Genuinely multilinear weighted estimates for singular integrals in product spaces
aa r X i v : . [ m a t h . C A ] N ov GENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULARINTEGRALS IN PRODUCT SPACES
KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINENA
BSTRACT . We prove genuinely multilinear weighted estimates for singular integrals inproduct spaces. The estimates complete the qualitative weighted theory in this setting.Such estimates were previously known only in the one-parameter situation. Extrapolationgives powerful applications – for example, a free access to mixed-norm estimates in thefull range of exponents.
1. I
NTRODUCTION
For given exponents < p , . . . , p n < ∞ and /p = P i /p i > , a natural form of aweighted estimate in the n -variable context has the form (cid:13)(cid:13)(cid:13) g n Y i =1 w i (cid:13)(cid:13)(cid:13) L p . n Y i =1 k f i w i k L pi for some functions f , . . . , f n and g . It is natural to initially assume that w p i i ∈ A p i , where A q stands for the classical Muckenhoupt weights. Even with this assumption the targetweight only satisfies Q ni =1 w pi ∈ A np ) A p making the case n ≥ have a different flavourthan the classical case n = 1 . Importantly, it turns out to be very advantageous – we getto the application later – to only impose a weaker joint condition on the tuple of weights ~w = ( w , . . . , w n ) rather than to assume individual conditions on the weights w p i i . Thisgives the problem a genuinely multilinear nature. For many fundamental mappings ( f , . . . , f n ) g ( f , . . . , f n ) , such as the n -linear maximal function, these joint conditionson the tuple ~w are necessary and sufficient for the weighted bounds.Genuinely multilinear weighted estimates were first proved for n -linear one-parameter singular integral operators (SIOs) by Lerner, Ombrosi, Pérez, Torres and Trujillo-Gonzálezin the extremely influential paper [29]. A basic model of an n -linear SIO T in R d is ob-tained by setting T ( f , . . . , f n )( x ) = U ( f ⊗ · · · ⊗ f n )( x, . . . , x ) , x ∈ R d , f i : R d → C , where U is a linear SIO in R nd . See e.g. Grafakos–Torres [16] for the basic theory. Esti-mates for SIOs play a fundamental role in pure and applied analysis – for example, L p estimates for the homogeneous fractional derivative D α f = F − ( | ξ | α b f ( ξ )) of a productof two or more functions, the fractional Leibniz rules , are used in the area of dispersiveequations, see e.g. Kato–Ponce [28] and Grafakos–Oh [15]. Mathematics Subject Classification.
Key words and phrases. singular integrals, multilinear analysis, multi-parameter analysis, weighted esti-mates, commutators.
In the usual one-parameter context of [29] there is a general philosophy that the maxi-mal function controls SIOs T – in fact, we have the concrete estimate k T ( f , . . . , f n ) w k L p . k M ( f , . . . , f n ) w k L p , p > , w p ∈ A ∞ . Thus, the heart of the matter of [29] reduces to the maximal function M ( f , . . . , f n ) = sup I I n Y i =1 h| f i |i I , where h| f i |i I = ffl I | f i | = | I | ´ I | f i | and the supremum is over cubes I ⊂ R d .In this paper we prove genuinely multilinear weighted estimates for multi-parameterSIOs in the product space R d = Q mi =1 R d i . For the classical linear multi-parameter theoryand some of its original applications see e.g. [4, 5, 18, 19, 20, 21, 22, 27]. Multilinear multi-parameter estimates arise naturally in applications whenever a multilinear phenomena,like the fractional Leibniz rules, are combined with product type estimates, such as thosethat arise when we want to take different partial fractional derivatives D αx D βx f . We referto our recent work [34] for a thorough general background on the subject.It is well-known that the multi-parameter SIO theory, even in the linear unweightedsetting, does not reduce to estimates for the maximal function. In fact, it is already known[13] that the multi-parameter maximal function ( f , . . . , f n ) sup R R Q ni =1 h| f i |i R , wherethe supremum is over rectangles R = Q mi =1 I i ⊂ Q mi =1 R d i with sides parallel to the axes,satisfies the desired genuinely multilinear weighted estimates. The corresponding SIOtheory is significantly more involved. In the paper [34] we developed the general theoryof bilinear bi-parameter SIOs including weighted estimates under the more restrictiveassumption w p i i ∈ A p i . In fact, we only reached these weighted estimates without anyadditional cancellation assumptions of the type T in [1].There are no genuinely multilinear weighted estimates for any multi-parameter SIOsin the literature – not even for the bi-parameter analogues (see e.g. [34, Appendix A])of Coifman–Meyer [6] type multilinear multipliers. Almost ten years after the maximalfunction result [13] we establish these missing bounds – not only for some special SIOs– but for a very general class of n -linear m -parameter SIOs. With weighted bounds pre-viously being known both in the linear multi-parameter setting [18, 19, 26] and in themultilinear one-parameter setting [29], we finally establish a holistic view completingthe theory of qualitative weighted estimates in the joint presence of multilinearity andproduct space theory.With the understanding that a Calderón–Zygmund operator (CZO) is an SIO satisfy-ing natural T type assumptions, our main result reads as follows.1.1. Theorem.
Suppose T is an n -linear m -parameter CZO in R d = Q mi =1 R d i . If < p , . . . , p n ≤∞ and /p = P ni =1 /p i > , we have k T ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi , w = n Y i =1 w i , for all n -linear m -parameter weights ~w = ( w , . . . , w n ) ∈ A ~p , ~p = ( p , . . . , p n ) . Here ~w ∈ A ~p if [ ~w ] A ~p := sup R h w p i p R n Y i =1 h w − p ′ i i i p ′ i R < ∞ , ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 3 where the supremum is over all rectangles R ⊂ R d . For the exact definitions, see the main text.Recent extrapolation methods are crucial both for the proof and for the applications.The extrapolation theorem of Rubio de Francia says that if k g k L p ( w ) . k f k L p ( w ) for some p ∈ (1 , ∞ ) and all w ∈ A p , then k g k L p ( w ) . k f k L p ( w ) for all p ∈ (1 , ∞ ) and all w ∈ A p .In [14] (see also [12]) a multivariable analogue was developed in the setting w p i i ∈ A p i , i = 1 , . . . , n . Such extrapolation results are already of fundamental use in proving otherestimates – often just to even deduce the full n -linear range of unweighted estimates Q nj =1 L p j → L p , P j /p j = 1 /p , < p j < ∞ , /n < p < ∞ , from some particular singletuple ( p , . . . , p n , p ) . Indeed, reaching p ≤ can often be a crucial challenge, particularlyso in multi-parameter settings where many other tools are completely missing.Very recently, in [31] it was shown that also the genuinely multilinear weighted esti-mates can be extrapolated. In the subsequent paper [32] (see also [39]) a key advantageof extrapolating using the general weight classes was identified: it is possible to bothstart the extrapolation and, importantly, to reach – as a consequence of the extrapolation– weighted estimates with p i = ∞ . See Theorem 3.12 for a formulation of these generalextrapolation principles. Moreover, extrapolation is flexible in the sense that one canextrapolate both -parameter and m -parameter, m ≥ , weighted estimates.These new extrapolation results are extremely useful e.g. in proving mixed-norm esti-mates – for example, in the bi-parameter case they yield that k T ( f , . . . , f n ) k L p ( R d ; L q ( R d )) . n Y i =1 k f i k L pi ( R d ; L qi ( R d )) , where < p i , q i ≤ ∞ , p = P i p i > and q = P i q i > . The point is that even all ofthe various cases involving ∞ become immediate. See e.g. [11, 32, 34] for some of theprevious mixed-norm estimates. Compared to [32] we can work with completely general n -linear m -parameter SIOs instead of bi-parameter tensor products of -parameter SIOs,and the proof is much simplified due to the optimal weighted estimates, Theorem 1.1.We also use extrapolation to give a new short proof of the boundedness of the multi-parameter n -linear maximal function [13] – see Proposition 4.1.On the technical level there is no existing approach to our result: the modern one-parameter tools (such as sparse domination in the multilinear setting, see e.g. [3]) aremissing and many of the bi-parameter methods [34] used in conjunction with the as-sumption that each weight individually satisfies w p i i ∈ A p i are of little use. Aside frommaximal function estimates, multi-parameter estimates require various square functionestimates (and combinations of maximal function and square function estimates). Simi-larly as one cannot use Q i M f i instead of M ( f , . . . , f n ) due to the nature of the multilin-ear weights, it is also not possible to use classical square function estimates separately forthe functions f i . Now, this interplay makes it impossible to decouple estimates to termslike k M f · w k L p k Sf · w k L p , since neither of them would be bounded separately as w p A p and w p A p . However, such decoupling of estimates has previously seemedalmost indispensable.Our proof starts with the reduction to dyadic model operators [2] (see also [10, 23, 35,37, 40]), which is a standard idea. After this we introduce a family of n -linear multi-parameter square function type objects A k . On the idea level, a big part of the proof KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINEN works by taking a dyadic model operator S and finding an appropriate square function A k so that k S ( f , . . . , f n ) w k L p . k A k ( f , . . . , f n ) w k L p . This requires different tools depending on the model operator in question and is a newway to estimate model operators that respects the n -linear structure fully. We then provethat all of our operators A k satisfy the genuinely n -linear weighted estimates k A k ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi . This is done with an argument that is based on using duality and lower square functionestimates iteratively until all of the cancellation present in these square functions hasbeen exploited.Aside from the full range of mixed-norm estimates, the weighted estimates immedi-ately give other applications as well. We present here a result on commutators, whichgreatly generalises [34]. Commutator estimates appear all over analysis implying e.g.factorizations for Hardy functions [7], certain div-curl lemmas relevant in compensatedcompactness, and were recently connected to the Jacobian problem
J u = f in L p (see[24]). For a small sample of commutator estimates in various other key setting see e.g.[22, 25, 26, 30].1.2. Theorem.
Suppose T is an n -linear m -parameter CZO in R d = Q mi =1 R d i , < p , . . . , p n ≤∞ and /p = P ni =1 /p i > . Suppose also that k b k bmo = sup R | R | ´ R | b − h b i R | < ∞ . Thenfor all ≤ k ≤ n we have the commutator estimate k [ b, T ] k ( f , . . . , f n ) w k L p . k b k bmo n Y i =1 k f i w i k L pi , [ b, T ] k ( f , . . . , f n ) := bT ( f , . . . , f n ) − T ( f , . . . , f k − , bf k , f k +1 , . . . , f n ) , for all n -linear m -parameter weights ~w = ( w , . . . , w n ) ∈ A ~p . Analogous results hold for iteratedcommutators. We note that we can also finally dispose of some of the sparse domination tools thatrestricted some of the theory of [34] to bi-parameter.
