Approximate weak factorizations and bilinear commutators
aa r X i v : . [ m a t h . C A ] F e b APPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS
TUOMAS OIKARIA
BSTRACT . By extending the approximate weak factorization method to the bilinear set-ting we identify testing conditions on the function b that characterize the L p × L q → L r boundedness of the commutator [ b, T ] ( f, g ) = bT ( f, g ) − T ( bf, g ) , for many exponents in the range < p, q < ∞ and r > , where T is a bilinear Calderón-Zygmund operator.
1. I
NTRODUCTION
The basic objects from which the bilinear commutators [ b, T ] ( f, g ) = bT ( f, g ) − T ( bf, g ) , b ∈ L ( R d ) , are built are the bilinear singular integral operators (SIOs) T of which a fine example isone of the bilinear Riesz transforms R ( f, g )( x ) = p.v. ˆ ˆ x i − y i ( | x − y | + | x − z | ) d +1 f ( y ) g ( z ) d y d z. (1.1)The bilinear Riesz transform is a good example in that not only is its kernel K ( x, y, z ) = x i − y i ( | x − y | + | x − z | ) d +1 (1.2)cancellative enough to be L × L → L bounded, but in that it is also large enough(non-degenerate) to imply commutator lower bounds.The study of commutator estimates have their roots in the work of Nehari [18] whereby complex analytic methods the boundedness of the linear commutator [ b, H ] of theHilbert transform was characterized via Hankel operators. Later, Coifman, Rochbergand Weiss in [3] developed real analytic methods and showed that(1.3) k b k BMO . d X j =1 k [ b, R j ] k L p ( R d ) → L p ( R d ) . k b k BMO := sup I I | b − h b i I | , p ∈ (1 , ∞ ) , where the supremum is taken over all cubes I ⊂ R d and h b i I = | I | ´ I b . The upperbound in (1.3) was proved in [3] for rather general Calderón-Zygmund operators T ,while the lower bound requires some non-degeneracy and there they worked with thelinear Riesz transforms. Later, the lower bound in (1.3) was improved separately by bothJanson in [11] and Uchiyama in [20] to k b k BMO . k [ b, T ] k L p ( R d ) → L p ( R d ) under certain non-degeneracy assumptions on the kernel of T which encompass any single Riesz transform Mathematics Subject Classification.
Key words and phrases.
Calderón–Zygmund operators, singular integrals, multilinear analysis,commutators. (in contrast with 1.3 involving all the d Riesz transforms). Janson’s proof also gives thefollowing off-diagonal characterization of the boundedness of the commutator k [ b, T ] k L p → L q ∼ k b k ˙ C ,α ∼ sup Q ℓ ( Q ) − α Q | b − h b i Q | , α := d (cid:16) p − q (cid:17) , (1.4)when < p < q < ∞ . The off-diagonal characterizations in the last open case < q
and < p, q < ∞ and our cases separate accordinglyto the following three conditions r < p + 1 q sub-diagonal , r = 1 p + 1 q diagonal , r > p + 1 q super-diagonal . (1.7)If r > then we are in the Banach range of exponents and if r ≤ then we are in the quasiBanach range of exponents. In Chaffee [2] the necessity of b ∈ BMO on the diagonal inthe Banach range of exponents was shown with kernels expandable locally as a Fourierseries. A unified approach to the diagonal and sub-diagonal was given in Guo, Lian andWu [9] which covers the diagonal in the whole quasi Banach range, however on the sub-diagonal they only treat linear commutators. Lastly, let us mention the recent article ofK. Li [16] where he proved the necessity of b ∈ BMO on the diagonal in the two weightsetting.
PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 3
In many aspects our result are more general than those recorded before: the Definition2.10 of non-degeneracy is weak, whereas the non-degeneracy assumptions in [15], [13],[2], [9] and [16] are rather strong; the awf scheme allows us to consider complex val-ued functions b, whereas [16] was limited to real valued functions; the full quasi-Banachrange is reached in the sub-diagonal case, whereas [2] is limited to the Banach range; andin that the awf method encompasses bilinear CZOs with both dini and rough kernels.Lastly, let us mention that the super-diagonal case is completely new in the bilinear set-ting and due to us studying the quasi Banach range, the arguments, in all of the cases,involve additional twists absent in [10],[1] and [19]. Our full results are recorded in the-orems 4.1, 4.2, 4.15 and 5.19; the following is a condensed version1.8. Theorem.
Let b ∈ L ( R d ; C ) , let T be a non-degenerate bilinear Calderón-Zygmundoperator, let < r < ∞ and < p, q < ∞ . Then, for i = 1 , , there holds that k [ b, T ] i k L p × L q → L r ∼ k b k BMO , if r = p + q k b k ˙ C α, , α = d (cid:0) ( p + q ) − r (cid:1) , if r < p + q k b k ˙ L s , r = s + p + q , if r > p + q , r ≥ . Acknowledgements.
I thank Henri Martikainen for helpful discussions and comments.2. B
ASIC DEFINITIONS AND PRELIMINARIES
Basic notation.
We let
Σ = Σ( R d ) denote the linear span of indicator functions ofcubes on R d . Similarly we denote L ( R d ) = L , ´ R d = ´ , and so on, mostly leavingout the ambient space if this information is obvious. We denote averages with h f i A = ffl A f := | A | ´ A f, where | A | denotes the Lebesgue measure of the set A . The indicatorfunction of a set A is denoted by A .In this paper we study the L p × L q → L r boundedness of [ b, T ] and hence it is usefulto denote σ ( p, q ) − = p − + q − ; the motivation for this notation comes from that forexponents < p, q < ∞ we may write Hölder’s inequality as k f g k L σ ( p,q ) ≤ k f k L p k g k L q . Lastly, we denote A . B , if A ≤ CB for some constant C > depending only onthe dimension of the underlying space, on integration exponents and on other absoluteconstants appearing in the assumptions that we do not care about. Then A ∼ B , if A . B and B . A. Subscripts on constants ( C a,b,c,... ) and quantifiers ( . a,b,c,... ) signifytheir dependence on those subscripts.2.2. Multilinear singular integrals.
We denote the diagonal with ∆ = { ( x, y, z ) ∈ ( R d ) : x = y = z } and say that a mapping K : ( R d ) \ ∆ → C is a bilinear Calderón-Zygmund kernel if itsatisfies the size estimate: | K ( x, y ) | ≤ C K (cid:0) n X i =1 | x − y i | (cid:1) − d , (2.1)and the regularity estimate:(2.2) | G ( x, y, z ) − G ( x ′ , y, z ) | ≤ ω (cid:0) | x − x ′ || x − y | + | x − z | (cid:1) ( | x − y | + | x − z | ) − d TUOMAS OIKARI for G ∈ { K, K ∗ , K ∗ } , whenever | x − x ′ | ≤ max( | x − y | , | x − z | ) . Here the function ω is increasing, subadditive, and such that ω (0) = 0 and k ω k Dini = ´ ω ( t ) d tt < ∞ . Wealso assume that the appearing constants C K , k ω k Dini are the best possible. We denotethe collection of all such kernels with CZ (2 , d, ω ) and define the norm k K k CZ (2 ,d,ω ) = C K + k ω k Dini . Definition.
A bilinear operator T : Σ → L is said to be a (variable kernel) bilinearSIO, if there exists a bilinear kernel K ∈ CZ (2 , d, ω ) so that h T ( f , f ) , g i = ˆ ˆ ˆ K ( x, y, z ) f ( y ) f ( z ) g ( x ) d y d z d x, ∩ i =1 spt( f i ) ∩ spt( g ) = ∅ for all f , f , g ∈ Σ . Definition.
A bilinear operator T Ω : Σ → L is said to be a (rough) bilinear SIO,whenever it is well-defined as T Ω ( f , f )( x ) = lim ε → ˆ ˆ max( | x − y | , | x − z | ) >ε K Ω ( x, y, z ) f ( y ) f ( z ) d y d z, where K Ω ( x, y, z ) = Ω(( x − y, x − z ) ′ ) | ( x − y, x − z ) | d , Ω( h ′ ) = Ω( h/ | h | ) , Ω ∈ L ( S d − ) . Definition.
A bilinear SIO T that is also a bounded operator L p × L q → L σ ( p,q ) forall exponents < p, q < ∞ is said to be a bilinear Calderón-Zygmund operator.2.6. Remark.
For operators as in Definition 2.3, the boundedness for one tuple of expo-nents p, q is equivalent with the boundedness for all tuples p, q > . In the rough case only lately a general L × L → L bound was shown in [5] underthe assumptions Ω ∈ L q ( S d − ) , for some q > , and ´ S d − Ω = 0 . The future goal in [5]is set out by ”We plan to pursue general L p × · · · × L p m → L p boundedness for manymultilinear operators in subsequent work.” Consequently, at the time of writing, noviable testing conditions on Ω seem to be available to make T Ω bounded simultaneouslyfor many exponents, p, q > , which is what we need in this paper – hence, Definition2.5.2.7. Definition.
