMO , p ∈ (1 , ∞ ) , for a wide class of singular integrals T . These estimates are called the commutator lowerbound and the commutator upper bound, respectively. Both estimates are non-trivialand proved quite differently. The upper bound should hold for all bounded SIOs, whilethe lower bound requires some suitable non-degenaracy. Given some operator A b , thedefinition of which depends naturally on some function b , the Bloom type questionsconcerns the validity of the estimate k A b k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k BMO( ν ) . Mathematics Subject Classification.
Key words and phrases.
Iterated commutators, Bloom’s inequality, product BMO, weighted BMO.
When studied in the natural commutator setting we may also, for suitable singular inte-grals T , hope to prove the appropriate lower bound, such as k b k BMO( ν ) . [ µ ] Ap , [ λ ] Ap k [ b, T ] k L p ( µ ) → L p ( λ ) . In the Hilbert transform case T = H Bloom [2] proved such a two-sided estimate.A renewed interest on the Bloom type estimates started from the recent works ofHolmes–Lacey–Wick [14, 15]. They proved Bloom’s upper bound for general boundedSIOs in R n using modern proof techniques, and considered the lower bound in the Rieszcase. Lerner–Ombrosi–Rivera-Ríos [30] further refined these results using sparse dom-ination. An iterated commutator of the form [ b, [ b, T ]] is studied by Holmes–Wick [17],when b ∈ BMO ∩ BMO( ν ) . It was also possible to prove this iterated case from the firstorder case via the so-called Cauchy integral trick of Coifman–Rochberg–Weiss [3], seeHytönen [21]. However, this works precisely because it is also assumed that b ∈ BMO . Itturns out that
BMO ∩ BMO( ν ) is not optimal. A fundamentally improved iterated caseis by Lerner–Ombrosi–Rivera-Ríos [31]. There b ∈ BMO( ν / ) ) BMO ∩ BMO( ν ) , andthis is a characterisation as lower bounds also hold by [31]. In the recent paper [22] byHytönen lower bounds with weak non-degeneracy assumptions on T (or its kernel K )are shown. On the other hand, multilinear Bloom type inequalities were initiated byKunwar–Ou [24].We are interested in Bloom type inequalities in the bi-parameter setting. Classical one-parameter kernels are “singular” (involve “division by zero”) exactly when x = y .In contrast, the multi-parameter theory is concerned with kernels whose singularity isspread over the union of all hyperplanes of the form x i = y i , where x, y ∈ R d are writtenas x = ( x i ) ti =1 ∈ R d × · · · × R d t for a fixed partition d = d + . . . + d t . The bi-parametercase d = d + d = n + m is already representative of many of the challenges arising in thiscontext. The prototype example is / [( x − y )( x − y )] , the product of Hilbert kernels inboth coordinate directions of R , but general two-parameter kernels are neither assumedto be of the product nor of the convolution form. A general bi-parameter SIO is a linearoperator with various kernel representations depending whether we have separationin both R n and R m (a full kernel representation), or just in R n or R m (a partial kernelrepresentation). See the bi-parameter representation theorem [34] by one of us. This isthe modern viewpoint but deals with the same class of SIOs as originally introduced byJourné [23] (see Grau de la Herrán [13]).In general, there is a vast difference between the techniques that are required in thebi-parameter (or more generally multi-parameter) setting and in the one-parameter the-ory. For example, there is only one sparse domination paper in this setting, namelyBarron–Pipher [1], but the sparse domination is quite a bit more restricted than in theone-parameter setting.Bloom theory in the bi-parameter setting is very recent. Bloom type upper bounds forcommutators of the form [ b, T ] are valid for bi-parameter Calderón–Zygmund operators T and they are established using the bi-parameter dyadic representation theorem [34]and carefully estimating the resulting commutators of various dyadic model operators(DMOs). Here we define a bi-parameter Calderón–Zygmund operator (CZO) to be a bi-parameter SIO such that T , T ∗ , T (1) , T ∗ (1) ∈ BMO prod , and certain additional weaktesting conditions hold. Here T is a partial adjoint of T in the first slot, h T ( f ⊗ f ) , g ⊗ g i = h T ( g ⊗ f ) , f ⊗ g i , and BMO prod is the product BMO of Chang and Fefferman
LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 3 [6, 7]. Holmes–Petermichl–Wick [16] proved the first bi-parameter Bloom type estimate k [ b, T ] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k bmo( ν ) . Here A p stands for bi-parameter weights (replace cubes by rectangles in the usual defi-nition) and bmo( ν ) is the weighted little BMO space defined using the norm k b k bmo( ν ) := sup R ν ( R ) ˆ R | b − h b i R | , where h b i R = 1 | R | ˆ R b, ν ( R ) = ˆ R ν, and the supremum is over all rectangles R = I × J ⊂ R n × R m . We refer to these typesof commutators as little BMO commutators.In the recent paper [33] we reproved the result of [16] in an efficient way based onimproved commutator decompositions from our bilinear bi-parameter theory [32]. Theclear structure of our proof allowed us to also handle the iterated little BMO commutatorcase and to prove the upper bound(1.1) k [ b k , · · · [ b , [ b , T ]] · · · ] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k Y i =1 k b i k bmo( ν θi ) , k X i =1 θ i = 1 . There is also a different type of commutator, which is equally fundamental in the bi-parameter setting, but which we have not yet discussed.