Acknowledgements.
K. Li was supported by the National Natural Science Foundationof China through project number 12001400. H. Martikainen and E. Vuorinen were sup-ported by the Academy of Finland through project numbers 294840 (Martikainen) and327271 (Martikainen, Vuorinen), and by the three-year research grant 75160010 of theUniversity of Helsinki. 2. P
RELIMINARIES
Throughout this paper A . B means that A ≤ CB with some constant C that wedeem unimportant to track at that point. We write A ∼ B if A . B . A . Sometimes wee.g. write A . ǫ B if we want to make the point that A ≤ C ( ǫ ) B . ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 5
Dyadic notation.
Given a dyadic grid D in R d , I ∈ D and k ∈ Z , k ≥ , we use thefollowing notation:(1) ℓ ( I ) is the side length of I .(2) I ( k ) ∈ D is the k th parent of I , i.e., I ⊂ I ( k ) and ℓ ( I ( k ) ) = 2 k ℓ ( I ) .(3) ch( I ) is the collection of the children of I , i.e., ch( I ) = { J ∈ D : J (1) = I } .(4) E I f = h f i I I is the averaging operator, where h f i I = ffl I f = | I | ´ I f .(5) ∆ I f is the martingale difference ∆ I f = P J ∈ ch( I ) E J f − E I f .(6) ∆ I,k f is the martingale difference block ∆ I,k f = X J ∈D J ( k ) = I ∆ J f. For an interval J ⊂ R we denote by J l and J r the left and right halves of J , respectively.We define h J = | J | − / J and h J = | J | − / (1 J l − J r ) . Let now I = I × · · · × I d ⊂ R d bea cube, and define the Haar function h ηI , η = ( η , . . . , η d ) ∈ { , } d , by setting h ηI = h η I ⊗ · · · ⊗ h η d I d . If η = 0 the Haar function is cancellative: ´ h ηI = 0 . We exploit notation by suppressingthe presence of η , and write h I for some h ηI , η = 0 . Notice that for I ∈ D we have ∆ I f = h f, h I i h I (where the finite η summation is suppressed), h f, h I i := ´ f h I .2.B. Multi-parameter notation.
We will be working on the m -parameter product space R d = Q mi =1 R d i . We denote a general dyadic grid in R d i by D i . We denote cubes in D i by I i , J i , K i , etc. Thus, our dyadic rectangles take the forms Q mi =1 I i , Q mi =1 J i , Q mi =1 K i etc.We usually denote the collection of dyadic rectangles by D = Q mi =1 D i .If A is an operator acting on R d , we can always let it act on the product space R d by setting A f ( x ) = A ( f ( · , x , . . . , x n ))( x ) . Similarly, we use the notation A i f if A isoriginally an operator acting on R d i . Our basic multi-parameter dyadic operators – mar-tingale differences and averaging operators – are obtained by simply chaining togetherrelevant one-parameter operators. For instance, an m -parameter martingale difference is ∆ R f = ∆ I · · · ∆ mI m f, R = m Y i =1 I i . When we integrate with respect to only one of the parameters we may e.g. write h f, h I i ( x , . . . , x n ) := ˆ R d f ( x , . . . , x n ) h I ( x ) d x or h f i I , ( x , . . . , x n ) := I f ( x , . . . , x n ) d x . Adjoints.
Consider an n -linear operator T on R d = R d × R d . Let f j = f j ⊗ f j , j = 1 , . . . , n + 1 . We set up notation for the adjoints of T in the bi-parameter situation.We let T j ∗ , j ∈ { , . . . , n } , denote the full adjoints, i.e., T ∗ = T and otherwise h T ( f , . . . , f n ) , f n +1 i = h T j ∗ ( f , . . . , f j − , f n +1 , f j +1 , . . . , f n ) , f j i . KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINEN
A subscript or denotes a partial adjoint in the given parameter – for example, wedefine h T ( f , . . . , f n ) , f n +1 i = h T j ∗ ( f , . . . , f j − , f n +1 ⊗ f j , f j +1 , . . . , f n ) , f j ⊗ f n +1 i . Finally, we can take partial adjoints with respect to different parameters in different slotsalso – in that case we denote the adjoint by T j ∗ ,j ∗ , . It simply interchanges the functions f j and f n +1 and the functions f j and f n +1 . Of course, we e.g. have T j ∗ ,j ∗ , = T j ∗ and T ∗ ,j ∗ , = T j ∗ , so everything can be obtained, if desired, with the most general notation T j ∗ ,j ∗ , . In any case, there are ( n + 1) adjoints (including T itself). Similarly, the bi-parameter dyadic model operators that we later define always have ( n + 1) differentforms. These notions have obvious extensions to m -parameters.2.D. Structure of the paper.
To avoid unnecessarily complicating the notation, we startby proving everything in the bi-parameter case m = 2 . Importantly, we present a proofwhich does not exploit this in a way that would not be extendable to m -parameters (e.g.,our proof for the partial paraproducts does not exploit sparse domination for the ap-pearing one-parameter paraproducts). At the end, we demonstrate for some key modeloperators how the general case can be dealt with.3. M ULTILINEAR BI - PARAMETER WEIGHTS
The following notions have an obvious extension to m -parameters. A weight w ( x , x ) (i.e. a locally integrable a.e. positive function) belongs to the bi-parameter weight class A p = A p ( R d × R d ) , < p < ∞ , if [ w ] A p := sup R h w i R h w − p ′ i p − R = sup R h w i R h w − p − i p − R < ∞ , where the supremum is taken over rectangles R – that is, over R = I × I where I i ⊂ R d i is a cube. Thus, this is the one-parameter definition but cubes are replaced by rectangles.We have(3.1) [ w ] A p ( R d × R d ) < ∞ iff max (cid:0) ess sup x ∈ R d [ w ( x , · )] A p ( R d ) , ess sup x ∈ R d [ w ( · , x )] A p ( R d ) (cid:1) < ∞ , and that max (cid:0) ess sup x ∈ R d [ w ( x , · )] A p ( R d ) , ess sup x ∈ R d [ w ( · , x )] A p ( R d ) (cid:1) ≤ [ w ] A p ( R d × R d ) , while the constant [ w ] A p is dominated by the maximum to some power. It is also usefulthat h w i I , ∈ A p ( R d ) uniformly on the cube I ⊂ R d . For basic bi-parameter weightedtheory see e.g. [26]. We say w ∈ A ∞ ( R d × R d ) if [ w ] A ∞ := sup R h w i R exp (cid:0) h log w − i R (cid:1) < ∞ . It is well-known that A ∞ ( R d × R d ) = [
We introduce the classes of multilinear Muckenhoupt weights that we will use.3.2.
Definition.
Given ~p = ( p , . . . , p n ) with ≤ p , . . . , p n ≤ ∞ we say that ~w =( w , . . . , w n ) ∈ A ~p = A ~p ( R d × R d ) , if < w i < ∞ , i = 1 , . . . , n, almost everywhere and [ ~w ] A ~p := sup R h w p i p R n Y i =1 h w − p ′ i i i p ′ i R < ∞ , where the supremum is over rectangles R , w := n Y i =1 w i and p = n X i =1 p i . If p i = 1 we interpret h w − p ′ i i i p ′ i R as ess sup R w − i , and if p = ∞ we interpret h w p i p R as ess sup R w .3.3. Remark. (1) It is important that the lower bound(3.4) h w p i p R n Y i =1 h w − p ′ i i i p ′ i R ≥ holds always. To see this recall that for α , α > we have by Hölder’s inequalitythat ≤ h w − α i α R h w α i α R . (3.5) Apply this with α = p and α = n − p . Then apply Hölder’s inequality with theexponents u i = (cid:16) n − p (cid:17) p ′ i to get D(cid:16) Q ni =1 w i (cid:17) − n − p E n − p R ≤ Q ni =1 h w − p ′ i i i p ′ i R .(2) Our definition is essentially the usual one-parameter definition [29] with the dif-ference that cubes are replaced by rectangles. However, we are also using therenormalised definition from [32] that works better with the exponents p i = ∞ .Compared to the usual formulation of [29] the relation is that [ w p , · · · , w p n n ] p A ~p with A ~p defined as in [29] agrees with our [ ~w ] A ~p when p i < ∞ .(3) The case p = · · · = p n = ∞ = p can be used as the starting point of extrapolation.This is rarely useful but we will find use for it when we consider the multilinearmaximal function.The following characterization of the class A ~p is convenient. The one-parameter re-sult with the different normalization is [29, Theorem 3.6]. We record the proof for theconvenience of the reader.3.6. Lemma.
Let ~p = ( p , . . . , p n ) with ≤ p , . . . , p n ≤ ∞ , /p = P ni =1 /p i ≥ , ~w =( w , . . . , w n ) and w = Q ni =1 w i . We have [ w − p ′ i i ] A np ′ i ≤ [ ~w ] p ′ i A ~p , i = 1 , . . . , n, KANGWEI LI, HENRI MARTIKAINEN, AND EMIL VUORINEN and [ w p ] A np ≤ [ ~w ] pA ~p . In the case p i = 1 the estimate is interpreted as [ w n i ] A ≤ [ ~w ] /nA ~p , and in the case p = ∞ we have [ w − n ] A ≤ [ ~w ] /nA ~p .Conversely, we have [ ~w ] A ~p ≤ [ w p ] p A np n Y i =1 [ w − p ′ i i ] p ′ i A np ′ i . Proof.