Given a bilinear operator T : Σ → L its adjoints T ∗ , T ∗ are given bythe identities h T ∗ ( g, f ) , f i = h T ( f , f ) , g i = h T ∗ ( f , g ) , f i , f , f , g ∈ Σ , and if T has a kernel K , then we denote the kernel of T i ∗ with K i ∗ . Truncations of bilinear SIOs.
We let K ∈ CZ (2 , d, ω ) and define the truncated op-erator T ε as T ε ( f, g )( x ) = ¨ max( | x − y | , | x − z | ) >ε K ( x, y, z ) f ( y ) g ( z ) d y d z, f, g ∈ Σ . (2.8)A particular case of Cotlar’s inequality in the bilinear setting states, see e.g. Grafakos,Torres [8], that for a bilinear CZO T with the kernel K ∈ CZ (2 , d, | · | δ ) , for δ ∈ (0 , , seee.g. [7], there holds that T ∗ ( f, g ) = sup ε> | T ε ( f, g ) | . | T ( f, g ) | + M f M g. PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 5
Since T, M are bounded, this shows especially that sup ε> k T ε k L p × L q → L σ ( p,q ) < ∞ forexponents such that < p, q < ∞ . For rough kernels a Cotlars inequality is not immediately available, at least the au-thor was unable to find one. Still, in this paper, we would only need the boundedness sup ε> k T Ω ,ε k L p × L q → L σ ( p,q ) < ∞ for some p, q > such that σ ( p, q ) ≥ where the trunca-tions are defined by T Ω ,ε ( f, g )( x ) = ¨ max( | x − y | , | x − z | ) >ε K Ω ( x, y, z ) f ( y ) g ( z ) d y d z, For example, it was very recently shown in Theorem 1.1. of [6] that under the assump-tions Ω ∈ L q ( S d − ) , for some q > , and ´ S d − Ω = 0 , there holds that k T ∗ Ω k L × L → L . k Ω k L q ( S d − ) and this is a uniform bound of the desired type.2.4. Bilinear non-degeneracy.
We recall the definition of non-degeneracy for linear ker-nels.2.9.
Definition.
A kernel K : R d × R d \ ∆ → C is said to be non-denegerate, if given y ∈ R d and r > there exists a point x such that | x − y | > r, | K ( x, y ) | & r − d . For bilinear SIOs with variable kernel we set the following.2.10.
Definition.
A kernel K : ( R d ) \ ∆ → C is said to be non-degenerate if both of thefollowing items hold: (1) for all points y, there exists two points x, z such that max a,b ∈{ x,y,z } | a − b | > r, | K ( x, y, z ) | & r − d . (2) for all points z, there exists two points x, y such that max a,b ∈{ x,y,z } | a − b | > r, | K ( x, y, z ) | & r − d . It is immediate from the size estimate that if e.g. y, r and x, z are given as in the item (1) of Definition 2.10 and max a,b ∈{ x,y,z } | a − b | = | c − d | then | c − d | ∼ r , indeed, to seethis, we simply check that r − d . | K ( x, y, z ) | . ( | x − y | + | x − z | ) − d . | c − d | − d , which shows the claim. Equally, it is straightforward that max a ∈{ x,z } | y − a | > r/ and max a ∈{ x,z } | y − a | ∼ r. Similarly for the item (2) . Remark.
We will use the assumption (1) to prove Theorem 1.8 for the index i = 1 and the assumption (2) for the index i = 2 . It follows that we will run the proofs of allof our results through with the assumption (1) of Definition 2.10 and it is clear how tomodify them to get the case i = 2 . For kernels of rough SIOs we set the following.2.12.
Definition.
A kernel K Ω is non-degenerate if Ω = 0 , i.e. it has at least one non-zeroLebesgue point θ = ( θ , θ ) ∈ S d − . TUOMAS OIKARI
Remark.
The kernel K of a bilinear Riesz transform as on the line (1.2) satisfies bothitems (1) and (2) in Definition 2.10 and is also non-degenerate when considered as arough bilinear SIO as in Definition 2.12.In [16] the following definition of non-degeneracy is given; to contrast it with the non-degeneracy we name it the strong non-degeneracy.2.14. Definition.
A kernel K : ( R d ) \ ∆ → C is said to be strongly non-degenerate if foreach given point y ∈ R d and r > there exists a point x ∈ B ( y, r ) c such that | K ( x, y, y ) | & r − d . It is straightforward that strong non-degeneracy is stronger than non-degeneracy.2.15.
Proposition.
Let K be a strongly non-degenerate kernel. Then, the kernel K is non-degenerate.Proof. We only show the point (1) from Definition . . Fix a point y ∈ R d , then bystrong non-degeneracy there exists a point x ∈ B ( y, r ) c so that | K ( x, y, y ) | & r − d . Con-sequently, it remains to write the previous estimate as | K ( x, y, z ) | & r − d , where z = y and to notice that x ∈ B ( y, r ) c . (cid:3) Definition.
We say that a bilinear singular integral operator T is non-degenerate ifits kernel K is non-degenerate. 3. B ILINEAR AWF
When proving the commutator lower bounds we do not need the full strength of thekernel assumption (2.2) and we will replace this with something significantly weaker bymerely assuming the function ω to satisfy ω (0) = 0 , be increasing, be subadditive andsuch that(3.1) | G ( x, y, z ) − G ( x ′ , y, z ) | ≤ ω (cid:0) | x − x ′ || x − y | + | x − z | (cid:1) ( | x − y | + | x − z | ) − d for G ∈ { K, K ∗ , K ∗ } , whenever | x − x ′ | ≤ max( | x − y | , | x − z | ) . Another strengtheningis by the fact that the awf method only requires us to consider the following off-supportinformation on the kernel
K,T ( f Q , g Q )( x ) = ˆ Q ˆ Q K ( x, y, z ) f ( x ) g ( z ) d x d z, y ∈ Q , where Q , Q , Q are cubes of the same size such that max i =1 , dist( Q i , Q ) ∼ ℓ ( Q ) . Topress the point, there is no reference whatsoever to the operator T and everything isdefined with the kernel K only. For further discussion on the awf method in the linearsetting we refer the reader to Hytönen [10].Recall that we only need the assumption (1) from Definition 2.10 to show the lowerbounds for the commutator [ b, T ] . Proposition.
Fix a constant A ≥ and let K either ( i ) be a non-degenerate bilinear kernel that satisfies the estimates (2.1) , (3.1) , or ( ii ) be a rough non-degenerate bilinear kernel. PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 7
Let Q ⊂ R d be a cube with centre point c Q and let D and D be arbitrary dyadic grids. Then,there exists cubes Q i ∈ D i and points c Q i ∈ Q i (not necessarily the centre point of Q i ), for i = 1 , , such that max a ∈{ , } | c Q − c Q a | ∼ Aℓ ( Q ) , ℓ ( Q ) ∼ ℓ ( Q ) ∼ ℓ ( Q ) . (3.3) And, for a constant A large enough, there holds, | K ( c Q , c Q , c Q ) | ∼ A − d | Q | − , (3.4) and for all x ∈ Q , y ∈ Q , z ∈ Q , ˆ Q ˆ Q | K ( x, y, z ) − K ( c Q , c Q , c Q ) | d x d z . ω ( A − ) A − d , (3.5) where ω ( A − ) → as A → ∞ , and (cid:12)(cid:12)(cid:12) ˆ Q ˆ Q K ( x, y, z ) d x d z (cid:12)(cid:12)(cid:12) ∼ ˆ Q ˆ Q | K ( x, y, z ) | d x d z ∼ A − d . (3.6) Moreover, the similar estimates to (3.5) and (3.6) hold where we always integrate over any twoof the cubes Q , Q , Q with the variables x, y, z. Remark.