In this paper we are interestedin Bloom type question for this commutator – the main point is that now the right space isnot the little bmo but the much harder product
BMO . If T n and T m are one-parameterCZOs (bounded SIOs) in R n and R m , respectively, then for the L p → L p norm of thecommutator [ T n , [ b, T m ]] f = T n ( bT m f ) − T n T m ( bf ) − bT m T n f + T m ( bT n f ) , where T n f ( x ) = T n ( f ( · , x ))( x ) , the right object is b ∈ BMO prod ( R n + m ) . We refer tothese commutators as product BMO type commutators. Classical references for commu-tators of both type in the Hilbert transform case include Ferguson–Sadosky [12] and thegroundbreaking paper Ferguson–Lacey [11]. In particular, [11] contains the extremelydifficult lower bound estimate k b k BMO prod ( R ) . k [ H , [ b, H ]] k L → L . See also Lacey–Petermichl–Pipher–Wick [27, 28, 29] for the higher dimensional Riesz set-ting and applications to div-curl lemmas. The corresponding upper bound is not easyeither – even for special operators such as the Riesz transforms. However, in [9] Dalencand Ou proved that for all CZOs k [ T n , [ b, T m ]] k L p → L p . k b k BMO prod using the modern approach via the one-parameter representation theorem of Hytönen[18].However, no Bloom type estimates have been considered. In this paper we prove theDalenc–Ou [9] type bound in the Bloom setting – i.e., we prove Bloom type upper boundsfor product BMO commutators. The following is our main result.1.2.
Theorem.
Let T n and T m be one-parameter CZOs in R n and R m , respectively, and let p ∈ (1 , ∞ ) , µ, λ ∈ A p and ν := µ /p λ − /p . We have the quantitative estimate k [ T n , [ b, T m ]] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) . KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINEN
Here
BMO prod ( ν ) is a weighted product BMO space as defined at least in [16, 33]. Wenote that bmo( ν ) ⊂ BMO prod ( ν ) (a proof can be found at least in [33]).This product BMO Bloom estimate is proved in Section 4. While we were inspired byour previous paper [33] dealing with the new approach to little BMO commutators, aquite different take on things is required by the current product BMO setting. The proofidea is outlined in the beginning of Section 4.It is probably possible to add more singular integrals to this Bloom bound and/or con-sider multi-parameter CZOs here, but we content with the most fundamental case here.However, we still mention the following. If we consider multi-parameter singular inte-grals in the product BMO type commutators, then some kind of little product
BMO as-sumptions concerning b are the right thing. Suppose e.g. that T and T are bi-parameterCZOs in R n × R n and R n × R n , respectively. Then according to Ou–Petermich–Strouse[35] and Holmes–Petermichl–Wick [16] we have k [ T , [ b, T ]] f k L ( Q i =1 R ni ) . max (cid:0) sup x ,x k b ( · , x , · , x ) k BMO prod , sup x ,x k b ( · , x , x , · ) k BMO prod , sup x ,x k b ( x , · , · , x ) k BMO prod , sup x ,x k b ( x , · , x , · ) k BMO prod (cid:1) k f k L ( Q i =1 R ni ) , where instead of T we could also similarly as above write T , highlighting the fact thathere it acts on the first two variables. In general, the field of multi-parameter commutatorestimates is again very active – we mention e.g. that recently the commutators of multi-parameter flag singular integrals were investigated by Duong–Li–Ou–Pipher–Wick [10].The product BMO Bloom type upper bound is the main contribution of this paper.However, we also complement our previous paper [33] and (1.1) by giving an easy lit-tle BMO iterated commutator lower bound proof using the so-called median method,previously used in the one-parameter setting in [31, 22] (see also [32]). In [33] we onlyrecorded the following remark regarding the Estimate (1.1). Choosing b = · · · = b k = b and θ = · · · = θ k = 1 /k in (1.1) we get a bi-parameter analog of [31], while choosing θ = 1 (and the rest zero) we get analogs of [17, 21]. However, the first is the better choiceas bmo( ν /k ) ⊃ bmo ∩ bmo( ν ) . Indeed, similarly as in the one-parameter case [31], thisis seen by using that h ν i θR . [ ν ] A h ν θ i R for all θ ∈ (0 , and rectangles R (this estimatefollows from [4, Theorem 2.1] by iteration). In this paper we actually prove the lowerbound showing the optimality of bmo( ν /k ) . The details are given quickly in Section 5. Acknowledgements.
K. Li was supported by Juan de la Cierva - Formación 2015 FJCI-2015-24547, by the Basque Government through the BERC 2018-2021 program and bySpanish Ministry of Economy and Competitiveness MINECO through BCAM SeveroOchoa excellence accreditation SEV-2017-0718 and through project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE) and acronym “HAQMEC”.H. Martikainen was supported by the Academy of Finland through the grants 294840and 306901, and by the three-year research grant 75160010 of the University of Helsinki.He is a member of the Finnish Centre of Excellence in Analysis and Dynamics Research.E. Vuorinen was supported by the Academy of Finland through the grant 306901, bythe Finnish Centre of Excellence in Analysis and Dynamics Research, and by Jenny andAntti Wihuri Foundation.
LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 5
2. D
EFINITIONS AND PRELIMINARIES
Basic notation.
We denote A . B if A ≤ CB for some constant C that can dependon the dimension of the underlying spaces, on integration exponents, and on variousother constants appearing in the assumptions. We denote A ∼ B if B . A . B .We work in the bi-parameter setting in the product space R n + m = R n × R m . In sucha context x = ( x , x ) with x ∈ R n and x ∈ R m . We often take integral pairingswith respect to one of the two variables only: If f : R n + m → C and h : R n → C , then h f, h i : R m → C is defined by h f, h i ( x ) = ´ R n f ( y , x ) h ( y ) d y . Dyadic notation, Haar functions and martingale differences.