We fix an arbitrary j ∈ { , . . . , n } for which we will show [ w − p ′ j j ] A np ′ j ≤ [ ~w ] p ′ j A ~p .Notice that(3.7) p + X i = j p ′ i = n − p j . We define q j via the identity q j = 1 n − p j · p and for i = j we set q i = 1 n − p j · p ′ i . From (3.7) we have that P i q i = 1 . By definition we have(3.8) [ w − p ′ j j ] A np ′ j = sup R h w − p ′ j j i R h w p ′ j np ′ j − j i np ′ j − R . Notice that p ′ j np ′ j − n − p ′ j = 1 n − p j . Using Hölder’s inequality with the exponents q , . . . , q n we have the desired estimate h w − p ′ j j i p ′ j R h w pqj j i qjp R = h w − p ′ j j i p ′ j R h w pqj Y i = j w − pqj i i qjp R ≤ h w p i p R Y i h w − p ′ i i i p ′ i R ≤ [ ~w ] A ~p . When p j = 1 this is ess sup R w − j h w n j i nR ≤ [ ~w ] A ~p , and so [ w n j ] nA ≤ [ ~w ] A ~p .We now move on to bounding [ w p ] A np . Notice that by definition(3.9) [ w p ] A np = sup R h w p i R h w − pnp − i np − R . We define s i via − pnp − · s i = − p ′ i and notice that then P i s i = 1 . Then, by Hölder’s inequality with the exponents s , . . . , s n we have h w p i R h w − pnp − i np − R ≤ h w p i R Y i h w − p ′ i i i (cid:0) pnp − (cid:1) p ′ i ( np − R = h h w p i p R Y i h w − p ′ i i i p ′ i R i p ≤ [ ~w ] pA ~p , ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 9 which is the desired bound for [ w p ] A np . Notice that in the case p = ∞ we get [ w − n ] nA = sup R (cid:10) w − n (cid:11) nR ess sup R w ≤ sup R h Y i h w − i i R i ess sup R w = [ ~w ] A ~p . We then move on to bounding [ ~w ] A ~p . It is based on the following inequality(3.10) ≤ h w − pnp − i n − p R Y i D w n −
1+ 1 pi i E n − pi R . Before proving this, we show how it implies the desired bound. We have [ ~w ] A ~p = sup R h w p i p R Y i h w − p ′ i i i p ′ i R ≤ sup R h h w p i R h w − pnp − i np − R i p Y i h h w − p ′ i i i R (cid:10) w p ′ i np ′ i − i (cid:11) np ′ i − R i p ′ i ≤ [ w p ] p A np Y i [ w − p ′ i i ] p ′ i A np ′ i , where in the last estimate we recalled (3.8) and (3.9).Let us now give the details of (3.10). We apply (3.5) with α = pnp − and α = n ( n − p to get ≤ h w − pnp − i n − p R D w n ( n − p E n ( n − p R . The first term is already as in (3.10). Define u i via n ( n −
1) + p u i = 1 n − p i and notice that by Hölder’s inequality with these exponents ( P i u i = 1 ) we have D w n ( n − p E n ( n − p R ≤ Y i D w n −
1+ 1 pi i E n − pi R , which matches the second term in (3.10). (cid:3) The following duality of multilinear weights is handy – see [36, Lemma 3.1]. We givethe short proof for convenience.3.11.
Lemma.
Let ~p = ( p , . . . , p n ) with < p , . . . , p n < ∞ and p = P ni =1 1 p i ∈ (0 , . Let ~w = ( w , . . . , w n ) ∈ A ~p with w = Q ni =1 w i and define ~w i = ( w , . . . , w i − , w − , w i +1 , . . . , w n ) ,~p i = ( p , . . . , p i − , p ′ , p i +1 , . . . , p n ) . Then we have [ ~w i ] A ~p i = [ ~w ] A ~p . Proof.
We take i = 1 for notational convenience. Notice that p ′ + P ni =2 1 p i = p ′ . Noticealso that w − Q ni =2 w i = w − . Therefore, we have [ ~w i ] A ~p i = h w − p ′ i p ′ R h w p i p R n Y i =2 h w − p ′ i i i p ′ i R = [ ~w ] A ~p . (cid:3) We now recall the recent extrapolation result of [32]. The previous version, which didnot yet allow exponents to be ∞ appeared in [31]. For related independent work see [39].The previous extrapolation results with the separate assumptions w p i i ∈ A p i appear in[14] and [12]. An even more general result than the one below appears in [32], but wewill not need that generality here. Finally, we note that the proof of this extrapolationresult can be made to work in m -parameters even though [32] provides the details onlyin the one-parameter case – we give more details later in Section 8.3.12. Theorem.
Let f , . . . , f n and g be given functions. Given ~p = ( p , . . . , p n ) with ≤ p , . . . , p n ≤ ∞ let p = P ni =1 1 p i . Assume that given any ~w = ( w , . . . , w n ) ∈ A ~p the inequality (3.13) k gw k L p . n Y i =1 k f i w i k L pi holds, where w := Q ni =1 w i . Then for all exponents ~q = ( q , . . . , q n ) , with < q , . . . , q n ≤ ∞ and q = P ni =1 1 q i > , and for all weights ~v = ( v , . . . , v n ) ∈ A ~q the inequality k gv k L q . n Y i =1 k f i v i k L qi holds, where v := Q ni =1 v i .Given functions f j , . . . , f jn and g j so that (3.13) holds uniformly on j , we have for the samefamily of exponents and weights as above, and for all exponents ~s = ( s , . . . , s n ) with the inequality (3.14) k ( g j v ) j k L q ( ℓ s ) . n Y i =1 k ( f ji v i ) j k L qi ( ℓ si ) . Remark.
Using Lemma 3.11 and extrapolation, Theorem 3.12, we see that the weightedboundedness of T transfers to the adjoints T j ∗ . Partial adjoints have to always be con-sidered separately, though.As a final thing in this section, we demonstrate the necessity of the A ~p condition forthe weighted boundedness of SIOs. We work in the m -parameter setting and let R d = R d × · · · × R d n . Let R j be the following version of the n -linear one-parameter Riesztransform in R d j : R j ( f , . . . , f n ) = p.v. ˆ R djn P ni =1 P d j k =1 ( x − y i ) k ( P ni =1 | x − y i | ) d j n +1 f ( y ) · · · f n ( y n ) d y · · · d y n , where ( x − y i ) k is the k -th coordinate of x − y i ∈ R d j . Consider the tensor product R ⊗ R ⊗ · · · ⊗ R m . Let ~w = ( w , . . . , w n ) be a multilinear weight, that is, < w i < ∞ a.e., and denote w = Q ni =1 w i . Suppose that for some exponents < p , . . . , p n ≤ ∞ with /p = P ni =1 /p i > the estimate k R ⊗ R ⊗ · · · ⊗ R m ( f , . . . , f n ) k L p, ∞ ( w p ) . n Y i =1 k f i w i k L pi holds for all f i ∈ L ∞ c . We show that ~w is an m -parameter A ~p weight. ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 11
Define σ i = w − p ′ i i . Let E ⊂ R d be an arbitrary set such that E σ i ∈ L ∞ c for all i =1 , . . . , n . Fix an m -parameter rectangle R = R × · · · × R m ⊂ R d , where each R j is a cube.Let R + = ( R ) + × · · · × ( R m ) + , where ( R j ) + := R j + ( ℓ ( R j ) , . . . , ℓ ( R j )) .Using the kernel of R ⊗ · · · ⊗ R m we have for all x ∈ R + that R ⊗ R ⊗ · · · ⊗ R m (1 E σ R , . . . , E σ n R )( x ) & n Y i =1 h E σ i i R . Hence w p ( R + ) p n Y i =1 h E σ i i R . n Y i =1 k E σ i R w i k L pi = n Y i =1 σ i ( E ∩ R ) pi , which gives that h w p i p R + Q ni =1 h E σ i i p ′ i R . . Since E was arbitrary this implies the esti-mate(3.16) h w p i p R + n Y i =1 h σ i i p ′ i R . . Similarly, we can show that(3.17) h w p i p R n Y i =1 h σ i i p ′ i R + . . By Hölder’s inequality we have that h w − pnp − i np − p R + ≤ n Y i =1 h σ i i p ′ i R + . Hence, (3.17) shows that h w p i p R h w − pnp − i np − p R + . . Therefore, h w p i p R h w p i p R + = h w p i p R h w − pnp − i np − p R + h w p i p R + h w − pnp − i np − p R + . , where the denominator in the middle term was ≥ . Thus, h w p i p R . h w p i p R + , whichtogether with (3.16) gives that h w p i p R Q ni =1 h σ i i p ′ i R . .
4. M
AXIMAL FUNCTIONS
It was proved in [13] that the multilinear bi-parameter (or multi-parameter) maximalfunction is bounded with respect to the genuinely multilinear bi-parameter weights. Wegive a new efficient proof of this. Let D = D × D be a fixed lattice of dyadic rectanglesand define M D ( f , . . . , f n ) = sup R ∈D n Y i =1 h| f i |i R R . Proposition. If < p , . . . , p n ≤ ∞ and /p = P ni =1 /p i we have k M D ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi for all multilinear bi-parameter weights ~w ∈ A ~p .Proof. Our proof is based on the proof of the case ~p = ( p , . . . , p n ) = ( ∞ , . . . , ∞ ) andextrapolation, Theorem 3.12. We have sup R h Y i h w − i i R i · ess sup R w = [ ~w ] A ~p , and therefore Y i h w − i i R . R w . For every R ∈ D let N R ⊂ R be such that | N R | = 0 and w ( x ) ≤ ess sup R w for all x ∈ R \ N R . Let N = S R ∈D N R . Then | N | = 0 and for every x ∈ R d \ N we have w ( x ) ≥ sup R ∈D R ( x )ess sup R w . Thus, we have M D ( f , . . . , f n )( x ) w ( x ) ≤ h Y i k f i w i k L ∞ i sup R ∈D h R ( x ) Y i h w − i i R i · w ( x ) . h Y i k f i w i k L ∞ i sup R ∈D h R ( x )ess sup R w i · w ( x ) ≤ Y i k f i w i k L ∞ almost everywhere, and so k M D ( f , . . . , f n ) w k L ∞ . Q i k f i w i k L ∞ as desired. (cid:3) If an average is with respect to a different measure µ than the Lebesgue measure, wewrite h f i µR := µ ( R ) ´ R f d µ and define M µ D f = sup R R h| f |i µR . The following is a result of R. Fefferman [20]. Recently, we also recorded a proof in [33,Appendix B].4.2.
Proposition.
Let λ ∈ A p , p ∈ (1 , ∞ ) , be a bi-parameter weight. Then for all s ∈ (1 , ∞ ) wehave k M λ D f k L s ( λ ) . [ λ ] /sA p k f k L s ( λ ) . We will formulate some vector-valued versions of Proposition 4.2 later – see Proposi-tion 8.5. We will have use for them. Everything in this section works easily in the generalmulti-parameter situation.
ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 13
5. S
QUARE FUNCTIONS
Let D = D × D be a fixed lattice of dyadic rectangles. We define the square functions S D f = (cid:16) X R ∈D | ∆ R f | (cid:17) / , S D f = (cid:16) X I ∈D | ∆ I f | (cid:17) / and define S D f analogously.The following lower square function estimate valid for A ∞ weights is important forus. The importance comes from the fact that by Lemma 3.6 some of the key weights w p and w − p ′ i i are at least A ∞ for the multilinear weights of Definition 3.2.5.1. Lemma.
There holds k f k L p ( w ) . k S i D i f k L p ( w ) . k S D f k L p ( w ) for all p ∈ (0 , ∞ ) and bi-parameter weights w ∈ A ∞ . For a proof of the one-parameter estimate see [41, Theorem 2.5]. The bi-parameterresults can be deduced using the following extremely useful A ∞ extrapolation result [8].5.2. Lemma.
Let ( f, g ) be a pair of non-negative functions. Suppose that there exists some < p < ∞ such that for every w ∈ A ∞ we have ˆ f p w . ˆ g p w. Then for all < p < ∞ and w ∈ A ∞ we have ˆ f p w . ˆ g p w. Remark.
We often use the lower square function estimate with the additional obser-vation that we e.g. have for all k = ( k , k ) ∈ { , , . . . } that S D f = (cid:16) X K = K × K ∈D | ∆ K,k f | (cid:17) / , ∆ K,k = ∆ K ,k ∆ K ,k . This simply follows from disjointness.For k = ( k , k ) we define the following family of n -linear square functions. First, weset A ( f , . . . , f n ) = A ,k ( f , . . . , f n ) = (cid:16) X K ∈D h| ∆ K,k f |i K n Y j =2 h| f j |i K K (cid:17) . In addition, we understand this so that A ,k can also take any one of the symmetric forms,where each ∆ iK i ,k i appearing in ∆ K,k = ∆ K ,k ∆ K ,k can alternatively be associatedwith any of the other functions f , . . . , f n . That is, A ,k can, for example, also take theform A ,k ( f , . . . , f n ) = (cid:16) X K ∈D h| ∆ K ,k f |i K h| ∆ K ,k f |i K n Y j =3 h| f j |i K K (cid:17) . For k = ( k , k , k ) we define A ,k ( f , . . . , f n )= (cid:16) X K ∈D (cid:16) X K ∈D h| ∆ K ,k f |i K h| ∆ K ,k f |i K h| ∆ K ,k f |i K n Y j =4 h| f j |i K K (cid:17) (cid:17) , (5.4)where we again understand this as a family of square functions. First, the appearingthree martingale blocks can be associated with different functions, too. Second, we canhave the K summation out and the K summation in (we can interchange them), butthen we have two martingale blocks with K and one martingale block with K .Finally, for k = ( k , k , k , k ) we define A ,k ( f , . . . , f n ) = X K ∈D h| ∆ K, ( k ,k ) f |i K h| ∆ K, ( k ,k ) f |i K n Y j =3 h| f j |i K K , where this is a family with two martingale blocks in each parameter, which can be movedaround.5.5. Theorem. If < p , . . . , p n ≤ ∞ and p = P ni =1 1 p i > we have k A j,k ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi , j = 1 , , , for all multilinear bi-parameter weights ~w ∈ A ~p .Proof. The proofs of all of the cases have the same underlying idea based on an iterativeuse of duality and the lower square function estimate until all of the cancellation hasbeen utilised. One can also realise that the result for A ,k follows using the above schemejust once if the result is first proved for A ,k and A ,k .We show the proof for A ,k with the explicit form (5.4). Fix some ~p = ( p , . . . , p n ) with < p i < ∞ and p > . This is enough by extrapolation, Theorem 3.12. Toestimate k A ,k ( f , . . . , f n ) w k L p we take a sequence ( f n +1 ,K ) K ⊂ L p ′ ( ℓ ) with a norm k ( f n +1 ,K ) K k L p ′ ( ℓ ) ≤ and look at(5.6) X K h| ∆ K ,k f | , K ih| ∆ K ,k f |i K h| ∆ K ,k f |i K n Y j =4 h| f j |i K h f n +1 ,K w i K . There holds that(5.7) h| ∆ K ,k f | , K i = h ∆ K ,k f , ϕ K ,f i = h f , ∆ K ,k ϕ K ,f i , | ϕ K ,f | ≤ K . We now get that (5.6) is less than k f w k L p multiplied by (cid:13)(cid:13)(cid:13) X K h f n +1 ,K w i K h| ∆ K ,k f |i K h| ∆ K ,k f |i K n Y j =4 h| f j |i K ∆ K ,k ϕ K ,f w − (cid:13)(cid:13)(cid:13) L p ′ . We will now apply the lower square function estimate k gw − k L p ′ . k S D ( g ) w − k L p ′ ,Lemma 5.1, with the weight w − p ′ ∈ A ∞ (see Lemma 3.6). Here we use the block form of ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 15
Remark 5.3. Using also that | ∆ K ,k ϕ K ,f | . K we get that the last norm is dominatedby (cid:13)(cid:13)(cid:13)(cid:16) X K (cid:16) X K h| f n +1 ,K | w i K h| ∆ K ,k f |i K h| ∆ K ,k f |i K n Y j =4 h| f j |i K K (cid:17) (cid:17) w − (cid:13)(cid:13)(cid:13) L p ′ . We still have cancellation to use in the form of the other two martingale differences andwill continue the process.We repeat the argument from above – this gives that the previous term is dominatedby k f w k L p multiplied by (cid:13)(cid:13)(cid:13)(cid:16) X K (cid:16) X K h| f n +1 ,K | w i K h| f ,K | w − i K h| ∆ K ,k f |i K n Y j =4 h| f j |i K K (cid:17) (cid:17) w − (cid:13)(cid:13)(cid:13) L p ′ where k ( f ,K ) K k L p ( ℓ ) ≤ . Running this argument one more time finally gives us thatthis is dominated by k f w k L p multiplied by (cid:13)(cid:13)(cid:13)(cid:16) X K (cid:16) X K h| f n +1 ,K | w i K h| f ,K | w − i K h| f ,K | w − i K n Y j =4 h| f j |i K K (cid:17) (cid:17) w − (cid:13)(cid:13)(cid:13) L p ′ ≤ (cid:13)(cid:13)(cid:13)(cid:16) X K (cid:16) X K M D ( f n +1 ,K w, f ,K w − , f ,K w − , f , . . . , f n ) (cid:17) (cid:17) w − (cid:13)(cid:13)(cid:13) L p ′ , where k ( f ,K ) K k L p ( ℓ ) ≤ .Using Lemma 3.11 three times (we dualized three times) shows that ( w − , w , w , w , . . . , w n ) ∈ A ( p ′ ,p ,p ,p ,...,p n ) . The maximal function satisfies the weighted L p ′ ( ℓ ∞ K ( ℓ K )) × L p ( ℓ ∞ K ( ℓ K )) × L p ( ℓ K ( ℓ ∞ K )) × L p × · · · × L p n → L p ′ ( ℓ K ( ℓ K )) estimate. This gives that the last norm above is dominated by k ( f n +1 ,K ww − ) K k L p ′ ( ℓ ) k ( f ,K w − w ) K k L p ( ℓ ) k ( f ,K w − w ) K k L p ( ℓ ) n Y i =4 k f i w i k L pi , where the first three norms are ≤ . This concludes the proof for A ,k and the rest of thecases are similar. (cid:3) We also record some linear estimates. We will need these when we deal with the mostcomplicated model operators – the partial paraproducts.5.8.
Proposition.
For u ∈ A ∞ and p, s ∈ (1 , ∞ ) we have (cid:13)(cid:13)(cid:13)h X m (cid:16) X K ∈D h| ∆ K,k f m |i K K h u i K (cid:17) s i s u p (cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:16) X m | f m | s (cid:17) s u − p ′ (cid:13)(cid:13)(cid:13) L p . Proof.
By (3.5) we have for all n ≥ that ≤ h u i K D u − n − E n − K . Simply using this we reduce to (cid:13)(cid:13)(cid:13)h X m (cid:16) X K ∈D h| ∆ K,k f m |i K D u − n − E n − K K (cid:17) s i s u p (cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13)h X m A ,k (cid:0) f m , u − n − , . . . , u − n − (cid:1) s i s u p (cid:13)(cid:13)(cid:13) L p , where A ,k is a suitable square function as in Theorem 5.5.We then fix n large enough so that u ∈ A n . We then notice that this implies that(5.9) (cid:0) u − p ′ , u n − , . . . , u n − (cid:1) ∈ A ( p, ∞ ,..., ∞ ) . To see this, notice that the target weight associated with this tuple is u − p ′ u = u p and thatthe target exponent is p , and so (cid:2)(cid:0) u − p ′ , u n − , . . . , u n − (cid:1)(cid:3) A ( p, ∞ ,..., ∞ ) = sup R h u i /pR h u i /p ′ R (cid:10) u − n − (cid:11) n − R = [ u ] A n < ∞ . It remains to use the weighted (with the weight (5.9)) vector-valued estimate L p ( ℓ s ) × L ∞ × · · · × L ∞ → L p ( ℓ s ) of A ,k , which follows by Theorem 5.5 and (3.14). (cid:3) Remark.
It is possible to prove the above proposition also directly with the dualityand lower square function strategy that was used in the proof of Theorem 5.5.6. D
YADIC MODEL OPERATORS
In this section we are working with a fixed set of dyadic rectangles D = D × D . Allthe model operators depend on this lattice, but it is not emphasised in the notation.6.A. Shifts.
Let k = ( k , . . . , k n +1 ) , where k j = ( k j , k j ) ∈ { , , . . . } . An n -linear bi-parameter shift S k takes the form h S k ( f , . . . , f n ) , f n +1 i = X K X R ,...,R n +1 R ( kj ) j = K a K, ( R j ) n +1 Y j =1 h f j , e h R j i . Here
K, R , . . . , R n +1 ∈ D = D × D , R j = I j × I j , R ( k j ) j := ( I j ) ( k j ) × ( I j ) ( k j ) and e h R j = e h I j ⊗ e h I j . Here we assume that for m ∈ { , } there exist two indices j m , j m ∈{ , . . . , n + 1 } , j m = j m , so that e h I mjm = h I mjm , e h I mjm = h I mjm and for the remaining indices j
6∈ { j m , j m } we have e h I mj ∈ { h I mj , h I mj } . Moreover, a K, ( R j ) = a K,R ,...,R n +1 is a scalarsatisfying the normalization(6.1) | a K, ( R j ) | ≤ Q n +1 j =1 | R j | / | K | n . Theorem.