Tied to the necessity of setting up Q i ∈ D i , i = 0 , , for the study of thesuper-diagonal case when r = 1 , is the fact that for rough kernels (item ( ii ) ) the points c Q i ∈ Q i can not be chosen to be the centre points; conversely, when Proposition 3.2 isused in the other cases, the extra structure of Q i ∈ D i is not needed and the points c Q i will be the centre points of Q i . Proof of the case ( i ) : As we will mostly manage without the property Q i ∈ D i , we firstfind any two cubes Q i , for i = 1 , , satisfying the rest of the claims.We fix a cube Q ⊂ R d and denote its centre point with c Q . Let r = A diam( Q ) / and by non-degeneracy find two points c Q , c Q so that c Q ∈ B ( c Q , r ) c (the case c Q ∈ B ( c Q , r ) c is completely symmetric) and | K ( c Q , c Q , c Q ) | ∗ ∼ r − d ∼ A − d | Q | − . (3.8)The fact that we have ∼ above where indicated by ∗ follows from the discussion afterDefinition 2.10. Hence the claim (3.4) holds. Then, we let Q = ( c Q − c Q ) + Q , Q = ( c Q − c Q ) + Q be the cubes respectfully with the centre points c Q and c Q . Now, by the above and that | c Q − c Q | ∼ r, the claims on the line (3.3) hold.Then, we estimate | K ( x, y, z ) − K ( c Q , c Q , c Q ) | ≤ | K ( x, y, z ) − K ( c Q , y, z ) | + | K ( c Q , y, z ) − K ( c Q , c Q , z ) | + | K ( c Q , c Q , z ) − K ( c Q , c Q , c Q ) | , TUOMAS OIKARI and further by the regularity estimate (3.1) and the sub-additivity of ω the first of thethree intermediate terms as | K ( x, y, z ) − K ( c Q , y, z ) | . ω (cid:0) | x − c Q || c Q − z | + | c Q − y | (cid:1) ( | c Q − z | + | c Q − y | ) − d ≤ ω (cid:0) / Q ) A/ Q ) (cid:1)(cid:0) A diam( Q ) (cid:1) − d . ω ( A − ) A − d | Q | − , where we used that for all points x ∈ Q , y ∈ Q , z ∈ Q , | x − c Q | ≤ | c Q − z | ≤
12 max( | c Q − y | , | c Q − z | ) , which follows immediately by c Q ∈ B ( x, Ar ) c , z ∈ Q and that A ≥ . The remainingtwo terms estimate similarly and consequently we find that | K ( x, y, z ) − K ( c Q , c Q , c Q ) | . ω ( A − ) A − d | Q | − . (3.9)Now, by choosing A large enough, subtracting and adding K ( c Q , c Q , c Q ) and using(3.8) and (3.9) we actually find that | K ( x, y, z ) | ∼ A − d | Q | − , (3.10)which is an improvement of (3.4). Similarly, by using (3.9) we find (3.5) from which theclaims (3.6) follow.We still need to argue that we can arrange Q i ∈ D i , for i = 0 , . Assume that wehave shown the claims for the triple of cubes e Q , Q , e Q with centre points c e Q , c Q , c e Q . Then, we let Q id ∈ D i be the largest cube such that Q id ⊂ e Q i . Now the cubes Q id clearlysatisfy the claims on the line (3.3), and as (3.10) is valid especially with the triple of points ( c Q d , c Q , c Q d ) , we find | K ( c Q d , c Q , c Q d ) | ∼ A − d | Q | − ∼ A − d | Q d | − (3.11)and hence (3.4) is checked. By (3.9), again using that c Q id ∈ Q id ⊂ e Q i , we find, | K ( x, y, z ) − K ( c Q d , c Q , c Q d ) | ≤ | K ( x, y, z ) − K ( c e Q , c Q , c e Q ) | + | K ( c e Q , c Q , c e Q ) − K ( c Q d , c Q , c Q d ) | . ω ( A − ) A − d | Q d | − . (3.12)By (3.11) and (3.12) the claims (3.5) and (3.6) follow. As all the estimates were based onpoint-wise estimates, also the last claim (we can integrate over any two of the cubes)follows. (cid:3) Proof the case ( ii ) : We first check the claims with balls in place of cubes. By the non-degeneracy assumption let θ = ( θ , θ ) ∈ S d − be a non-zero Lebesgue point of thefunction Ω . Then, fix a ball B with centre c B and radius r. Let the points c B , c B bedefined by the following identities, c B − c B = rAθ , c B − c B = rAθ , PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 9 and let B i be a ball with centre c B i and radius r, for i = 0 , . It is then clear that (3.3)holds and that K Ω ( c B , c B , c B ) = Ω(( rAθ , rAθ ) ′ ) | ( rAθ , rAθ ) | d ∼ A − d | B | − d | Ω( θ , θ ) | , hence (3.4) holds.It remains to check (3.5) and (3.6). Let x ∈ B , y ∈ B , z ∈ B be arbitrary and write x = c B + ru x , y = c B − rAθ + ru y , z = c B − rAθ + u z , with some u a ∈ B (0 , depending on the parameter a ∈ { x, y, z } . To ease notation wewrite Ω( h ′ ) = Ω( h ) . Then, we have, K ( x, y, z ) − K ( c B , c B , c B ) = Ω( x − y, x − z ) | ( x − y, x − z ) | d − Ω( c B − c B , c B − c B ) | ( c B − c B , c B − c B ) | d = Ω (cid:0) rAθ + r ( u x − u y ) , rAθ + r ( u x − u z ) (cid:1)(cid:12)(cid:12)(cid:0) rAθ + r ( u x − u y ) , rAθ + r ( u x − u z ) (cid:1)(cid:12)(cid:12) d − Ω (cid:0) Arθ , Arθ (cid:1)(cid:12)(cid:12) Arθ , Arθ (cid:12)(cid:12) d = (cid:0) Ar (cid:1) − d (cid:16) Ω (cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12)(cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12) − d − Ω( θ , θ ) (cid:17) = (cid:0) Ar (cid:1) − d (cid:0) I + II (cid:1) , where I = (cid:16) Ω (cid:0) θ + u x − u y A , θ + u x − u z A (cid:1) − Ω( θ , θ ) (cid:17) − (cid:12)(cid:12)(cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12) d and II = Ω( θ , θ ) (cid:16)(cid:12)(cid:12)(cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12) − d − (cid:17) . Since | u x − u y | + | u x − u z | . , we find with a choice of A large enough that | II | . Ω( θ ,θ ) (cid:12)(cid:12) − (cid:12)(cid:12)(cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12) d (cid:12)(cid:12) = (cid:12)(cid:12) | ( θ , θ ) | d − (cid:12)(cid:12)(cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12) d (cid:12)(cid:12) ∗ . (cid:12)(cid:12) | ( θ , θ ) | − (cid:12)(cid:12)(cid:0) θ + u x − u y A , θ + u x − u z A (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:0) u x − u y A , u x − u z A (cid:1)(cid:12)(cid:12) . A − , where as indicated by ∗ we used the intermediate value theorem, and hence we find that ˆ B ˆ B ( Ar ) − d | II | d x d z . ω II ( A − ) A − d , ω II ( A − ) = A − . (3.13) With a fixed y, the point u x − u y varies over B (0 , and with a fixed y, x the point u x − u z varies over B (0 , . Hence, we estimate ˆ B ˆ B ( Ar ) − d | I | d x d z . A − d B B (cid:12)(cid:12) Ω (cid:0) θ + u x − u y A , θ + u x − u z A (cid:1) − Ω( θ , θ ) (cid:12)(cid:12) d x d z ≤ A − d B B (0 , (cid:12)(cid:12) Ω (cid:0) θ + u x − u y A , θ + tA (cid:1) − Ω( θ , θ ) (cid:12)(cid:12) d x d z ≤ A − d B (0 , B (0 , (cid:12)(cid:12) Ω (cid:0) θ + sA , θ + tA (cid:1) − Ω( θ , θ ) (cid:12)(cid:12) d s d t = A − d B (0 , A ) B (0 , A ) (cid:12)(cid:12) Ω (cid:0) θ + s, θ + t (cid:1) − Ω( θ , θ ) (cid:12)(cid:12) d s d t = A − d ω I ( A − ) , (3.14)where ω I ( A − ) = B (0 , A ) B (0 , A ) (cid:12)(cid:12) Ω (cid:0) θ + s, θ + t (cid:1) − Ω( θ , θ ) (cid:12)(cid:12) d xs d t → as A → ∞ by ( θ , θ ) being a Lebesgue point of Ω . Having the ultimate estimate togetherwith (3.13) shows (3.5), ˆ Q ˆ Q | K ( x, y, z ) − K ( c B , c B , c B ) | d y d z . ˆ B ˆ B ( Ar ) − d | I + II | d x d z . (cid:0) ω I ( A − ) + ω II ( A − ) (cid:1) A − d = ω ( A − ) A − d , and (3.5) implies (3.6).Lastly, let Q be the maximal cube with centre point c Q = c B such that Q ⊂ B andlet Q id be the maximal dyadic cube in D i such that c B i ∈ Q id ⊂ B i , for i = 1 , . We denote c Q id = c B i and note that then c Q id is not necessarily the centre point of Q id . It is then clearfrom the setup that the triple of cubes Q d , Q , Q d and the points ( c Q d , c Q , c Q d ) satisfy theclaims (3.3) and (3.4). The claim (3.5) follows by using the just shown result for balls, ˆ Q d ˆ Q d | K ( x, y, z ) − K ( c Q d , c Q , c Q d ) | d x d z ≤ ˆ B ˆ B | K ( x, y, z ) − K ( c B , c B , c B ) | d x d z . ω ( A − ) A − d , and (3.5) together with (3.4) implies (3.6). The last claim (we can integrate over any twoof the cubes) follows by noting that the estimate for the term II was point-wise andinspecting the estimate (3.14), for the term I. (cid:3) From now on whenever we fix a cube Q , the associated cubes Q and Q will standfor cubes generated through Proposition 3.2. Also, if a function has support in the cube Q i then it has the subscript i or the subscript Q i e.g. if spt( g ) ⊂ Q i then we write g = g i = g Q i . Remark.