We denote a dyadicgrid in R n by D n and a dyadic grid in R m by D m . If I ∈ D n , then I ( k ) denotes theunique dyadic cube S ∈ D n so that I ⊂ S and ℓ ( S ) = 2 k ℓ ( I ) . Here ℓ ( I ) stands for sidelength. Also, ch ( I ) denotes the dyadic children of I , i.e., I ′ ∈ ch( I ) if I ′ ∈ D n , I ′ ⊂ I and ℓ ( I ′ ) = ℓ ( I ) / . We often write D = D n × D m .When I ∈ D n we denote by h I a cancellative L normalised Haar function. Thismeans the following. Writing I = I × · · · × I n we can define the Haar function h ηI , η = ( η , . . . , η n ) ∈ { , } n , by setting h ηI = h η I ⊗ · · · ⊗ h η n I n , where h I i = | I i | − / I i and h I i = | I i | − / (1 I i,l − I i,r ) for every i = 1 , . . . , n . Here I i,l and I i,r are the left and right halves of the interval I i respectively. The reader shouldcarefully notice that h I is the non-cancellative Haar function for us and that in someother papers a different convention is used. If η ∈ { , } n \ { } the Haar function iscancellative: ´ h ηI = 0 . We usually suppress the presence of η and simply write h I forsome h ηI , η ∈ { , } n \ { } . Then h I h I can stand for h η I h η I , but we always treat such aproduct as a non-cancellative function i.e. use only its size.For I ∈ D n and a locally integrable function f : R n → C , we define the martingaledifference ∆ I f = X I ′ ∈ ch ( I ) (cid:2)(cid:10) f (cid:11) I ′ − (cid:10) f (cid:11) I (cid:3) I ′ . Here (cid:10) f (cid:11) I = | I | ´ I f . We also write E I f = (cid:10) f (cid:11) I I . Now, we have ∆ I f = P η =0 h f, h ηI i h ηI ,or suppressing the η summation, ∆ I f = h f, h I i h I , where h f, h I i = ´ f h I . A martingaleblock is defined by ∆ K,i f = X I ∈D n I ( i ) = K ∆ I f, K ∈ D n , i ∈ N . Next, we define bi-parameter martingale differences. Let f : R n × R m → C be locallyintegrable. Let I ∈ D n and J ∈ D m . We define the martingale difference ∆ I f : R n + m → C , ∆ I f ( x ) := ∆ I ( f ( · , x ))( x ) . Define ∆ J f analogously, and also define E I and E J similarly. We set ∆ I × J f : R n + m → C , ∆ I × J f ( x ) = ∆ I (∆ J f )( x ) = ∆ J (∆ I f )( x ) . KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINEN
Notice that ∆ I f = h I ⊗ h f, h I i , ∆ J f = h f, h J i ⊗ h J and ∆ I × J f = h f, h I ⊗ h J i h I ⊗ h J (suppressing the finite η summations). Martingale blocks are defined in the natural way ∆ i,jK × V f = X I : I ( i ) = K X J : J ( j ) = V ∆ I × J f = ∆ K,i (∆ V,j f ) = ∆ V,j (∆ K,i f ) . Weights.
A weight w ( x , x ) (i.e. a locally integrable a.e. positive function) belongsto bi-parameter A p ( R n × R m ) , < p < ∞ , if [ w ] A p ( R n × R m ) := sup R (cid:10) w (cid:11) R (cid:10) w − p ′ (cid:11) p − R < ∞ , where the supremum is taken over R = I × J , where I ⊂ R n and J ⊂ R m are cubes withsides parallel to the axes (we simply call such R rectangles). We have [ w ] A p ( R n × R m ) < ∞ iff max (cid:0) ess sup x ∈ R n [ w ( x , · )] A p ( R m ) , ess sup x ∈ R m [ w ( · , x )] A p ( R n ) (cid:1) < ∞ , and that max (cid:0) ess sup x ∈ R n [ w ( x , · )] A p ( R m ) , ess sup x ∈ R m [ w ( · , x )] A p ( R n ) (cid:1) ≤ [ w ] A p ( R n × R m ) ,while the constant [ w ] A p is dominated by the maximum to some power. Of course, A p ( R n ) is defined similarly as A p ( R n × R m ) – just take the supremum over cubes Q .For the basic theory of bi-parameter weights consult e.g. [16].2.4. Square functions and maximal functions.
Given f : R n + m → C and g : R n → C wedenote the dyadic maximal functions by M D n g := sup I ∈D n I | I | ˆ I | g | and M D f := sup R ∈D R | R | ˆ R | f | . We also set M D n f ( x , x ) = M D n ( f ( · , x ))( x ) . The operator M D m is defined similarly.Define the square functions S D f = (cid:16) X R ∈D | ∆ R f | (cid:17) / , S D n f = (cid:16) X I ∈D n | ∆ I f | (cid:17) / and S D m f = (cid:16) X J ∈D m | ∆ J f | (cid:17) / . Define also S D ,M f = (cid:16) X I ∈D n I | I | ⊗ [ M D m h f, h I i ] (cid:17) / and S D ,M f = (cid:16) X J ∈D m [ M D n h f, h J i ] ⊗ J | J | (cid:17) / . We record the following standard estimates, which are used repeatedly below in an im-plicit manner. Some similar estimates that follow from the ones below are also used.2.1.
Lemma.
For p ∈ (1 , ∞ ) and w ∈ A p = A p ( R n × R m ) we have the weighted square functionestimates k f k L p ( w ) ∼ [ w ] Ap k S D f k L p ( w ) ∼ [ w ] Ap k S D n f k L p ( w ) ∼ [ w ] Ap k S D m f k L p ( w ) . Moreover, for p, s ∈ (1 , ∞ ) we have the Fefferman–Stein inequality (cid:13)(cid:13)(cid:13)(cid:16) X j | M f j | s (cid:17) /s (cid:13)(cid:13)(cid:13) L p ( w ) . [ w ] Ap (cid:13)(cid:13)(cid:13)(cid:16) X j | f j | s (cid:17) /s (cid:13)(cid:13)(cid:13) L p ( w ) . Here M can e.g. be M D n or M D . Finally, we have k S D ,M f k L p ( w ) + k S D ,M f k L p ( w ) . [ w ] Ap k f k L p ( w ) . LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 7
One easy way to show such estimates is to reduce to p = 2 via standard extrapolation.When p = 2 it is especially easy to use one-parameter results iteratively. See e.g. [5, 8] forone-parameter square function results and their history.2.5.