Suppose S k is an n -linear bi-parameter shift, < p , . . . , p n ≤ ∞ and p = P ni =1 1 p i > . Then we have k S k ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 17 for all multilinear bi-parameter weights ~w ∈ A ~p . The implicit constant does not depend on k .Proof. We use duality to always reduce to one of the operators of type A in Theorem 5.5.Performing the proof like this has the advantage that the form of the shift really playsno role – it just affects which type of A operator we get. For example, we consider theexplicit case S k ( f , . . . , f n ) = X K A K ( f , . . . , f n ) , where A K ( f , . . . , f n ) = X R ,...,R n +1 R ( kj ) j = K a K, ( R j ) h f , h R i n Y j =2 h f j , e h R j i h R n +1 . Fix some ~p = ( p , . . . , p n ) with < p i < ∞ and p > , which is enough by extrapola-tion. We will dualise using f n +1 with k f n +1 w − k L p ′ ≤ . The normalization of the shiftcoefficients gives the direct estimate X K |h A K ( f , . . . , f n ) , f n +1 i| ≤ (cid:13)(cid:13)(cid:13) X K h| ∆ K,k f |i K n Y j =2 h| f j |i K h| ∆ K,k n +1 f n +1 |i K K (cid:13)(cid:13)(cid:13) L . Notice that(6.3) ( w , · · · , w n , w − ) ∈ A ( p , ··· ,p n ,p ′ ) , w = n Y i =1 w i . The target weight associated to this data is ww − = 1 and the target exponent is /p +1 /p ′ = 1 . By using Theorem 5.5 with a suitable A ( f , . . . , f n +1 ) and the above weight wecan directly dominate this by h n Y i =1 k f i w i k L pi i · k f n +1 w − k L p ′ ≤ n Y i =1 k f i w i k L pi . We are done. (cid:3)
Partial paraproducts.
Let k = ( k , . . . , k n +1 ) , where k j ∈ { , , . . . } . An n -linear bi-parameter partial paraproduct ( Sπ ) k with the paraproduct component on R d takes theform(6.4) h ( Sπ ) k ( f , . . . , f n ) , f n +1 i = X K = K × K X I ,...,I n +1 ( I j ) ( kj ) = K a K, ( I j ) n +1 Y j =1 h f j , e h I j ⊗ u j,K i , where the functions e h I j and u j,K satisfy the following. There are j , j ∈ { , . . . , n + 1 } , j = j , so that e h I j = h I j , e h I j = h I j and for the remaining indices j
6∈ { j , j } we have e h I j ∈ { h I j , h I j } . There is j ∈ { , . . . , n + 1 } so that u j ,K = h K and for the remaining indices j = j we have u j,K = K | K | . Moreover, the coefficients are assumed to satisfy(6.5) k ( a K, ( I j ) ) K k BMO = sup K ∈D (cid:16) | K | X K ⊂ K | a K, ( I j ) | (cid:17) / ≤ Q n +1 j =1 | I j | | K | n . Of course, ( πS ) k is defined symmetrically.The following H - BMO duality type estimate is well-known and elementary:(6.6) X K | a K || b K | . k ( a K ) k BMO (cid:13)(cid:13)(cid:13)(cid:16) X K | b K | K | K | (cid:17) / (cid:13)(cid:13)(cid:13) L . Such estimates have natural multi-parameter analogues also, and the proofs in all pa-rameters are analogous. See e.g. [38, Equation (4.1)].Our result for the partial paraproducts has a significantly more difficult proof than forthe other model operators. It is also more inefficient in that is produces an exponential– although crucially with an arbitrarily small exponent – dependence on the complexity.This has some significance for the required kernel regularity of CZOs, but a standard t t α type continuity modulus will still suffice.6.7. Theorem.
Suppose ( Sπ ) k is an n -linear partial paraproduct, < p , . . . , p n ≤ ∞ and p = P ni =1 1 p i > . Then, for every < β ≤ we have k ( Sπ ) k ( f , . . . , f n ) w k L p . β max j k j β n Y i =1 k f i w i k L pi for all multilinear bi-parameter weights ~w ∈ A ~p .Proof. Recall that ( Sπ ) k is of the form (6.4). Recall also the indices j and j , which saythat e h I j = h I j at least for j ∈ { j , j } , and the index j , which specifies the place of h K in the second parameter. It makes no difference for the argument what the indices j and j are, so we assume that j = 1 and j = 2 . It makes a small difference whether j ∈ { j , j } or j
6∈ { j , j } , so we do not specify j yet. To make the following formulaeshorter we write e h I j for every j but keep in mind that these are cancellative at least for j ∈ { , } . We define A K ( g , . . . , g n +1 ) = n +1 Y j =1 h g j , u j,K i and write ( Sπ ) k in the form h ( Sπ ) k ( f , . . . , f n ) , f n +1 i = X K = K × K X I ,...,I n +1 ( I j ) ( kj ) = K a K, ( I j ) A K ( h f , e h I i , . . . , h f n +1 , e h I n +1 i ) . Fix some ~p = ( p , . . . , p n ) with < p i < ∞ and p > , which is enough by extrapola-tion. We will dualise using f n +1 with k f n +1 w − k L p ′ ≤ . We may assume f j ∈ L ∞ c . The ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 19 H - BMO duality (6.6) gives that |h ( Sπ ) k ( f , . . . , f n ) , f n +1 i| . X K X I ,...,I n +1 ( I j ) ( kj ) = K " Q n +1 j =1 | I j | | K | n ˆ R d (cid:16) X K | A K ( h f , e h I i , . . . , h f n +1 , e h I n +1 i ) | K | K | (cid:17) . (6.8)Suppose j ∈ { , . . . , n + 1 } is such that e h I j = h I j and k j > , that is, we have non-cancellative Haar functions and non-zero complexity. We expand | I j | − h f j , h I j i = h f i I j , = h f j i K , + k j X i j =1 h ∆ I j ) ( ij ) f j i ( I j ) ( ij − , . For convenience, we further write that h ∆ I j ) ( ij ) f j i ( I j ) ( ij − , = h h ( I j ) ( ij ) i ( I j ) ( ij − h f j , h ( I j ) ( ij ) i , where we are suppressing the summation over the d − different Haar functions. Weperform these expansions inside the operators A K , and take the sums out of the ℓ K norm. This gives that the right hand side of (6.8) is less than a sum of at most Q nj =3 (1+ k j ) terms of the form X K X I ,...,I n +1 ( I j ) ( kj ) = K " Q n +1 j =1 | I j || ( I j ) ( i j ) | − | K | n ˆ R d (cid:16) X K | A K ( h f , e h ( I ) ( i i , . . . , h f n +1 , e h ( I n +1 ) ( in +1) i ) | K | K | (cid:17) . Here we have the following properties. If j in an index such that we did not do theexpansion related to j , then i j = 0 . Thus, at least i = i = 0 . We also remind that e h ( I j ) ( ij ) = h ( I j ) ( ij ) for j = 1 , . If i j < k j , then e h ( I j ) ( ij ) = h ( I j ) ( ij ) . If i j = k j , then e h ( I j ) ( ij ) ∈ { h K , h K } . We can further rewrite this as(6.9) X K X L ,...,L n +1 ( L j ) ( lj ) = K Q n +1 j =1 | L j | | K | n ˆ R d (cid:16) X K | A K ( h f , e h L i , . . . , h f n +1 , e h L n +1 i ) | K | K | (cid:17) . This is otherwise analogous to the right hand side of (6.8) except for the key differencethat if a non-cancellative Haar function appears, then the related complexity is zero.We turn to estimate (6.9). We show that(6.10) (6.9) . β max j k j β h n Y j =1 k f j w j k L pj i k f n +1 w − k L p ′ . Recalling that k f n +1 w − k L p ′ ≤ this implies that the left hand side of (6.8) satisfies LHS (6.8) . β (1 + max j k j ) n − max j k j β n Y j =1 k f j w j k L pj . β max j k j β n Y j =1 k f j w j k L pj , which proves the theorem.Let ( v , . . . , v n +1 ) ∈ A (2 ,..., and v = Q n +1 j =1 v j . We will prove the ( n + 1) -linear estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X K X L ,...,L n +1 ( L j ) ( lj ) = K " Q n +1 j =1 | L j | | K | n K | K | (cid:16) X K | A K ( h f , e h L i , . . . , h f n +1 , e h L n +1 i ) | K | K | (cid:17) v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L n +1 . max j k j β n +1 Y j =1 k f j v j k L . (6.11)Extrapolation, Theorem 3.12, then gives that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X K X L ,...,L n +1 ( L j ) ( lj ) = K Q n +1 j =1 | L j | | K | n K | K | (cid:16) X K | A K ( h f , e h L i , . . . , h f n +1 , e h L n +1 i ) | K | K | (cid:17) v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q . max j k j β n +1 Y j =1 k f j v j k L qj for all q , . . . , q n +1 ∈ (1 , ∞ ] such that q = P n +1 j =1 1 q j > and for all ( v , . . . , v n +1 ) ∈ A ( q ,...,q n +1 ) . Applying this with the exponent tuple ( p , . . . , p n , p ′ ) and the weight tuple ( w , . . . , w n , w − ) ∈ A ( p ,...,p n ,p ′ ) gives (6.10).It remains to prove (6.11). We denote σ j = v − j . The A (2 ,..., condition gives that h v n +1 i n +1 K n +1 Y j =1 h σ j i K . . Using this we have | A K ( h f , e h L i , . . . , h f n +1 , e h L n +1 i ) | . h v n +1 i n +1 K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A K h f , e h L i h σ i K , . . . , h f n +1 , e h L n +1 i h σ n +1 i K !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For the moment we abbreviate the last | A K ( · · · ) | as c K, ( L j ) . There holds that h v n +1 i n +1 K c K, ( L j ) = " h v n +1 i K D c n +1 K, ( L j ) K v − n +1 v n +1 E K n +1 ≤ (cid:16) M v n +1 D (cid:16) c n +1 K, ( L j ) K v − n +1 (cid:17) ( x ) (cid:17) n +1 for all x ∈ K . ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 21