If we were dealing only with the integrability exponents p, q, r ∈ (1 , ∞ ) thefollowing Proposition 3.16 would be somewhat easier to state as then we could choose PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 11 the appearing functions g i simply as Q i , however, due to the fact that we allow r ∈ (0 , , quite arbitrary functions g i have to be allowed, see the point ( iii ) in the statement.3.16. Proposition.
Let K be a non-degenerate bilinear kernel that satisfies the conclusions ofProposition 3.2. Also,(i) Let Q be a cube and let Q , Q stand for the cubes generated by Proposition 3.2 above.(ii) Let f be a locally bounded function with zero mean supported on the cube Q .(iii) Let g i be functions such that spt( g i ) ⊂ Q i and h| g i |i Q i ∼ k g i k ∞ & . Then, for all A ≥ large enough (depending only on the above data C, K ), the function f canbe written as f = (cid:2) h T ∗ ( g , g ) − g T ( h , g ) (cid:3) + (cid:2) h T ( g , g ) − g T ∗ ( h , g ) (cid:3) + e f (3.17) and we have the size and support localization information | h | . A d | f | , | h | . A d ω ( A − ) k f k ∞ | g | , | e f | . ω ( A − ) k f k ∞ | g | , (3.18) where the implicit constants on the line (3.18) depend only on the implicit constants in the point ( iii ) and are otherwise independent of the functions g i , i = 0 , , . Moreover, there holds that ´ Q e f = 0 . Proof.
We proceed to write out the function f as f = h T ∗ ( g , g ) − g T ( h , g ) + e w, h = fT ∗ ( g , g ) , e w = g T ( h , g ) . We first check that the function h is well-defined. We denote with K Q the constant K ( c Q , c Q , c Q ) , where c Q i ∈ Q i are the points given by Proposition 3.2. Let y ∈ spt( f ) ⊂ Q and split into two parts, T ∗ ( g , g )( y ) = (cid:0) T ∗ ( g , g )( y ) − K Q ˆ Q ˆ Q g ( x ) g ( z ) d x d z (cid:1) + K Q ˆ Q ˆ Q g ( x ) g ( z ) d x d z = I ( y ) + II.
By Proposition (3.2) we have | I ( y ) | . ˆ Q ˆ Q (cid:12)(cid:12) K ( x, y, z ) − K ( c Q , c Q , c Q ) (cid:12)(cid:12) | g ( x ) || g ( x ) | d x d z . ω ( A − ) A − d k g k ∞ k g k ∞ and that | II | = | K Q || Q | |h g i Q ||h g i Q | ∼ A − d k g k ∞ k g k ∞ , and consequently by the triangle inequality, after choosing A sufficiently large, it followsthat | T ∗ ( g , g )( y ) | ∼ A − d k g k ∞ k g k ∞ & A − d , and hence, that the function h is well-defined. The first claim on the line (3.18) is then also clear. Next, we control the term e w. We expand T ( h , g ) = T (cid:16) h − fK Q ˜ g g , g (cid:17) + T (cid:16) fK Q ˜ g g , g (cid:17) . We estimate the left term on the right-hand side first and for this fix a point y ∈ spt( h − fK Q ˜ g g ) = spt( f ) ⊂ Q . By Proposition 3.2 we have (cid:12)(cid:12)(cid:12) h ( y ) − f ( y ) K Q ˜ g g (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) f ( y ) ˆ Q ˆ Q (cid:0) K Q − K ( x, y, z ) (cid:1) g ( x ) g ( z ) d x d z (cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) T ( g , g )( y ) K Q ¨ g g (cid:12)(cid:12)(cid:12) − . | f ( y ) |k g k ∞ k g k ∞ ω ( A − ) A − d × (cid:16) A − d h| g |i Q h| g |i Q · A − d |h g i Q ||h g i Q | (cid:17) − . ω ( A − ) A d h| g |i − Q | f ( y ) | , and consequently, for x ∈ spt( g ) ⊂ Q there holds that (cid:12)(cid:12)(cid:12) T (cid:16) h − fK Q ˜ g g , g (cid:17) ( x ) (cid:12)(cid:12)(cid:12) ≤ ˆ Q ˆ Q | K ( x, y, z ) | (cid:12)(cid:12)(cid:12) h ( y ) − f ( y ) K Q ˜ g g (cid:12)(cid:12)(cid:12) | g ( z ) | d y d z . ω ( A − ) A d h| g |i − Q ˆ Q ˆ Q | K ( x, y, z ) || f ( y ) || g ( z ) | d y d z . ω ( A − ) k f k ∞ . For the other term, we use ´ Q f = 0 to estimate, (cid:12)(cid:12)(cid:12) T (cid:16) fK Q ˜ g g , g (cid:17) ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ Q ˆ Q (cid:0) K ( x, y, z ) − K Q (cid:1) f ( y ) g ( z ) d y d z (cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) K Q ¨ g g (cid:12)(cid:12)(cid:12) − . ˆ Q ˆ Q (cid:12)(cid:12)(cid:0) K ( x, y, z ) − K Q (cid:1) f ( y ) g ( z ) (cid:12)(cid:12) d y d z × ( |h g i Q ||h g i Q | A − d ) − . ω ( A − ) A − d k f k ∞ k g k ∞ × ( |h g i Q ||h g i Q | A − d ) − . ω ( A − ) k f k ∞ . By having the above two estimates together it follows that | e w ( x ) | . ω ( A − ) k f k ∞ | g ( x ) | . We also have by the definition of the adjoint that ˆ e w = ˆ g T (cid:16) fT ∗ ( g , g ) , g (cid:17) = ˆ fT ∗ ( g , g ) T ∗ ( g , g ) = ˆ f = 0 . All the properties of the function f on the cube Q that allowed us to run throughthe first iteration of the decomposition, are enjoyed by the function e ω on the cube Q . Also, for the kernel K ∗ of T ∗ we have | K ∗ ( c Q , c Q , c Q ) | = | K ( c Q , c Q , c Q ) | & r − d . By these remarks we write out the function e ω as e w = h T ( g , g ) − g T ∗ ( h , g ) + e f , h = e wT ( g , g ) , e f = g T ∗ ( h , g ) . Repeating the above arguments, we find that | h | . A d | e w | . A d ω ( A − ) k f k ∞ | g | PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 13 and | e f | . ω ( A − ) k e w k ∞ | g | . ω ( A − ) k f k ∞ | g | . Consequently we have checked the remaining claims on the line (3.18) and now it re-mains to check that ´ e f = 0 , however, this follows by using the adjoints similarly as itdid for the function e w . (cid:3) Definition.
The oscillation of a function b ∈ L over a cube Q is osc( b ; Q ) = Q | b − h b i Q | . (3.20)3.21. Definition.
Let γ ∈ (0 , . A subset F ′ ⊂ F is said to be a γ -major subset, if | F ′ | >γ | F | . Proposition.