BMO spaces.
Let b : R n + m → C be locally integrable.We define the weighted product BMO space. Given ν ∈ A ( R n × R m ) set k b k BMO prod ( ν, D ) := sup Ω (cid:16) ν (Ω) X R ∈D R ⊂ Ω |h b, h R i| h ν i − R (cid:17) / , where h R := h I ⊗ h J and the supremum is taken over those sets Ω ⊂ R n + m such that | Ω | < ∞ and such that for every x ∈ Ω there exist R ∈ D so that x ∈ R ⊂ Ω . Thenon-dyadic product BMO space BMO prod ( ν ) is defined using the norm defined by thesupremum over all dyadic grids of the above dyadic norms.We also say that b belongs to the weighted dyadic little BMO space bmo( ν, D ) if k b k bmo( ν, D ) := sup R ∈D ν ( R ) ˆ R | b − h b i R | < ∞ . The non-dyadic variant bmo( ν ) has the obvious definition. There holds bmo( ν, D ) ⊂ BMO prod ( ν, D ) (this is proved explicitly at least in [33]). We do not need this fact in thispaper, however.For a sequence ( a I ) I ∈D n we define k ( a I ) k BMO( D n ) = sup I ∈D n (cid:16) | I | X I ⊂ I | a I | (cid:17) / .
3. M
ARTINGALE DIFFERENCE EXPANSIONS OF PRODUCTS
Let D n and D m be some fixed dyadic grids in R n and R m , respectively, and write D = D n × D m . In what follows we sum over I ∈ D n and J ∈ D m . In general, in thispaper always K, I, I , I ∈ D n and V, J, J , J ∈ D m . Paraproduct operators.
The product BMO type paraproducts are A ( b, f ) = X I,J ∆ I × J b ∆ I × J f, A ( b, f ) = X I,J ∆ I × J bE I ∆ J f,A ( b, f ) = X I,J ∆ I × J b ∆ I E J f, A ( b, f ) = X I,J ∆ I × J b (cid:10) f (cid:11) I × J . The little BMO type paraproducts are A ( b, f ) = X I,J E I ∆ J b ∆ I × J f, A ( b, f ) = X I,J E I ∆ J b ∆ I E J f,A ( b, f ) = X I,J ∆ I E J b ∆ I × J f, A ( b, f ) = X I,J ∆ I E J bE I ∆ J f. The “illegal” bi-parameter paraproduct is W ( b, f ) = X I,J (cid:10) b (cid:11) I × J ∆ I × J f. KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINEN
Things make sense at least with the a priori assumptions that b is bounded and f isbounded and compactly supported. For us b ∈ BMO prod ( ν ) so that only the A , . . . A are good as standalone operators. If we would have b ∈ bmo( ν ) ⊂ BMO prod ( ν ) , thenalso A , . . . , A would be good (but this is not the case here). So we can only rely on thefollowing boundedness property. See [16, 33].3.1. Lemma.
Suppose b ∈ BMO prod ( ν ) , where ν = µ /p λ − /p , µ, λ ∈ A p and p ∈ (1 , ∞ ) .Then for i = 1 , . . . , we have k A i ( b, · ) k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) . We formally have bf = X i =1 A i ( b, f ) + W ( b, f ) . When we write like this we say that we decompose bf in the bi-parameter sense. Whenwe employ this decomposition in practice we have to form suitable differences of theproblematic operators with some other terms to get something that is bounded with theproduct BMO assumption. The operator W has a special role.Next, we define operators related to one-parameter type decompositions. Define theone-parameter paraproducts a ( b, f ) = X I ∆ I b ∆ I f, a ( b, f ) = X I ∆ I bE I f. Define also the “illegal” one-parameter paraproduct w ( b, f ) = X I E I b ∆ I f. The operators a i ( b, · ) would be bounded if b ∈ bmo( ν ) ⊂ BMO prod ( ν ) , but again this isnot the case here. The operators a ( b, f ) , a ( b, f ) and w ( b, f ) are defined analogously.We formally have bf = X i =1 a i ( b, f ) + w ( b, f ) = X i =1 a i ( b, f ) + w ( b, f ) . In this case we say that we decomposed bf in the one-parameter sense (either in R n or R m ). 4. T HE B LOOM TYPE PRODUCT
BMO
UPPER BOUND
Theorem.
Let T n and T m be one-parameter CZOs in R n and R m , respectively, and let p ∈ (1 , ∞ ) , µ, λ ∈ A p and ν := µ /p λ − /p . We have the quantitative estimate k [ T n , [ b, T m ]] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) . Remark.
Here we are proving only the right quantitative bound, and prefer to un-derstand the inequality so that b is nice to begin with – we at least make the purelyqualitative a priori assumption b ∈ L ∞ ( R n + m ) . LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 9
Proof.