We substitute this into the left hand side of (6.11). This gives that the term there isdominated by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X K X L ,...,L n +1 ( L j ) ( lj ) = K Q n +1 j =1 | L j | | K | n +1 (cid:16) X K M v n +1 D (cid:16) c n +1 K, ( L j ) K v − n +1 (cid:17) n +1) | K | (cid:17) v (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L n +1 . We use the L ( ℓ n +1 ( ℓ n +1) ) boundedness of the maximal function M v n +1 D , see Proposi-tion 8.5. This gives that the last norm is dominated by(6.12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X K X L ,...,L n +1 ( L j ) ( lj ) = K Q n +1 j =1 | L j | | K | n +1 K (cid:16) X K c K, ( L j ) K | K | (cid:17) v − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L n +1 . Now, we recall what the numbers c K, ( L j ) are. At this point it becomes relevant whichof the Haar functions e h L j are cancellative and what is the form of the operators A K . Weassume that e h L j = h L j for j = 1 , . . . , n and e h L n +1 = h L n +1 = h K n +1 , which is a goodrepresentative of the general case. First, we assume that the index j , which specifies theplace of h K in A K , satisfies j ∈ { , . . . , n } . The point is that then e h L j = h L j . Forconvenience of notation we assume that j = 1 . With these assumptions there holds that(6.13) c K, ( L j ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h f , h L ⊗ h K ih σ i K n Y j =2 D f j , h L j ⊗ K | K | E h σ j i K · D f n +1 , h K ⊗ K | K | E h σ n +1 i K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For j = 2 , . . . , n we estimate that (cid:12)(cid:12)(cid:12)D f j , h L j ⊗ K | K | E(cid:12)(cid:12)(cid:12) h σ j i K = (cid:12)(cid:12)(cid:12)D h f j , h L j i h σ j i − K , h σ j i K , , K | K | E(cid:12)(cid:12)(cid:12) hh σ j i K , i K ≤ M h σ j i K , D ( h f j , h L j i h σ j i − K , )( x ) (6.14)for all x ∈ K . Also, there holds that (cid:12)(cid:12)(cid:12)D f n +1 , h K ⊗ K | K | E(cid:12)(cid:12)(cid:12) h σ n +1 i K ≤ | K | M σ n +1 D ( f n +1 σ − n +1 )( x ) for all x ∈ K . These give (recall that L n +1 = K ) that X L ,...,L n +1 ( L j ) ( lj ) = K Q n +1 j =1 | L j | | K | n +1 K (cid:16) X K c K, ( L j ) K | K | (cid:17) ≤ n Y j =1 F j,K · M σ n +1 D ( f n +1 σ − n +1 ) , where(6.15) F ,K = 1 K X ( L ) ( l = K | L | | K | (cid:16) X K |h f , h L ⊗ h K i| h σ i K K | K | (cid:17) and(6.16) F j,K = 1 K X ( L j ) ( lj ) = K | L j | | K | M h σ j i K , D ( h f j , h L j i h σ j i − K , ) for j = 2 , . . . , n .We use these pointwise estimates in (6.12). Since v − = Q n +1 j =1 v − j , we have that(6.12) . n Y j =1 (cid:13)(cid:13)(cid:13)(cid:16) X K F j,K (cid:17) v − j (cid:13)(cid:13)(cid:13) L (cid:13)(cid:13) M σ n +1 D ( f n +1 σ − n +1 ) v − n +1 (cid:13)(cid:13) L . Since σ j = v − j there holds by Proposition 4.2 that k M σ n +1 D ( f n +1 σ − n +1 ) v − n +1 k L = k M σ n +1 D ( f n +1 σ − n +1 ) k L ( σ n +1 ) . k f n +1 v n +1 k L . It remains to estimate the norms for j = 1 , . . . , n .We begin with j = 1 . If l = 0 , then we directly have that (cid:16) X K F ,K (cid:17) = (cid:16) X K |h f , h K i| h σ i K K | K | (cid:17) . Since |h f , h K i|| K | − ≤ h| ∆ K f |i K , we may use Proposition 5.8 to have that (cid:13)(cid:13)(cid:13)(cid:16) X K F ,K (cid:17) v − (cid:13)(cid:13)(cid:13) L . k f σ − k L = k f v k L . Suppose then l > . There holds that (cid:13)(cid:13)(cid:13)(cid:16) X K F ,K (cid:17) v − (cid:13)(cid:13)(cid:13) L = (cid:16) X K k F ,K v − k L (cid:17) . Let s ∈ (1 , ∞ ) be such that d /s ′ = β/ (2 n ) . Then F ,K ≤ l β n K (cid:18) X ( L ) ( l = K | L | s | K | s (cid:16) X K |h f , h L ⊗ h K i| h σ i K K | K | (cid:17) s (cid:19) s . Therefore, k F ,K v − j k L is less than(6.17) l βn (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X ( L ) ( l = K | L | s | K | s (cid:16) X K |h f , h L ⊗ h K i| h σ i K K | K | (cid:17) s (cid:19) s h σ i K , (cid:13)(cid:13)(cid:13)(cid:13) L | K | . Notice that |h f , h L ⊗ h K i|| K | − ≤ h| ∆ K h f , h L i |i K . Therefore, the one-parametercase of Proposition 5.8 gives that(6.17) . l βn (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) X ( L ) ( l = K | L | s | K | s |h f , h L i | s (cid:17) s h σ i − K , (cid:13)(cid:13)(cid:13)(cid:13) L | K |≤ l βn (cid:13)(cid:13)(cid:13)(cid:13) X ( L ) ( l = K | L | | K | |h f , h L i |h σ i − K , (cid:13)(cid:13)(cid:13)(cid:13) L | K | . (6.18) ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 23
Notice that X ( L ) ( l = K | L | | K | |h f , h L i | ≤ h| ∆ K ,l f |i K , . Thus, summing the right hand side of (6.18) over K leads to l βn ˆ R d X K h| ∆ K ,l f |i K , h σ i − K , | K | = 2 l βn ˆ R d X K h| ∆ K ,l f |i K , K h σ i K , σ . l βn ˆ R d | f | v , where we used Proposition 5.8 again.Finally, we estimate the norms related to j = 2 , . . . , n , which are all similar. We assumethat l j > . It will be clear how to do the case l j = 0 . As above we have F j,K ≤ ljβ n K (cid:18) X ( L j ) ( lj ) = K | L j | s | K | s M h σ j i K , D ( h f j , h L j i h σ j i − K , ) s (cid:19) s . Therefore, we get that k F j,K v − j k L ≤ ljβn (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X ( L j ) ( lj ) = K | L j | s | K | s M h σ j i K , D ( h f j , h L j i h σ j i − K , ) s (cid:19) s h σ j i K , (cid:13)(cid:13)(cid:13)(cid:13) L | K | . ljβn (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X ( L j ) ( lj ) = K | L j | s | K | s |h f j , h L j i h σ j i − K , | s (cid:19) s h σ j i K , (cid:13)(cid:13)(cid:13)(cid:13) L | K |≤ ljβn (cid:13)(cid:13)(cid:13)(cid:13) X ( L j ) ( lj ) = K | L j | | K | |h f j , h L j i |h σ j i − K , (cid:13)(cid:13)(cid:13)(cid:13) L | K | , where we applied the one-parameter version of Proposition 8.5. The last norm is likethe last norm in (6.18), and therefore the estimate can be concluded with familiar steps.Combining the estimates we have shown that n Y j =1 (cid:13)(cid:13)(cid:13)(cid:16) X K F j,K (cid:17) v − j (cid:13)(cid:13)(cid:13) L . n Y j =1 ljβ n k f j v j k L ≤ max k j β n Y j =1 k f j v j k L . Above, we assumed that the index j related to the form of the paraproduct satisfied j = 1 , see the discussion before (6.13). It remains to comment on the case j = n + 1 . Inthis case the formula corresponding to (6.13) is c K, ( L j ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 D f j , h L j ⊗ K | K | E h σ j i K · h f n +1 , h K ⊗ h K ih σ n +1 i K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For j = 1 , . . . , n we do the estimate (6.14). Also, there holds that |h f n +1 , h K ⊗ h K i|h σ n +1 i K = | K | (cid:12)(cid:12)(cid:10) h f n +1 , h K i h σ n +1 i − K , h σ n +1 i K , (cid:11) K (cid:12)(cid:12) hh σ n +1 i K , i K ≤ | K | M h σ n +1 i K , D ( h f n +1 , h K i h σ n +1 i − K , )( x ) for any x ∈ K . With the pointwise estimates we proceed as above. Related to f j , j = 1 , . . . , n , this leads to terms which we know how to estimate. Related to f n +1 we geta similar term except that the parameters are in opposite roles. We are done. (cid:3) Full paraproducts. An n -linear bi-parameter full paraproduct Π takes the form(6.19) h Π( f , . . . , f n ) , f n +1 i = X K = K × K a K n +1 Y j =1 h f j , u j,K ⊗ u j,K i , where the functions u j,K and u j,K are like in (6.4). The coefficients are assumed tosatisfy k ( a K ) k BMO prod = sup Ω (cid:16) | Ω | X K ⊂ Ω | a K | (cid:17) / ≤ , where the supremum is over open sets Ω ⊂ R d = R d × R d with < | Ω | < ∞ . As al-ready discussed the H - BMO duality works also in bi-parameter (see again [38, Equation(4.1)]):(6.20) X K | a K || b K | . k ( a K ) k BMO prod (cid:13)(cid:13)(cid:13)(cid:16) X K | b K | K | K | (cid:17) / (cid:13)(cid:13)(cid:13) L . We are ready to bound the full paraproducts.6.21.
Theorem.