Let K be a bilinear non-degenerate kernel satisfying the conclusions of Propo-sition 3.16, let b ∈ L , let γ ∈ (0 , , fix a cube Q and let g i = 1 E Qi , for i = 0 , , , where E Q i ⊂ Q i is a γ -major subset. Then, there holds that, | Q | osc( b ; Q ) . | (cid:10) [ b, T ] ( h , g ) , g (cid:11) | + | (cid:10) [ b, T ] ( g , g ) , h (cid:11) | , (3.23) and we have the following size and support localization information, | h | . Q , | h | . ω ( A − ) | g | , (3.24) where the implicit constants depend only on γ. Proof. By b − h b i Q having zero mean on the cube Q and duality find a function f withthe properties ´ Q f = 0 , k f k L ∞ ≤ , such that | Q | osc( b ; Q ) ≤ ˆ Q bf. By Proposition 3.16 we write out the function f to arrive at ˆ Q bf = ˆ b (cid:2) h T ∗ ( g , g ) − g T ( h , g ) (cid:3) + ˆ b (cid:2) h T ( g , g ) − g T ∗ ( h , g ) (cid:3) + ˆ Q b e f = − (cid:10) [ b, T ] ( h , g ) , g (cid:11) − (cid:10) [ b, T ] ( g , g ) , h (cid:11) + ˆ Q b e f . The claims on the line (3.24) follow immediately from k f k L ∞ ≤ , the choice of the func-tions g i and the corresponding information in Proposition 3.16. Then, by ´ e f = 0 and thebound (3.18) on the error term, we estimate (cid:12)(cid:12)(cid:12) ˆ Q b e f (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ Q ( b − h b i Q ) e f (cid:12)(cid:12)(cid:12) ≤ k e f k L ∞ | Q | osc( b ; Q ) . ω ( A − ) | Q | osc( b ; Q ); consequently, we find that | Q | osc( b ; Q ) . | (cid:10) [ b, T ] ( h , g ) , g (cid:11) | + | (cid:10) [ b, T ] ( g , g ) , h (cid:11) | + ω ( A − ) | Q | osc( b ; Q ) . Now, as b ∈ L , the common term shared on both sides of the estimate is finite, andhence, by choosing A large enough, we absorb it to the left-hand side and the claimfollows. (cid:3) The off-support norms that model the commutator norm will be given next.
Definition.
Let p, q, r ∈ (0 , ∞ ) . Let K be a kernel that is locally bounded outsidethe diagonal and let b ∈ L . Then, we define the off-support norm O Ap,q,r ( b ; K ) = sup (cid:12)(cid:12)(cid:12) ˆ Q ˆ Q ˆ Q ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) f ( z ) f ( x ) d y d z d x (cid:12)(cid:12)(cid:12) × | Q | − (1 /p +1 /q +1 /r ′ ) , where the supremum is taken over all triples of cubes Q , Q , Q of the same size suchthat max a,b ∈{ , , } dist( Q a , Q b ) ∼ A diam( Q ) and over all functions f a such that | f a | ≤ Q a for a ∈ { , , } . Remark.
For exponents ≤ r < ∞ , it is immediate by Hölder’s inequality that O Ap,q,r ( b ; K ) ≤ k [ b, T ] k L p × L q → L r , whenever the commutator is well-defined, e.g. for r = 1 and r ′ = ∞ we have, O Ap,q,r ( b ; K ) = sup (cid:12)(cid:12) h f Q , [ b, T ] ( f , f ) i (cid:12)(cid:12) × | Q | − (1 /p +1 /q ) ≤ sup k f k ∞ k [ b, T ] ( f , f ) k L | Q | − (1 /p +1 /q ) ≤ k [ b, T ] k L p × L q → L . When < r < we will use the following off-support norm.3.27. Definition.
Let p, q, r ∈ (0 , ∞ ) , let K be any kernel that is locally bounded outsidethe diagonal and let b ∈ L . Then, we define the weak off-support norm O ∞ ,Ap,q,r ( b ; K ) tobe the smallest constant C such that for all triples of cubes Q , Q , Q of the same sizeand functions f , f such that max a,b ∈{ , , } dist( Q a , Q b ) ∼ A diam( Q ) , | f | ≤ Q , | f | ≤ Q , there exists a major subset F ′ ⊂ Q such that for all functions satisfying | f | ≤ F ′ thereholds that (cid:12)(cid:12)(cid:12) ˆ Q ˆ Q ˆ Q ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) f ( z ) f ( x ) d y d z d x (cid:12)(cid:12)(cid:12) ≤ C | Q | /p +1 /q +1 /r ′ . We will now fix the constant A to be so large that all the above propositions where itappears are applicable. Hence, we will also drop the superscript A from the off-supportnorms 3.25 and 3.27 and only write O ∞ p,q,r , O p,q,r . As O ∞ p,q,r ( b ; K ) ≤ O p,q,r ( b ; K ) , also O ∞ p,q,r is a reasonable off-support norm in the Banach range of exponents. Before connecting theoff-support norms to the commutator, we remark the following a priori upper bound.3.28. Remark. If K is a bilinear kernel satisfying the size estimate (2.1), then we have thefollowing upper bound O p,q,r ( b ; K ) . sup Q ℓ ( Q ) α Q | b − h b i Q | . This can be quickly seen as follows: fix triples Q i , f i for i ∈ { , , } as in the Definition3.25 and let e Q be a minimal cube such that Q , Q ⊂ e Q. Then, by the triangle inequalitywe see that it is enough to control two symmetric terms of which the other one is and is
PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 15 controlled as, | Q | − (cid:0) /p +1 /q +1 /r ′ (cid:1) ˆ Q ˆ Q ˆ Q | b ( x ) − h b i e Q || K ( x, y, z ) f ( y ) f ( z ) f ( x ) | d y d z d x . | Q | − (cid:0) /p +1 /q +1 /r ′ (cid:1) ( Aℓ ( Q )) − d | Q | ˆ Q | b ( x ) − h b i e Q | d x ∼ ℓ ( e Q ) − α e Q | b ( x ) − h b i e Q | d x. Next we relate the weak off-support norm O ∞ p,q,r ( b ; K ) to the commutator. For this, werecall that a function f belongs to the space L s, ∞ ( R d ) , < s < ∞ , if k f k L s, ∞ ( R d ) := sup λ> λ |{ x ∈ R d : | f ( x ) | > λ }| /s < ∞ . Also, recall that k f k L s, ∞ ≤ k f k L s , for all s > . The following Lemma 3.29 is standard,see e.g. Section 2.4. Dualization of quasi-norms in the book [17] of Muscalu and Schlag.3.29.
Lemma.
Let s ∈ (0 , ∞ ) and fix a constant C > . Then, the following are equivalent:(i) There holds that k ψ k L s, ∞ ( R d ) . C. (ii) For each set F with | F | ∈ (0 , ∞ ) , there exists a major subset F ′ ⊂ F such that for allfunctions | g | ≤ F ′ , there holds that |h ψ, g i| . C | F | /s ′ . Taken together, the following two propositions control the oscillation with the com-mutator.3.30.
Proposition.
Let p, q, r ∈ (0 , ∞ ) be arbitrary exponents. Then, there holds that O ∞ p,q,r ( b ; K ) . k [ b, T ] k L p × L q → L r, ∞ , (3.31) whenever the commutator T is well-defined and has the kernel K. Proof.
Consider a triple Q , Q , Q and functions f , f as in the Definition 3.27 of O ∞ p,q,r ( b ; K ) . Clearly we may assume that the right-hand side of (3.31) is finite and hencethat k [ b, T ] ( f , f ) k L r, ∞ < ∞ . Then, denote F = Q and let F ′ ⊂ F be the major subsetgiven by the item (2) in Lemma 3.29 such that for all functions | f | ≤ F ′ there holds that | ˆ Q ˆ Q ˆ Q ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) f ( z ) f ( x ) d y d z d x | = | (cid:10) [ b, T ] ( f , f ) , f (cid:11) | . k [ b, T ]( f , f ) k L r, ∞ | F | /r ′ . k [ b, T ] k L p × L q → L r, ∞ | Q | /p +1 /q +1 /r ′ , which implies the claim. (cid:3) Proposition.
Let < p, q, r < ∞ be arbitrary exponents and let K be a bilinear non-degenerate kernel. Then, for all cubes Q ⊂ R d there holds that osc( b ; Q ) . O ∞ p,q,r ( b ; K ) | Q | /p +1 /q − /r . Proof.
Fix a cube Q and let Q , Q be the cube given by Proposition 3.16 and let g =1 Q , g = 1 Q . Then, according to the definition of O ∞ p,q,r ( b ; K ) let F ′ ⊂ Q be a majorsubset and define g = 1 F ′ . Then, by Proposition 3.22, we find that | Q | osc( b ; Q ) . | (cid:10) [ b, T ] ( h , g ) , g (cid:11) | + | (cid:10) [ b, T ] ( g , g ) , h (cid:11) | . O ∞ p,q,r ( b ; K ) | Q | /p +1 /q +1 /r ′ . from which after dividing with | Q | the claim follows. (cid:3)
4. T
HE CASES r − ≤ σ ( p, q ) − In this section we will be either on the diagonal, meaning that r − = σ ( p, q ) − , or onthe sub-diagonal, meaning that r − < σ ( p, q ) − . In both cases the lower bounds formu-late simultaneously in Theorem 4.1 and the upper bounds in theorems 4.2 and 4.15.4.1.
Theorem.