It is enough to prove the Bloom type inequality for [ U n , [ b, U m ]] , where U n and U m are DMOs (dyadic model operators) appearing in the representation theorem [18, 19].This means that U n ∈ { S n , π n } , where S n is a dyadic shift in R n and π n is a dyadicparaproduct in R n (we will recall what these mean as we go). We only have to maintaina polynomial dependence on the complexity of the shift.Let f : R n + m → C be bounded and compactly supported. How we expand variousproducts of functions (in the bi-parameter sense, in the one-parameter sense, or not atall) depends on the structure of the model operators: a cancellative Haar function in acorrect position is required to tricker an expansion in the corresponding parameter. Ingeneral, the model operators have the form U n f = X KI ( ki ) i = K a K, ( I i ) ˜ h I ⊗ h f, ˜ h I i , U m f = X VJ ( vj ) j = V a V, ( J j ) h f, ˜ h J i ⊗ ˜ h J , where k , k , v , v ≥ , K, I , I ∈ D n , V, J , J ∈ D m , a K, ( I i ) , a V, ( J j ) are appropriateconstants and ˜ h I i ∈ { h I i , I i / | I i |} , ˜ h J j ∈ { h J j , J j / | J j |} . We will write [ U n , [ b, U m ]] f = I − II − III + IV, where I = U n ( bU m f ) , II = U n U m ( bf ) , III = bU m U n f and IV = U m ( bU n f ) . In all of the terms
I, . . . , IV the appearing product b · U m f , b · f , b · U m U n f or b · U n f ,respectively, is expanded. We now explain which of the appearing Haar functions deter-mine the expansion strategy in each of the terms. For I the determining Haar functionsare ˜ h I , ˜ h J , for II they are ˜ h I , ˜ h J , for III they are ˜ h I , ˜ h J and for IV they are ˜ h I , ˜ h J . Ifboth of the determining Haar functions are cancellative, we expand in the bi-parametersense using the paraproducts A i ( b, · ) , W ( b, · ) . If one of them is cancellative we expand inthe corresponding parameter k ∈ { , } using the paraproducts a ki ( b, · ) , w k ( b, · ) . In addi-tion, when a non-cancellative Haar function appears we sometimes add and subtract anaverage to ease the upcoming expansions. The case [ S n , [ b, S m ]] . We study the case that S n f = X KI ( ki ) i = K a K, ( I i ) h I ⊗ h f, h I i , S m f = X VJ ( vj ) j = V a V, ( J j ) h f, h J i ⊗ h J . Here k , k , v , v ≥ , K, I , I ∈ D n and V, J , J ∈ D m . Only finitely many of theconstants a K, ( I i ) are non-zero and | a K, ( I i ) | ≤ | I | / | I | / | K | , and similarly for a V, ( J j ) .We denote ( S n S m ) b, , f := X KI ( ki ) i = K X VJ ( vj ) j = V h b i I × J a K, ( I i ) a V, ( J j ) h f, h I ⊗ h J i h I ⊗ h J . So here , refers to cubes over which we average b over, namely h b i I × J . Define ( S n S m ) b, , etc. analogously. We expand the appearing product in the bi-parameter sensein all of the terms I = S n ( bS m f ) , II = S n S m ( bf ) , III = bS m S n f and IV = S m ( bS n f ) . The term related to the paraproduct W ( b, · ) e.g. yields ( S n S m ) b, , in I , and so we get [ S n , [ b,S m ]] f = I − II − III + IV = E + X i =1 (cid:2) S n ( A i ( b, S m f )) − S n S m ( A i ( b, f )) − A i ( b, S m S n f ) + S m ( A i ( b, S n f )) (cid:3) , where E := ( S n S m ) b, , − ( S n S m ) b, , − ( S n S m ) b, , + ( S n S m ) b, , . We start looking at the sum over i . For i ≤ even all the individual terms are bounded.This is because of Lemma 3.1 and basic weighted bounds of DMOs: k S n f k L p ( w ) . [ w ] Ap k f k L p ( w ) . For i ≥ the term S n ( A i ( b, S m f )) should be paired with one of the terms with a minus.For i = 5 , we pair with − A i ( b, S m S n f ) and for i = 7 , we pair with − S n S m ( A i ( b, f )) .Let us take i = 5 . Notice that it is enough to study S n ( A ( b, f )) − A ( b, S n f ) , since S n S m = S m S n and S m is bounded.Now, the term S n ( A ( b, f )) − A ( b, S n f ) equals X KI ( ki ) i = K X J a K, ( I i ) (cid:2) hh b, h J i i I − hh b, h J i i I (cid:3) h f, h I ⊗ h J i h I ⊗ h J h J . Dualising and using that for a function p in R n we have h p i I − h p i K = P k k =1 h ∆ I ( k )1 p i I ,we reduce to estimating, for a fixed k = 1 , . . . , k , as follows: X KI ( ki ) i = K X J | a K, ( I i ) || I ( k )1 | − / |h b, h I ( k )1 ⊗ h J i||h f, h I ⊗ h J i|h|h g, h I i |i J = X K X I ( k − k ) = KJ X I ( k )1 = II ( k = K | a K, ( I i ) || I | − / |h b, h I ⊗ h J i||h f, h I ⊗ h J i|h|h g, h I i |i J ≤ X K X I ( k − k ) = KJ |h b, h I ⊗ h J i|| I | / | J | / h| ∆ k , K × J f |i I × J h| g |i K × J . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) ¨ R n + m (cid:16) X K,J [ M D ∆ k , K × J f ] (cid:17) / M D g · ν . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) k f k L p ( µ ) k g k L p ′ ( λ − p ′ ) . LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 11
Besides the various standard weighted estimates, we also used the weighted H - BMO prod duality estimate: X I,J |h b, h I ⊗ h J i|| c I,J | . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν, D ) ¨ R n + m (cid:16) X I,J | c I,J | I ⊗ J | I || J | (cid:17) / ν. For this see Proposition 4.1 in [16] (this is what is also used to prove Lemma 3.1). Noticealso that there was even too much cancellation in this term (we were able to throw awaythe cancellative Haar function h I ). We have shown k S n ( A ( b, f )) − A ( b, S n f ) k L p ( λ ) . [ µ ] Ap , [ λ ] Ap k k b k BMO prod ( ν ) k f k L p ( µ ) . The other terms are similar, and so we are done with the i summation.It remains to study E := ( S n S m ) b, , − ( S n S m ) b, , − ( S n S m ) b, , + ( S n S m ) b, , . Afternoting that h b i I × J − h b i I × J − h b i I × J + h b i I × J = k X k =1 v X v =1 h ∆ I ( k )1 × J ( v )2 b i I × J − k X k =1 v X v =1 h ∆ I ( k )1 × J ( v )1 b i I × J − k X k =1 v X v =1 h ∆ I ( k )2 × J ( v )2 b i I × J + k X k =1 v X v =1 h ∆ I ( k )2 × J ( v )1 b i I × J we can reduce the term E to terms, which are very similar to what has been estimatedabove. Finally, we get k [ S n , [ b, S m ]] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap (1 + max k i )(1 + max v i ) k b k BMO prod ( ν ) . The case [ π n , [ b, π m ]] . Here we study the case that π n f = X K a K h K ⊗ h f i K, , π m f = X V a V h f i V, ⊗ h V . Here K ∈ D n and V ∈ D m , k ( a K ) k BMO( D n ) ≤ and a K = 0 for only finitely many K , and similarly for a V . Paraproducts also have the dual form (unlike shifts which arecompletely symmetric) – this is taken into account and we comment on it later on.For I = π n ( bπ m f ) we perform a one-parameter expansion in R m so that I = X i =1 π n ( a i ( b, π m f )) + π n ( w ( b, π m f )) , where by adding and subtracting an average we get π n ( w ( b, π m f )) = X K,V a K a V h [ h b i V, − h b i K × V ] h f i V, i K h K ⊗ h V + ( π n π m ) b f. Here we have denoted ( π n π m ) b f = P K,V h b i K × V a K a V h f i K × V h K × h V . For the term II = π n π m ( bf ) we only add and subtract an average to the end that II = X K,V a K a V h [ b − h b i K × V ] f i K × V h K ⊗ h V + ( π n π m ) b f. The term
III = bπ m π n f warrants a bi-parameter decomposition so that we get III = X i =1 A i ( b, π m π n f ) + ( π n π m ) b f. Finally, the term IV = π m ( bπ n f ) is treated symmetrically to I so that IV = X i =1 π m ( a i ( b, π n f )) + X K,V a K a V h [ h b i K, − h b i K × V ] h f i K, i V h K ⊗ h V + ( π n π m ) b f. Now, we get [ π n , [ b, π m ]] f = I − II − III + IV = − X i =1 A i ( b, π n π m f ) + n X i =1 π n ( a i ( b, π m f )) + X i =1 π m ( a i ( b, π n f )) − X i =5 A i ( b, π m π n f ) o + X K,V h a K a V h [ h b i V, − h b i K × V ] h f i V, i K h K ⊗ h V + a K a V h [ h b i K, − h b i K × V ] h f i K, i V h K ⊗ h V − a K a V h [ b − h b i K × V ] f i K × V h K ⊗ h V i . (4.3)The terms with A i , i = 1 , . . . , , are readily in control. The terms inside the bracket arepaired in the natural way: a goes with A , a goes with A , a goes with A and a goeswith A . We study π n ( a ( b, π m f )) − A ( b, π m π n f ) = π n ( a ( b, π m f )) − A ( b, π n π m f ) . As usual it is enough to study π n ( a ( b, f )) − A ( b, π n f ) . We see that it equals X K,J a K h [ h b, h J i − hh b, h J i i K ] h f, h J i i K h K ⊗ h J h J . Expanding the product inside the average we see that this further equals X K a K X I ⊂ KJ h h I h I i K h b, h I ⊗ h J ih f, h I ⊗ h J i h K ⊗ h J h J . Dualising this against a function g we are left with estimating as follows: X K | a K || K | X I ⊂ KJ |h b, h I ⊗ h J i||h f, h I ⊗ h J i|h|h g, h K i |i J . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) ˆ R m X K | a K | M D m h g, h K i h S D f · ν i K, . k b k BMO prod ( ν ) ¨ R n × R m M D n ( S D f · ν ) S D ,M g . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) k f k L p ( µ ) k g k L p ′ ( λ − p ′ ) . The three other natural terms coming from this bracket are estimated in a similar way.
LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 13
Regarding the last term in (4.3) we note that after expanding the products it simplyequals X K,V a K a V X I ⊂ KJ ⊂ V h h I h I i K h h J h J i V h b, h I ⊗ h J ih f, h I ⊗ h J i h K ⊗ h V . We dualise and estimate as follows: X K,V | a K || K | | a V || V | |h g, h K ⊗ h V i| X I ⊂ KJ ⊂ V |h b, h I ⊗ h J i||h f, h I ⊗ h J i| . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) X K,V | a K || a V |h S D f · ν i K × V |h g, h K ⊗ h V i| . k b k BMO prod ( ν ) ¨ R n × R m (cid:16) X K,V h S D f · ν i K × V |h g, h K ⊗ h V i| K ⊗ V | K || V | (cid:17) / ≤ k b k BMO prod ( ν ) ¨ R n × R m M D ( S D f · ν ) S D g . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) k f k L p ( µ ) k g k L p ′ ( λ − p ′ ) . We have shown k [ π n , [ b, π m ]] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) . The case [ S n , [ b, π m ]] . Here we consider the mixed case that we have a one-parametershift and a paraproduct like above. Expanding I = S n ( bπ m f ) , II = S n π m ( bf ) , III = bπ m S n f and IV = π m ( bS n f ) similarly as above we get [ S n , [ b, π m ]] = I − II − III + IV = X i =1 S n ( A i ( b, π m f )) − X i =1 A i ( b, π m S n f )+ n X i =5 S n ( A i ( b, π m f )) + X i =1 π m ( a i ( b, S n f )) − X i =5 A i ( b, π m S n f ) − X i =1 S n π m ( a i ( b, f )) o + X V a V X KI ( ki ) i = K X J ⊂ V h h J h J i V a K, ( I i ) [ hh b, h J i i I − hh b, h J i i I ] h f, h I ⊗ h J i h I ⊗ h V . We handle the last term first. As usual, we reduce to estimating, for a fixed k =1 , . . . , k , as follows: X V | a V || V | X K X I ( k − k ) = KJ ⊂ V |h b, h I ⊗ h J i|| I | − / X I ( k )1 = II ( k = K | a K, ( I i ) ||h f, h I ⊗ h J i||h g, h I ⊗ h V i|≤ X V | a V || V | − / X K X I ( k − k ) = KJ ⊂ V |h b, h I ⊗ h J i|| I | / | J | / h| ∆ J f |i I × J h| ∆ k , K × V g |i K × V . This is further dominated in the . [ µ ] Ap , [ λ ] Ap sense by k b k BMO prod ( ν ) multiplied with ˆ R n X V | a V || V | / D(cid:16) X J [ M D ∆ J f ] (cid:17) / ν E V, (cid:16) X K h| ∆ k , K × V g |i K × V K (cid:17) / . ¨ R n × R m M D m (cid:16)(cid:16) X J [ M D ∆ J f ] (cid:17) / ν (cid:17)(cid:16) X K,V [ M D ∆ k , K × V g ] (cid:17) / . [ µ ] Ap , [ λ ] Ap k f k L p ( µ ) k g k L p ′ ( λ − p ′ ) . The sums where i = 1 , . . . , are obviously bounded. So it remains to bound the termsinside the bracket. This requires appropriately grouping the terms into pairs of differ-ences. We simply pair a with A , a with A , and then we pair the remaining A termstogether and the A terms together. This easily (after taking operators as common factorsas usually) reduces to terms that we have already bounded. Therefore, we are done: k [ S n , [ b, π m ]] k L p ( µ ) → L p ( λ ) . [ µ ] Ap , [ λ ] Ap (1 + max k i ) k b k BMO prod ( ν ) . Rest of the cases.