Let b ∈ L and let < r, p, q < ∞ be such that r − ≤ σ ( p, q ) − and let α := d (cid:0) σ ( p, q ) − − r − (cid:1) . Then, if K is a bilinear non-degenerate CZ-kernel, then sup Q ℓ ( Q ) − α Q | b − h b i Q | ∼ (cid:26) k b k BMO , if r − = σ ( p, q ) − k b k ˙ C ,α , if r − < σ ( p, q ) − (cid:27) . O ∞ p,q,r ( b ; K ) . Proof.
The claim follows immediately by Proposition 3.32, the known oscillatory charac-terization of ˙ C ,α and the definition of BMO . (cid:3) The following upper bound is well-known and is recorded e.g. in [16].4.2.
Theorem.
Let < r < ∞ , < p, q < ∞ and T ∈ CZO (2 , d, ω ) . Then, k [ b, T ] k L p × L q → L r . k b k BMO . The sub-diagonal upper bounds in Theorem 4.15 require some preparation consistingof extending parts from the linear theory to the bilinear setting and we refer the readerto Grafakos [4] for a complete account of the corresponding linear theory.4.3.
Proposition.
Let
U, T : Σ × Σ → L be bilinear SIOs with the same kernel. Then, thereexists a function m ∈ L so that ( U − T )( f , f ) = mf f for all f , f ∈ Σ . In addition, if
U, T are CZOs, then the identity ( U − T )( f , f ) = mf f extends to allfunctions f , f ∈ L ∞ c and the function m is bounded.Proof. We will first show the ‘consistency’ condition: Let Q ⊂ R d be a cube and f , f ∈ Σ , then almost everywhere, ( U − T )(1 Q f , Q f ) = 1 Q ( U − T )( f , f ) . (4.4)We reduce this to two parts, clearly (4.4) follows if we show that ( U − T )( f , Q f ) = 1 Q ( U − T )( f , f ) , (4.5)and ( U − T )(1 Q f , f ) = 1 Q ( U − T )( f , f ) . (4.6)We only show the claim (4.6), the claim (4.5) being similar.As the operators U, T share the kernel K, i.e. U ε = T ε , the claim (4.6) follows if weshow that for H ∈ { U, T } and all points x ∈ R d , there exists ε > such that ( H − H ε )(1 Q f , f )( x ) = 1 Q ( x )( H − H ε )( f , f )( x ) . (4.7)Assume first that x ∈ ( Q ∪ ∂Q ) c (the claim is made modulo sets of measure zero andhence we remove the boundary). Then, choose ε = dist( x, ∂Q ) and notice that forall points y ∈ Q there holds max( | x − y | , | x − y | ) ≥ | x − y | > ε , and consequently as x spt(1 Q f ) by the definition of K being the kernel of H that H (1 Q f , f )( x ) = ˆ R d ˆ R d K ( x, y, z )1 Q ( y ) ( y ) f ( z ) d y d z = ˆ ˆ max( | x − y | , | x − y | ) >ε K ( x, y, z ) f ( y ) f ( z ) d y d z = H ε (1 Q f , f )( x ) . PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 17
Hence, in this case both sides of (4.7) are zero. Then, let x ∈ Q \ ∂Q and again fix ε = dist( x, ∂Q ) . Then, as above, we see that ( H − H ε )(1 Q c f , f )( x ) = 0 . Consequently,for x ∈ Q \ ∂Q there holds that ( H − H ε )(1 Q f , f )( x ) = ( H − H ε )(1 R d f , f )( x ) − ( H − H ε )(1 Q c f , f )( x )= ( H − H ε )( f , f )( x ) −
0= 1 Q ( x )( H − H ε )( f , f )( x ) . Hence, the identity (4.6) holds almost everywhere.Then, we define the function m by m Q ( x ) = 1 Q ( x )( U − T )(1 Q , Q )( x ) , x ∈ Q (4.8)and, as the intersection of two cubes is a cube, the property (4.4) shows that this is well-defined.Then, let f i = P n i k c i,k Qi,k , for i = 1 , , be simple and let x ∈ R d . Fix a cube Q suchthat spt( f ) ∪ spt( f ) ∪ { x } ⊂ Q. Then, by (4.4) and linearity there holds that ( U − T )(1 Q f , Q f )( x ) = f ( x )( U − T )(1 Q , Q f )( x )= f ( x ) f ( x )( U − T )(1 Q , Q )( x ) = f ( x ) f ( x ) m ( x ) . Consequently, we have shown that U − T = m on Σ × Σ , and this also gives m ∈ L bytesting against simple functions.If U, T are bounded operators, say L × L → L , the identity ( U − T )( f , f ) = mf f follows by approximating L functions with those in the class Σ (for which the identityholds), as L ∞ c ⊂ L the desired identity follows. Also, by testing against simple func-tions, it follows by the boundedness of U, T that necessarily k m k ∞ ≤ k U k L × L → L + k T k L × L → L . (cid:3) Proposition.
Let K be a kernel such that sup ε> k T ε k L p × L q → L σ ( p,q ) < ∞ for some expo-nents such that < p, q < ∞ and ≤ σ ( p, q ) . Then, there exists a bounded bilinear operator T : L p × L q → L σ ( p,q ) with the kernel K and a sequence ε k → such that lim ε k → (cid:10) T ε k ( f , f ) , f (cid:11) = (cid:10) T ( f , f ) , f (cid:11) , (4.10) for all f , f , f ∈ L ∞ c . In addition, if T is a CZO with the kernel K , then (by Proposition 4.3)there exists a bounded function m such that T = T + m. Proof.
We will show the argument with the exponents p = q = 3 and σ ( p, q ) = . Let F be a countable dense subset of L . By the bound sup ε> k T ε k L × L → L < ∞ , Hölder’sinequality and a diagonalization argument, we find a sequence ε k → such that for all f , f ∈ F , Λ f ,f ( f ) = lim ε k → (cid:10) T ε k ( f , f ) , f (cid:11) (4.11)defines a bounded linear functional on L ∩ F with norm k Λ f ,f k L ∩F→ C ≤ sup ε> k T ε k L × L → L k f k L k f k L . By Cauchy sequences this extends as a bounded linear functional to the whole of L andthen the Riesz representation theorem gives a function ψ ( f , f ) ∈ L = (cid:0) L (cid:1) ∗ such that Λ f ,f ( f ) = ˆ ψ ( f , f ) f , k ψ ( f , f ) k L ≤ k Λ f ,f k L → C . Then, we define the operator T ( f , f ) = ψ ( f , f ) in the dense class F × F and clearly T : F × F → L is a bounded bilinear operator with the kernel K that satisfies (4.10) forfunctions f , f ∈ F and f ∈ L . Again, by Cauchy sequences T extends as a boundedbilinear functional to the whole L × L and then it remains to argue that T has the kernel K and that (4.10) holds for f , f , f ∈ L . That T has the kernel K follows by how T was extended to L × L via Cauchy sequences, the kernel representation being valid in F × F and the dominated convergence theorem. Similarly we find that (4.10) holds for f , f , f ∈ L . As L ∞ c ⊂ L , we are done. (cid:3) Proposition.
Let T be a SIO with a kernel K such that sup ε> k T ε k L p × L q → L σ ( p,q ) < ∞ for one tuple of exponents σ ( p, q ) ≥ , and let f , f ∈ L ∞ c and b ∈ ˙ C ,α . Then, there holds that (4.13) [ b, T ] ( f, g )( x ) = ˆ ˆ ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) g ( z ) d y d z. Proof. As b ∈ ˙ C ,α ⊂ L ∞ loc , bf ∈ L ∞ c , and then by Proposition 4.9 we have, (cid:10) [ b, T ] ( f , f ) , f (cid:11) = (cid:10) [ b, T − m ] ( f , f ) , f (cid:11) = (cid:10) [ b, T ] ( f , f ) , f (cid:11) = (cid:10) lim ε k → [ b, T ε k ]( f , f ) , f (cid:11) ∗ = (cid:10) ˆ ˆ ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) f ( z ) d y d z, f ( x ) (cid:11) x , (4.14)where the last step marked with ∗ follows by the dominated convergence theorem afterthe following estimate uniform in ε k , | ¨ max( | x − y | , | x − z | ) >ε k ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) f ( z ) d y d z |≤ ˆ ˆ | ( b ( x ) − b ( y )) K ( x, y, z ) f ( y ) f ( z ) | d y d z . k b k ˙ C ,α ( R d ) ˆ ˆ | x − y | α (cid:0) | x − y | + | x − z | (cid:1) d | f ( y ) || f ( z ) | d y d z < ∞ , where the finiteness follows simply by the fact that f, g ∈ L ∞ c and that the appearingsingularity is weak enough to be locally integrable, see also the last estimate in the proofof the following Theorem 4.15. Now as (4.14) holds for all test functions f the claim onthe line (4.13) follows. (cid:3) Theorem.