By duality and symmetry the only case we have to still consider is [ π n , [ b, π m ]] , where this time one of the paraproducts is in the dual form and one is not: π n f = X K a K K | K | ⊗ h f, h K i , π m f = X V a V h f i V, ⊗ h V . When we decompose [ π n , [ b, π m ]] , the usual terms with the A i , a i and a i produce nosurprises. What is left can be written in the form E + E , where E = π n ( W ( b, π m f )) − π n π m ( w ( b, f ))= − X K,V a K a V h [ h b i K, − h b i K × V ] h f, h K i i V K | K | ⊗ h V = − X K,V X J ⊂ V a K a V h h J h J i V h f, h K ⊗ h J ihh b, h J i i K K | K | ⊗ h V and E = − w ( b, π n π m f ) + π m ( bπ n f )= X K,V a K a V h ( b − h b i V, ) h f, h K i i V, K | K | ⊗ h V = X K,V X J ⊂ V a K a V h h J h J i V h f, h K ⊗ h J ih b, h J i K | K | ⊗ h V . Therefore, we have E + E = X K,V X J ⊂ V a K a V h h J h J i V h f, h K ⊗ h J i [ h b, h J i − hh b, h J i i K ] 1 K | K | ⊗ h V = X K,V X I ⊂ KJ ⊂ V a K a V h h J h J i V | K | − h b, h I ⊗ h J ih f, h K ⊗ h J i h I ⊗ h V . We now estimate this by duality, and first arrive at the obvious upper bound(4.4) k b k BMO prod ( ν ) X K,V | a K || K | | a V || V | ¨ K × V [ S D n h g, h V i ⊗ S D m h f, h K i ] ν. LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 15
Define the auxiliary functions ϕ S ( f ) = X K h K ⊗ S D m h f, h K i and ϕ S ( g ) = X V S D n h g, h V i ⊗ h V . Define also e π n f to be the same function as π n f except that a K is replaced by | a K | . Define e π m g via the formula e π m g = X V | a V |h g, h V i ⊗ V | V | . That is, this is not π m where a V is replaced by | a V | , but rather the dual of π m where a V isreplaced by | a V | (using the definitions of paraproducts that are valid in this subsection).Notice that (4.4) equals k b k BMO prod ( ν ) ¨ R n × R m e π n ( ϕ S ( f )) e π m ( ϕ S ( g )) ν ≤ k b k BMO prod ( ν ) k e π n ( ϕ S ( f )) k L p ( µ ) k e π m ( ϕ S ( g )) k L p ′ ( λ − p ′ ) . [ µ ] Ap , [ λ ] Ap k b k BMO prod ( ν ) k f k L p ( µ ) k g k L p ′ ( λ − p ′ ) . Here we used the weighted boundedness of the paraproducts and the operators ϕ iS . Forthe latter, notice e.g. that S D n ϕ S f = S D f . We are done with the proof. (cid:3)
5. L
OWER BOUNDS FOR COMMUTATORS [ b, · · · [ b, [ b, T ]] · · · ] Let K be a standard bi-parameter full kernel as in [34]. We also assume that K isuniformly non-degenerate. In our setup this means that for all y ∈ R n + m and r , r > there exists x ∈ R n + m such that | x − y | > r , | x − y | > r and(5.1) | K ( x, y ) | & r n r m . For example, we can have K ( x, y ) = K i,j ( x, y ) = x ,i − y ,i | x − y | n +1 x ,j − y ,j | x − y | m +1 . That is, K = K i,j is the full kernel of the bi-parameter Riesz transform R ni ⊗ R mj , i =1 , . . . , n , j = 1 , . . . , m . Regarding the assumed Hölder conditions of the kernel K , simi-larly as in [22], a weaker modulus of continuity should be enough, but we do not pursuethis.We record that (5.1) implies the following: given a rectangle R = I × J there exists arectangle ˜ R = ˜ I × ˜ J such that ℓ ( I ) = ℓ ( ˜ I ) , ℓ ( J ) = ℓ ( ˜ J ) , d ( I, ˜ I ) ∼ ℓ ( I ) , d ( J, ˜ J ) ∼ ℓ ( J ) andsuch that for some σ ∈ C with | σ | = 1 we have for all x ∈ ˜ R and y ∈ R that Re σK ( x, y ) & | R | . This can be seen as follows. Let, for a big enough constant A , the centre c ˜ R of ˜ R to be thepoint x given by (5.1) applied to y = c R , r = Aℓ ( I ) and r = Aℓ ( J ) . Choose σ so that σK ( c ˜ R , c R ) = | K ( c ˜ R , c R ) | . Finally, use mixed Hölder and size estimates repeatedly. Let k ≥ and b ∈ L k loc ( R n + m ) be real-valued. Let p > and µ, λ ∈ A p . Define Γ = Γ(