Let b ∈ L , let < r, p, q < ∞ be such that r − ≤ σ ( p, q ) − , let α := d (cid:0) σ ( p, q ) − − r − (cid:1) , and let T be a bilinear SIO such that sup ε> k T ε k L p × L q → L σ ( p ,q < ∞ for one tuple of exponents < p , q < ∞ with σ ( p , q ) ≥ . Then, k [ b, T ] k L p × L q → L r . sup Q ℓ ( Q ) − α Q | b − h b i Q | . PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 19
Proof.
Clearly we may assume that k b k ˙ C ,α < ∞ , as otherwise the claimed estimate isimmediate, and by density, it is enough to prove the claim for functions f , f ∈ L ∞ c . Then, by Proposition 4.12 we write the commmutator in a closed form, and then estimateit as, | [ b, T ] ( f , f )( x ) | . k b k ˙ C ,α ( R d ) ˆ ˆ | x − y | α (cid:0) | x − y | + | x − z | (cid:1) d | f ( y ) || f ( z ) | d y d z = k b k ˙ C ,α ( R d ) I α ( | f | , | f | )( x ) . Now, the operator I α is the multilinear fractional integral of Kenig and Stein, see [12],where its boundedness is fully characterized: it satisfies exactly the claimed estimates. (cid:3)
5. T
HE CASE r − > σ ( p, q ) − Now we are on the super-diagonal, meaning that r − > σ ( p, q ) − and we define theexponent s by r − = σ ( s, p, q ) − . The following proposition shows that the membershipof b ∈ ˙ L s is sufficient for commutator boundedness.5.1. Proposition.
Let / < r < ∞ and < p, q < ∞ be such that r − > σ ( p, q ) − and definethe exponent s by r − = σ ( s, p, q ) − . Also, let T be a bounded bilinear operator, T : L p × L q → L σ ( p,q ) , T : L σ ( s,p ) × L q → L r . Then, there holds that k [ b, T ] k L p × L q → L r . k b k ˙ L s . Proof.
We first estimate k [ b, T ] ( f, g ) k L r = k [ b − c, T ] ( f, g ) k L r ≤ k ( b − c ) T ( f, g ) k L r + k T (( b − c ) f, g ) k L r . Then, by Hölder’s inequality and the boundedness of T we estimate the first term on theright-hand side as k ( b − c ) T ( f, g ) k L r ≤ k b − c k L s k T ( f, g ) k L σ ( p,q ) ≤ k b − c k L s k T k L p × L q → L σ ( p,q ) k f k L p k g k L q . For the second term by the boundedness of T and Hölder’s inequality we have k T (( b − c ) f, g ) k L r ≤ k T k L σ ( s,p ) × L q → L r k ( b − c ) f k L σ ( s,p ) k g k L q ≤ k T k L σ ( s,p ) × L q → L r k b − c k L s k f k L p k g k L q . Taking the infimum over all c ∈ C shows the claim. (cid:3) If r ≥ , then σ ( s, p ) > and bilinear CZOs T are bounded as in the assumptions.5.1. Dyadic systems and notation.
Definition.
A dyadic grid on R d is a collection D of cubes satisfying,(i) for each k ∈ Z the collection D k = (cid:8) Q ∈ D : ℓ ( Q ) = 2 k (cid:9) is a disjoint cover of R d , (ii) for Q, P ∈ D we have Q ∩ P ∈ (cid:8) Q, P, ∅ (cid:9) . Given a cube and a grid such that Q ∈ D , we denote D Q = { P ∈ D : P ⊂ Q } . Definition.
We say that a collection of sets S is γ -sparse, if there exists a pairwisedisjoint collection of γ -major subsets, S E = { E Q ⊂ Q : Q ∈ S } . Principal stopping time families.
Our sparse collections will be built by splittinginto dyadic scales. Let f be locally integrable function and let Q ∈ D be a cube, then weset S ( f ; Q ) = (cid:8) P ∈ D : P ⊂ Q is a maximal cube s.t. h| f |i P > h| f |i Q (cid:9) and form the principal stopping time family S ⊂ D Q by S = ∪ k S k , S k +1 = ∪ P ∈ S k S ( f ; P ) , S = { Q } . For an arbitrary collection S ⊂ D of dyadic cubes and for each Q ∈ S we let ch S ( Q ) consist of the maximal cubes P ∈ S such that P ( Q. For a given cube Q ∈ S wedenote E Q = Q \ ∪ P ∈ ch S Q P and for each Q ∈ D we let Π Q := Π S Q denote the minimalcube P in S such that Q ⊂ P ()on the condition that it exists). With this notation then, ch S ( P ) = { Q ∈ S : Q ( P, Π Q = P } . We also denote ch k +1 S ( P ) = ∪ Q ∈ ch k S ( P ) ch S Q when ch S ( P ) = ch S ( P ) . Lastly, given a cube Q ∈ D , we denote ∆ Q f = X P ∈ ch D ( Q ) (cid:0) h f i P − h f i Q (cid:1) P . The following Lemma is recorded e.g. in [10].5.4.
Lemma.
Fix a cube Q ∈ D and a function f ∈ L . Then, the principal cube family S issparse: | E Q | > | Q | and E Q ∩ E P = ∅ if Q = P. In addition, if f ∈ L ∞ ( Q ) and ´ Q f = 0 , then, we split the function f according to the partition S = ∪ Nk =0 S k , S k = ch k S ( Q ) , where the number N is finite and depends only on k f k L ∞ ( Q ) , as f = N X k =0 X P ∈ S k f P , f P = X Π S Q = P ∆ Q f, (5.5) and the functions f P satisfy:(3) ´ f P = 0 , (4) k f P k ∞ . h| f |i P , (5) P Nk =0 P P ∈ S k k f P k s ∞ P . ( M f ) s , for all s > . Lemma.
Let S be a sparse collection, let γ > and let D be a dyadic grid. To each cube Q ∈ S associate another cube e Q ∈ D such that dist( Q, e Q ) . γℓ ( Q ) and ℓ ( e Q ) ∼ ℓ ( Q ) . Then,the collection f S = { e Q : Q ∈ S } is sparse.Proof. Let e P , e H ∈ f S be such that e H ( e P .
Then, from that dist( H, e H ) . γℓ ( H ) and ℓ ( H ) . ℓ ( e H ) , it follows that there exists a constant β ∼ γ so that H ⊂ β e P .
Consequently,we find that X e H ∈ e S e H ( e P | e H | ≤ X e H ∈ e S H ( β e P | e H | . X H ∈ S H ( β e P | H | . X H ∈ S H ( β e P | E H | ≤ | β e P | . | e P | , where we used ℓ ( e H ) . ℓ ( H ) in the second estimate. We have shown that the reflectedcollection f S is Carleson and as the Carleson condition is equivalent with sparseness fordyadic collections, for this fact see e.g. the book of Lerner and Nazarov [14], the claimfollows. (cid:3) PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 21
Lemma.
Let p ∈ (1 , ∞ ) and S be a sparse collection. Then, for any constants a Q thereholds that k X Q ∈ S a Q Q k L p . k X Q ∈ S | a Q | E Q k L p . Proof.
The claim follows by duality and the following estimate (cid:10) X Q ∈ S a Q Q , g (cid:11) ≤ X Q ∈ S | Q || a Q |h| g |i Q . X Q ∈ S | E Q || a Q |h| g |i Q . ˆ M g X Q ∈ S | a Q | E Q . k X Q ∈ S | a Q | E Q k L p k g k L p ′ . (cid:3) Definition.
Let b ∈ L , ≤ r, p, q < ∞ and let K be locally bounded away from thediagonal. Then, we define the super-diagonal off-support norm O Σ p,q,r ( b ; K ) = sup N X k =0 (cid:12)(cid:12)(cid:12) ˆ ˆ ˆ ( b ( x ) − b ( y )) K ( x, y, z ) f ,k ( y ) f ,k ( z ) f ,k ( x ) d y d z d x (cid:12)(cid:12)(cid:12) × N p,q,r ′ ( ~f ) − , where we have denoted N p,q,r ′ ( ~f ) = (cid:13)(cid:13) N X k =0 k f ,k k ∞ spt( f ,k ) (cid:13)(cid:13) L p ( R d ) (cid:13)(cid:13) N X k =0 k f ,k k ∞ spt( f ,k ) (cid:13)(cid:13) L q ( R d ) × (cid:13)(cid:13) N X k =0 k f ,k k ∞ spt( f ,k ) (cid:13)(cid:13) L r ′ ( R d ) , and where the supremum is taken over all finite collections of triples of cubes of the samediameter such that max i,j ∈{ , , } dist( Q ik , Q jk ) ∼ A diam( Q k ) and over all functions suchthat | f i,k | ≤ Q ik and | spt( f i,k ) | > . Remark. If r > , then we can replace the entries spt( f i,k ) , in the three terms of N p,q,r ′ ( ~f ) , with Q ik , for i = 0 , , . Remark.