K, b, µ, λ, k, p ) via the formula Γ = sup 1 µ ( R ) /p (cid:13)(cid:13)(cid:13) x ˜ R ( x ) ˆ A ( b ( x ) − b ( y )) k K ( x, y ) d y (cid:13)(cid:13)(cid:13) L p, ∞ ( λ ) , where the supremum is taken over all rectangles R, ˜ R with ℓ ( I ) = ℓ ( ˜ I ) , ℓ ( J ) = ℓ ( ˜ J ) , d ( I, ˜ I ) ∼ ℓ ( I ) and d ( J, ˜ J ) ∼ ℓ ( J ) , and all subsets A ⊂ R .The moral is simply the following. Notice that ˜ R ( x ) ˆ A ( b ( x ) − b ( y )) k K ( x, y ) d y = 1 ˜ R ( x )[ b, · · · [ b, [ b, T ]] · · · ](1 A )( x ) , if T is a bi-parameter singular integral with a full kernel K . If this iterated commutatormaps L p ( µ ) → L p, ∞ ( λ ) , then Γ is dominated by this norm. However, the constant Γ issignificantly weaker and depends only on the kernel K and some off-diagonal assump-tions.The following proposition supplies the lower bound related to [33] in the case b = · · · = b k = b . See Equation (1.1) in the Introduction.5.2. Proposition.
Suppose K is a uniformly non-degenerate bi-parameter full kernel, k ≥ and b ∈ L k loc ( R n + m ) is real-valued. Let p > , µ, λ ∈ A p and ν = µ /p λ − /p . Then for Γ = Γ(
K, b, µ, λ, k, p ) we have k b k bmo( ν /k ) . [ µ ] Ap , [ λ ] Ap Γ /k . Proof.
Let R = I × J be a fixed rectangle. As we saw above we can find ˜ R = ˜ I × ˜ J suchthat ℓ ( I ) = ℓ ( ˜ I ) , ℓ ( J ) = ℓ ( ˜ J ) , d ( I, ˜ I ) ∼ ℓ ( I ) , d ( J, ˜ J ) ∼ ℓ ( J ) and such that for some σ ∈ C with | σ | = 1 we have for all x ∈ ˜ R and y ∈ R that Re σK ( x, y ) & | R | . For t ∈ R let t + = max( t, . For an arbitrary α ∈ R and x ∈ ˜ R ∩ { b ≥ α } we have (cid:16) | R | ˆ R ( α − b ) + (cid:17) k ≤ | R | ˆ R ∩{ b ≤ α } ( b ( x ) − b ( y )) k d y . Re σ ˆ R ∩{ b ≤ α } ( b ( x ) − b ( y )) k K ( x, y ) d y. Now, let α be a median of b on ˜ R so that min {| ˜ R ∩ { b ≤ α }| , | ˜ R ∩ { b ≥ α }|} ≥ | ˜ R | | R | . In particular, it follows that λ ( ˜ R ∩ { b ≥ α } ) λ ( ˜ R ) ≥ [ λ ] − A p (cid:16) | ˜ R ∩ { b ≥ α }|| ˜ R | (cid:17) p & [ λ ] Ap . As λ ( M R ) . M, [ λ ] Ap λ ( R ) we have λ ( R ) ∼ [ λ ] Ap λ ( ˜ R ) . We then get λ ( R ) /p (cid:16) | R | ˆ R ( α − b ) + (cid:17) k . [ λ ] Ap λ ( ˜ R ∩ { b ≥ α } ) /p (cid:16) | R | ˆ R ( α − b ) + (cid:17) k LOOM TYPE UPPER BOUNDS IN THE PRODUCT BMO SETTING 17 . (cid:13)(cid:13)(cid:13) x ˜ R ( x ) ˆ R ∩{ b ≤ α } ( b ( x ) − b ( y )) k K ( x, y ) d y (cid:13)(cid:13)(cid:13) L p, ∞ ( λ ) ≤ Γ µ ( R ) /p . We have (cid:10) ν k ( k +1) · k ν − k +1 (cid:11) k +1 R ≤ (cid:10) ν /k (cid:11) kR h ν − i R ≤ (cid:10) ν /k (cid:11) kR h λ i /pR h µ − p ′ i /p ′ R ≤ (cid:10) ν /k (cid:11) kR [ µ ] /pA p h µ i − /pR h λ i /pR , and so µ ( R ) /p λ ( R ) − /p . [ µ ] Ap (cid:10) ν /k (cid:11) kR . Combining everything we get (cid:16) | R | ˆ R ( α − b ) + (cid:17) k . [ µ ] Ap , [ λ ] Ap (cid:10) ν /k (cid:11) kR Γ . So we have proved ´ R ( α − b ) + . [ µ ] Ap , [ λ ] Ap ν /k ( R )Γ /k , and the bound ´ R ( b − α ) + . [ µ ] Ap , [ λ ] Ap ν /k ( R )Γ /k is proved analogously. The claim k b k bmo( ν /k ) . [ µ ] Ap , [ λ ] Ap Γ /k follows. (cid:3) R EFERENCES [1] A. Barron, J. Pipher, Sparse domination for bi-parameter operators using square functions, preprint,arXiv:1709.05009, 2017.[2] S. Bloom, A commutator theorem and weighted
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