Using that for bilinear operators U there holds that N X i =1 h U ( f i , g i ) , h i i = E ′ E D U (cid:0) N X i =1 ε i ε ′ i f i , N X j =1 ε ′ j g j (cid:1) , N X l =1 ε l h l E , where ε i , ε ′ i are independent random signs, over some probability spaces with expecta-tions denoted respectively as E , E ′ , meaning that E ε i ε j = E ′ ε ′ i ε ′ j = 1 { i = j } ( i, j ) , it followsby Hölder’s inequality that for r ≥ we have O Σ p,q,r ( b ; K ) ≤ k [ b, T ] k L p × L q → L r , and con-sequently, that O Σ p,q,r is a reasonable off-support constant for r ≥ . Proposition.
Let K be a non-degenerate bilinear kernel that satisfies the conclusions ofProposition 3.16, let b ∈ L and let ≤ r, s, p, q < ∞ be exponents such that /r > /p + 1 /q and r − = σ ( s, p, q ) − . Then, there holds that k b k ˙ L s . O Σ p,q,r ( b ; K ) . Proof.
Let Q be an arbitrary dyadic cube and let D be a dyadic grid containing the cube Q . Fix a constant
M > and let f be a function such that Q f = f, ˆ Q f = 0 , k f k ∞ ≤ M, k f k L s ′ ≤ . (5.12)Note that s > r and hence s, s ′ > . Then, let S ⊂ D Q denote the sparse collection ofcubes we obtain through Lemma 5.4. We write the function f as on the line (5.5) and byProposition 3.16 factorize each of the terms f P , P ∈ S , as in (3.17), to arrive at f P = (cid:2) h P T ∗ ( g P , g P ) − g P T ( h P , g P ) (cid:3) + (cid:2) h P T ( g P , g P ) − g P T ∗ ( h P , g P ) (cid:3) + e f P , where we have written h i = h P i and g i = g P i . Next, we will specify how the cubes P , P and the functions g P i will be chosen.Recall, that by Proposition 3.2 we can assume P , P ∈ D . Then, by Lemma 5.6 thecollection g S = { P : P ∈ S } ⊂ D is sparse and we will denote the pairwise dis-joint major subsets with E P . Recall, that by Proposition 3.16 we are free to choose thefunctions g P i under the condition h| g i |i Q i & k g i k ∞ & and clearly the following choicessatisfy this condition, g P = 1 E P , g P = 1 P , g P = 1 P . (5.13)Now, the off-support norm only controls finite sums, but the collection S is potentiallyinfinite. Hence, we empty the collection S through an increasing chain of finite subcol-lections S ⊂ S · · · ⊂ S . Then, we have | ˆ Q bf | ≤ lim n →∞ X P ∈ S n |h [ b, T ] ( h P , g P ) , g P i| + X P ∈ S n |h [ b, T ] ( g P , g P ) , h P i| + | ˆ b e f Σ | , where we denote e f Σ = P P ∈ S e f P , and the implicit change of integration and summa-tion is easily justified after the subsequent estimates. We shall analyse the second sumon the right-hand side, the first one being similar. By trilinearity there trivially holds that (cid:10) [ b, T ] ( g P , g P ) , h P (cid:11) = (cid:10) [ b, T ] ( α P g P , α P g P ) , α P h P (cid:11) (5.14)for any constants such that α P α P α P = 1 . Hence, by the relation r − = σ ( s, p, q ) − , for(5.14), we take α P = k f P k s ′ r ′ − ∞ , α P = k f P k s ′ p ∞ , α P = k f P k s ′ q ∞ , s ′ r ′ −
1) + s ′ p + s ′ q . (5.15)Then, with the choices (5.15), also using | h P | . k f P k ∞ | g P | = k f P k ∞ E P , we findthat X P ∈ S n |h [ b, T ] ( α P g P , α P g P ) , α P h P i|≤ O Σ p,q,r ( b ; K ) (cid:13)(cid:13) X P ∈ S n k f P k s ′ p ∞ P (cid:13)(cid:13) L p ( R d ) (cid:13)(cid:13) X P ∈ S n k f P k s ′ q ∞ P (cid:13)(cid:13) L q ( R d ) × (cid:13)(cid:13) X P ∈ S n k f P k s ′ r ′ ∞ E P (cid:13)(cid:13) L r ′ ( R d ) = RHS.
PPROXIMATE WEAK FACTORIZATIONS AND BILINEAR COMMUTATORS 23
Now the proof splits into the cases r > and r = 1 . We first consider the case r > in which all the three terms of RHS are estimatedsimilarly. From that dist( P , P i ) . ℓ ( P i ) and ℓ ( P ) . ℓ ( P i ) it follows that there existsan absolute constant C > such that CP i ⊃ P , and consequently, that the collections { CP i : P ∈ S } are sparse with the major subsets E P . Hence, by Lemma 5.7, for i ∈ { , , } and v ∈ (1 , ∞ ) and u ∈ (0 , ∞ ) , there holds that (cid:13)(cid:13) X P ∈ S n k f P k u ∞ P i (cid:13)(cid:13) L v ( R d ) ≤ (cid:13)(cid:13) X P ∈ S n k f P k u ∞ CP i (cid:13)(cid:13) L v ( R d ) . (cid:13)(cid:13) X P ∈ S n k f P k u ∞ E P (cid:13)(cid:13) L v ( R d ) ≤ (cid:13)(cid:13) X P ∈ S n k f P k u ∞ P (cid:13)(cid:13) L v ( R d ) . (cid:13)(cid:13) ( M f ) u (cid:13)(cid:13) L v ( R d ) , where in the last estimate we used the point-wise estimate (5) from Lemma 5.4. Now, wefind that RHS . O Σ p,q,r ( b ; K ) (cid:13)(cid:13) M f s ′ p (cid:13)(cid:13) L p ( R d ) (cid:13)(cid:13) M f s ′ q (cid:13)(cid:13) L q ( R d ) (cid:13)(cid:13) M f s ′ r ′ (cid:13)(cid:13) L r ′ ( R d ) . O Σ p,q,r ( b ; K ) (cid:13)(cid:13) f (cid:13)(cid:13) s ′ p L s ′ ( R d ) (cid:13)(cid:13) f (cid:13)(cid:13) s ′ q L s ′ ( R d ) (cid:13)(cid:13) f (cid:13)(cid:13) s ′ r ′ L s ′ ( R d ) ≤ O Σ p,q,r ( b ; K ) , where we used s ′ > in the second estimate for the boundedness of the maximal func-tion.In the case r = 1 the first two terms of RHS estimate the same as in the case r > andthe last term estimates differently, RHS . O Σ p,q,r ( b ; K ) (cid:13)(cid:13) X P ∈ S n k f P k s ′ r ′ ∞ E P (cid:13)(cid:13) L r ′ ( R d ) = O Σ p,q,r ( b ; K ) (cid:13)(cid:13) X P ∈ S n E P (cid:13)(cid:13) L ∞ ( R d ) ≤ O Σ p,q,r ( b ; K ) k k L ∞ ( R d ) = O Σ p,q,r ( b ; K ) , the crucial step here was the disjointness of the sets E P . The just shown estimates also hold for the other term, and as the estimates are uniformin n , it follows that, | ˆ bf | . O Σ p,q,r ( b ; K ) + | ˆ b e f Σ | . By Lemma 5.4 and Proposition 3.16 we have | e f Σ | ≤ X P ∈ S k e f P k ∞ P . ω ( A − ) X P ∈ S k f P k ∞ P . ω ( A − ) M f (5.16)and as also Q e f Σ = e f Σ and ´ Q e f Σ = 0 , the function e f Σ satisfies the conditions on theline (5.12) but now with the additional decay . ω ( A − ) . Consequently, we draw theconclusion that sup (5.12) | ˆ bf | . O Σ p,q,r ( b ; K ) + ω ( A − ) sup (5.12) | ˆ bf | . (5.17)The common term on both sides of the estimate (5.17) is finite (recall that b ∈ L and foreach f as in the supremum k f k ∞ < M ), and hence by choosing A sufficiently large, by absorbing the common term to the left-hand side we conclude that sup (5.12) | ˆ Q bf | . O Σ p,q,r ( b ; K ) . (5.18)Then, as s > , the proof is concluded with exactly the same argument by Riesz’ repre-sentation theorem as in [10]. (cid:3) Having propositions 5.1 and 5.11 together gives us5.19.
Theorem.
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