Suppose Π is an n -linear bi-parameter full paraproduct, < p , . . . , p n ≤ ∞ and /p = P ni =1 /p i > . Then we have k Π( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi for all multilinear bi-parameter weights ~w ∈ A ~p .Proof. We use duality to always reduce to one of the operators of type A in Theorem 5.5.Fix some ~p = ( p , . . . , p n ) with < p i < ∞ and p > , which is enough by extrapolation.We will dualise using f n +1 with k f n +1 w − k L p ′ ≤ . The particular form of Π does notmatter – it only affects the form of the operator A we will get. We may, for example,look at Π( f , . . . , f n ) = X K = K × K a K D f , h K ⊗ K | K | ED f , K | K | ⊗ h K E n Y j =3 h f j i K · K | K | . We have |h Π( f , . . . , f n ) , f n +1 i| ≤ X K | a K | (cid:12)(cid:12)(cid:12)D f , h K ⊗ K | K | ED f , K | K | ⊗ h K E(cid:12)(cid:12)(cid:12) n +1 Y j =3 h| f j |i K . ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 25
We now apply the unweighted H - BMO duality estimate from above to bound this with (cid:13)(cid:13)(cid:13)(cid:16) X K h| ∆ K f |i K h| ∆ K f |i K n +1 Y j =3 h| f j |i K K (cid:17) (cid:13)(cid:13)(cid:13) L . Recalling (6.3) it remains to apply Theorem 5.5 with a suitable A ( f , . . . , f n +1 ) . (cid:3)
7. S
INGULAR INTEGRALS
Let ω be a modulus of continuity: an increasing and subadditive function with ω (0) =0 . A relevant quantity is the modified Dini condition(7.1) k ω k Dini α := ˆ ω ( t ) (cid:16) t (cid:17) α dtt , α ≥ . In practice, the quantity (7.1) arises as follows:(7.2) ∞ X k =1 ω (2 − k ) k α = ∞ X k =1 ˆ − k +1 − k ω (2 − k ) k α dtt . ˆ ω ( t ) (cid:16) t (cid:17) α dtt . Consider an n -linear operator T on R d = R d × R d . We define what it means for T to bean n -linear bi-parameter SIO. Let ω i be a modulus of continuity on R d i . Let f j = f j ⊗ f j , j = 1 , . . . , n + 1 . Bi-parameter SIOs.
Full kernel representation.
Here we assume that given m ∈ { , } there exists j , j ∈{ , . . . , n + 1 } so that spt f mj ∩ spt f mj = ∅ . In this case we demand that h T ( f , . . . , f n ) , f n +1 i = ˆ R ( n +1) d K ( x n +1 , x , . . . , x n ) n +1 Y j =1 f j ( x j ) d x, where K : R ( n +1) d \ { ( x , . . . , x n +1 ) ∈ R ( n +1) d : x = · · · = x n +1 or x = · · · = x n +1 } → C is a kernel satisfying a set of estimates which we specify next.The kernel K is assumed to satisfy the size estimate | K ( x n +1 , x , . . . , x n ) | . Y m =1 (cid:16) P nj =1 | x mn +1 − x mj | (cid:17) d m n . We also require the following continuity estimates. For example, we require that wehave | K ( x n +1 , x , . . . , x n ) − K ( x n +1 , x , . . . , x n − , ( c , x n )) − K (( x n +1 , c ) , x , . . . , x n ) + K (( x n +1 , c ) , x , . . . , x n − , ( c , x n )) | . ω (cid:16) | x n − c | P nj =1 | x n +1 − x j | (cid:17) (cid:16) P nj =1 | x n +1 − x j | (cid:17) d n × ω (cid:16) | x n +1 − c | P nj =1 | x n +1 − x j | (cid:17) (cid:16) P nj =1 | x n +1 − x j | (cid:17) d n whenever | x n − c | ≤ − max ≤ i ≤ n | x n +1 − x i | and | x n +1 − c | ≤ − max ≤ i ≤ n | x n +1 − x i | .Of course, we also require all the other natural symmetric estimates, where c can be inany of the given n + 1 slots and similarly for c . There are ( n + 1) different estimates.Finally, we require the following mixed continuity and size estimates. For example,we ask that | K ( x n +1 , x , . . . , x n ) − K ( x n +1 , x , . . . , x n − , ( c , x n )) | . ω (cid:16) | x n − c | P nj =1 | x n +1 − x j | (cid:17) (cid:16) P nj =1 | x n +1 − x j | (cid:17) d n · (cid:16) P nj =1 | x n +1 − x j | (cid:17) d n whenever | x n − c | ≤ − max ≤ i ≤ n | x n +1 − x i | . Again, we also require all the othernatural symmetric estimates. Partial kernel representations.
Suppose now only that there exists j , j ∈ { , . . . , n + 1 } sothat spt f j ∩ spt f j = ∅ . Then we assume that h T ( f , . . . , f n ) , f n +1 i = ˆ R ( n +1) d K ( f j ) ( x n +1 , x , . . . , x n ) n +1 Y j =1 f j ( x j ) d x , where K ( f j ) is a one-parameter ω -Calderón–Zygmund kernel but with a constant de-pending on the fixed functions f , . . . , f n +1 . For example, this means that the size esti-mate takes the form | K ( f j ) ( x n +1 , x , . . . , x n ) | ≤ C ( f , . . . , f n +1 ) 1 (cid:16) P nj =1 | x n +1 − x j | (cid:17) d n . The continuity estimates are analogous.We assume the following T type control on the constant C ( f , . . . , f n +1 ) . We have C (1 I , . . . , I ) . | I | and C ( a I , I , . . . , I ) + C (1 I , a I , I , . . . , I ) + · · · + C (1 I , . . . , I , a I ) . | I | for all cubes I ⊂ R d and all functions a I satisfying a I = 1 I a I , | a I | ≤ and ´ a I = 0 .Analogous partial kernel representation on the second parameter is assumed when spt f j ∩ spt f j = ∅ for some j , j .7.3. Definition. If T is an n -linear operator with full and partial kernel representationsas defined above, we call T an n -linear bi-parameter ( ω , ω ) -SIO. ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 27
Bi-parameter CZOs.
We say that T satisfies the weak boundedness property if |h T (1 R , . . . , R ) , R i| . | R | for all rectangles R = I × I ⊂ R d = R d × R d .An SIO T satisfies the diagonal BMO assumption if the following holds. For all rect-angles R = I × I ⊂ R d = R d × R d and functions a I i with a I i = 1 I i a I i , | a I i | ≤ and ´ a I i = 0 we have |h T ( a I ⊗ I , R , . . . , R ) , R i| + · · · + |h T (1 R , . . . , R ) , a I ⊗ I i| . | R | and |h T (1 I ⊗ a I , R , . . . , R ) , R i| + · · · + |h T (1 R , . . . , R ) , I ⊗ a I i| . | R | . An SIO T satisfies the product BMO assumption if it holds S (1 , . . . , ∈ BMO prod for all the ( n + 1) adjoints S = T j ∗ ,j ∗ , . This can be interpreted in the sense that k S (1 , . . . , k BMO prod = sup D = D ×D sup Ω (cid:16) | Ω | X R = I × I ∈D R ⊂ Ω |h S (1 , . . . , , h R i| (cid:17) / < ∞ , where the supremum is over all dyadic grids D i on R d i and open sets Ω ⊂ R d = R d × R d with < | Ω | < ∞ , and the pairings h S (1 , . . . , , h R i can be defined, in a natural way,using the kernel representations.7.4. Definition. An n -linear bi-parameter ( ω , ω ) -SIO T satisfying the weak bounded-ness property, the diagonal BMO assumption and the product BMO assumption is calledan n -linear bi-parameter ( ω , ω ) -Calderón–Zygmund operator ( ( ω , ω ) -CZO).In the paper [2], among other things, a dyadic representation theorem for n -linear bi-parameter CZOs was proved. The minimal regularity required is ω i ∈ Dini , but then thedyadic representation is in terms of certain modified versions of the model operators wehave presented, and bounded, in this paper. It appears to be difficult to prove weightedbounds for the modified operators with the optimal dependency on the complexity. In-stead, we will rely on a lemma, which says that all of the modified operators can bewritten as suitable sums of the standard model operators. This step essentially outrightloses of kernel regularity, and puts us in competition to obtain our weighted boundswith ω i ∈ Dini . The bilinear bi-parameter representation theorem with the usual Höldertype kernel regularity w i ( t ) = t α i appeared first in [34]. We now state a representationtheorem that we will rely on.There is a natural probability space Ω = Ω × Ω , the details of which are not relevantfor us here, so that to each σ = ( σ , σ ) ∈ Ω we can associate a random collection ofdyadic rectangles D σ = D σ × D σ . We denote the expectation over the probability space Ω by E σ . A consequence of [2, Theorem 5.35] and [2, Lemma 5.12] is the following.7.5. Proposition.
Suppose T is an n -linear bi-parameter ( ω , ω ) -CZO. Then we have h T ( f , . . . , f n ) , f n +1 i = C T E σ X u =( u ,u ) ∈ N ω (2 − u ) ω (2 − u ) h U u,σ ( f , . . . , f n ) , f n +1 i , where C T enjoys a linear bound with respect to the CZO quantities and U u,σ denotes some n -linear bi-parameter dyadic operator (defined in the grid D σ ) with the following property. We havethat U u = U u,σ can be decomposed using the standard dyadic model operators as follows: (7.6) U u = C u − X i =0 u − X i =0 V i ,i , where each V = V i ,i is a dyadic model operator (a shift, a partial paraproduct or a full paraprod-uct) of complexity k mj,V , j ∈ { , . . . , n + 1 } , m ∈ { , } , satisfying k mj,V ≤ u m . We were able to prove a complexity free weighted estimate for the shifts. On the con-trary, the weighted estimate for the partial paraproducts is even exponential, however,with an arbitrarily small power. For these reasons, we can prove a weighted estimatewith mild kernel regularity for paraproduct free T , and otherwise we will deal with thestandard kernel regularity ω i ( t ) = t α i . By paraproduct free we mean that the paraprod-ucts in the dyadic representation of T vanish, which could also be stated in terms of(both partial and full) “ T ” type conditions (only the partial paraproducts, and notthe full paraproducts, are problematic in terms of kernel regularity, of course). In theparaproduct free case the reader can think of convolution form SIOs.7.7. Theorem.
Suppose T is an n -linear bi-parameter ( ω , ω ) -CZO. For < p , . . . , p n ≤ ∞ and /p = P i /p i > we have k T ( f , . . . , f n ) w k L p . Y i k f i w i k L pi for all multilinear bi-parameter weights ~w ∈ A ~p , if one of the following conditions hold.(1) T is paraproduct free and ω i ∈ Dini .(2) We have ω i ( t ) = t α i for some α i ∈ (0 , .Proof. Notice that in the paraproduct free case (1) by our results for the shifts we alwayshave k U u,σ ( f , . . . , f n ) w k L p . (1 + u )(1 + u ) Y i k f i w i k L pi , where the complexity dependency comes only from the decomposition (7.6). We thentake some < p , . . . , p n < ∞ with p ∈ (1 , ∞ ) , use the dyadic representation theoremand conclude that T satisfies the weighted bound with these fixed exponents – recall (7.2)and that ω i ∈ Dini . Finally, we extrapolate using Theorem 3.12.The case of a completely general CZO with the standard kernel regularity is provedcompletely analogously. Just choose the exponent β in the exponential complexity de-pendendency of the partial paraproducts to be small enough compared to α and α . (cid:3)
8. D
ETAILS ON EXTRAPOLATION AND THE R UBIO DE F RANCIA ALGORITHM
This section is devoted to providing more details about Theorem 3.12 in the multi-parameter setting. We also obtain the proof of the vector-valued version of Proposition4.2 that we have used – see Proposition 8.5 below. We give the details in the bi-parametercase with the general case being similar.
ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 29
We begin with the following definitions. Given µ ∈ A ∞ ( R n + m ) , we say w ∈ A p ( µ ) if w > a.e. and [ w ] A p ( µ ) := sup R h w i µR (cid:16)(cid:10) w − p − (cid:11) µR (cid:17) p − < ∞ , < p < ∞ . And we say w ∈ A ( µ ) if w > a.e. and [ w ] A ( µ ) = sup R h w i µR ess sup R w − . We begin with the following auxiliary result needed to build the required machinery.This is an extension of Proposition 4.2.8.1.
Remark.
The so-called three lattice theorem states that there are lattices D mj in R d m , m ∈ { , } , j ∈ { , . . . , d m } , such that for every cube Q m ⊂ R d m there exists a j and I m ∈ D mj so that Q m ⊂ I m and | I m | ∼ | Q m | . Given λ ∈ A ∞ we have in particular that λ is doubling: λ (2 R ) . λ ( R ) for all rectangles R . It then follows that also the non-dyadicvariant M λ satisfies Proposition 4.2.8.2. Lemma.
Let µ ∈ A ∞ and w ∈ A p ( µ ) , < p < ∞ . Then we have k M µ f k L p ( wµ ) . k f k L p ( wµ ) . Proof.
Fix x and f ≥ and denote σ = w − p − . For an arbitrary rectangle R ⊂ R d with x ∈ R we have h f i µR = h σ i µR (cid:0) h w i µR (cid:1) p − (cid:0) h w i µR (cid:1) − p − σµ ( R ) ˆ R f µ ≤ [ w ] p − A p ( µ ) (cid:0) M wµ (cid:0) [ M σµ ( f σ − )] p − w − (cid:1) ( x ) (cid:1) p − . If wµ, σµ ∈ A ∞ , then by (the non-dyadic version of) Proposition 4.2 we have k M µ f k L p ( wµ ) . (cid:13)(cid:13)(cid:13)(cid:0) M wµ (cid:0) [ M σµ ( f σ − )] p − w − (cid:1)(cid:1) p − (cid:13)(cid:13)(cid:13) L p ( wµ ) . (cid:13)(cid:13)(cid:13)(cid:0) [ M σµ ( f σ − )] p − w − (cid:1) p − (cid:13)(cid:13)(cid:13) L p ( wµ ) = k M σµ ( f σ − ) k L p ( σµ ) . k f k L p ( wµ ) . Therefore, it remains to check that wµ, σµ ∈ A ∞ . We only explicitly prove that wµ ∈ A ∞ , since the other one is symmetric. First of all, write h w i µR (cid:16)(cid:10) w − p − (cid:11) µR (cid:17) p − ≤ [ w ] A p ( µ ) in the form h wµ i R (cid:10) w − p − µ (cid:11) p − R ≤ [ w ] A p ( µ ) h µ i pR . Then by the Lebesgue differentiation theorem, we have for all cubes I ⊂ R d that h wµ i I , ( x ) (cid:10) w − p − µ (cid:11) p − I , ( x ) ≤ [ w ] A p ( µ ) h µ i pI , ( x ) , x ∈ R d \ N I , where | N I | = 0 . By standard considerations there exists N so that | N | = 0 and for allcubes I ⊂ R d we have h wµ i I , ( x ) (cid:10) w − p − µ (cid:11) p − I , ( x ) ≤ [ w ] A p ( µ ) h µ i pI , ( x ) , x ∈ R d \ N. In other words, w ( · , x ) ∈ A p ( µ ( · , x )) (uniformly) for all x ∈ R m \ N . As µ ∈ A ∞ thereexists s < ∞ so that µ ∈ A s . Then for all cubes I ⊂ R d and arbitrary E ⊂ I we have | E || I | . (cid:16) µ ( · , x )( E ) µ ( · , x )( I ) (cid:17) s . (cid:16) wµ ( · , x )( E ) wµ ( · , x )( I ) (cid:17) ps , a . e . x ∈ R d , where the implicit constant is independent from x . This means wµ ( · , x ) ∈ A ∞ ( R d ) uniformly for a.e. x ∈ R d . Likewise we can show that wµ ( x , · ) ∈ A ∞ ( R d ) uniformlyfor a.e. x ∈ R d . This completes the proof and we are done. (cid:3) Now we are ready to formulate the following version of Rubio de Francia algorithm.8.3.
Lemma.
Let µ ∈ A ∞ and p ∈ (1 , ∞ ) . Let f be a non-negative function in L p ( wµ ) for some w ∈ A p ( µ ) . Let M µk be the k -th iterate of M µ , M µ f = f , and k M µ k L p ( wµ ) := k M µ k L p ( wµ ) → L p ( wµ ) be the norm of M µ as a bounded operator on L p ( wµ ) . Define Rf ( x ) = ∞ X k =0 M µk f (2 k M µ k L p ( wµ ) ) k . Then f ( x ) ≤ Rf ( x ) , k Rf k L p ( wµ ) ≤ k f k L p ( wµ ) , and Rf is an A ( µ ) weight with constant [ Rf ] A ( µ ) ≤ k M µ k L p ( wµ ) .Proof. The statements f ( x ) ≤ Rf ( x ) and k Rf k L p ( wµ ) ≤ k f k L p ( wµ ) are obvious. Since M µ ( Rf ) ≤ ∞ X k =0 M µk +1 f (2 k M µ k L p ( wµ ) ) k ≤ k M µ k L p ( wµ ) Rf, we have [ Rf ] A ( µ ) ≤ sup R (cid:0) inf R M µ ( Rf ) (cid:1)(cid:0) ess inf R Rf (cid:1) − ≤ k M µ k L p ( wµ ) . We are done. (cid:3)
With the above Rubio de Francia algorithm at hand, we are able to prove the bi-parameter version of [31, Theorem 3.1] and the corresponding endpoint cases similarlyas in [32, Theorem 2.3]. On the other hand, the key technical lemma [32, Lemma 2.14]can be extended to the bi-parameter setting very easily. Using these as in [32] we obtainTheorem 3.12.The above Rubio de Francia algorithm, of course, also yields the following standardlinear extrapolation. Let µ ∈ A ∞ and assume that(8.4) k g k L p ( wµ ) . k f k L p ( wµ ) for all w ∈ A p ( µ ) . Then the same inequality holds for all p ∈ (1 , ∞ ) and w ∈ A p ( µ ) .Using this we also obtain the vector-valued version of Proposition 4.2. We state the fol-lowing version with two sequence spaces – of course, a version with arbitrarily manyalso works.8.5. Proposition.
Let µ ∈ A ∞ , w ∈ A p ( µ ) and < p, s, t < ∞ . Then we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { M µ f ij } (cid:13)(cid:13) ℓ s (cid:13)(cid:13)(cid:13) ℓ t (cid:13)(cid:13)(cid:13) L p ( wµ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { f ij } (cid:13)(cid:13) ℓ s (cid:13)(cid:13)(cid:13) ℓ t (cid:13)(cid:13)(cid:13) L p ( wµ ) . In particular, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { M µ f ij } (cid:13)(cid:13) ℓ s (cid:13)(cid:13)(cid:13) ℓ t (cid:13)(cid:13)(cid:13) L p ( µ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) { f ij } (cid:13)(cid:13) ℓ s (cid:13)(cid:13)(cid:13) ℓ t (cid:13)(cid:13)(cid:13) L p ( µ ) . ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 31
9. T
HE MULTI - PARAMETER CASE
One can approach the multi-parameter case as follows.(1) What is the definition of an SIO/CZO? The definition of an n -linear m -parameterSIO is straightforward using the style above – however, it becomes extremelylengthy. For the linear case n = 1 in higher parameters one can consult [40].Alternatively, it would be possible to adapt the equivalent (see [17, 34, 40]) Jour ´ne[27] style definition – this kind of vector-valued definition style is shorter to state.(2) Is there a representation theorem in this generality? The multilinear representa-tion theorems [2, 34] are stated in the bi-parameter setting but can rather straight-forwardly be generalised to m -parameters. For the linear case, see again [40].(3) How do the model operators look like? Studying the above presented bi-parametermodel operators, one realises that the philosophies in each parameter are inde-pendent of each other – for example, if one has a shift type of philosophy in agiven parameter, one needs at least two cancellative Haar functions in that pa-rameter. With this logic it is clear how to define the m -parameter analogues justby working parameter by parameter. Alternatively, one can take all possible m -fold tensor products of one-parameter n -linear model operators, and then justreplace the appearing product coefficients by general coefficients. This yields theform of the model operators.(4) Finally, is it more difficult to show the genuinely multilinear weighted estimatesfor the m -parameter, m ≥ , model operators compared to the bi-parametermodel operators? When it comes to shifts and full paraproducts, there is no es-sential difference – their boundedness always reduces to Theorem 5.5, which hasan obvious m -parameter version. With out current proof, the answer for the par-tial paraproducts is more complicated. Thus, we will elaborate on how to provethe weighted estimates for m -parameter partial paraproducts. Notice that previ-ously e.g. in [34] we could only handle bi-parameter partial paraproducts as ourproof exploited the one-parameter nature of the paraproducts by sparse domina-tion. Here we have already disposed of sparse domination, but the proof is stillcomplicated and leads to some new philosophies in higher parameters.We now discuss how to prove a tri-parameter analogue of Theorem 6.7. We can havea partial paraproduct with a bi-parameter paraproduct component and a one-parametershift component, or the other way around. Regardless of the form, the initial stages ofthe proof of Theorem 6.7 can be used to reduce to estimating the weighted L norms ofcertain functions which are analogous to (6.15) and (6.16). Most of these norms can beestimated with similar steps as in the bi-parameter case. However, also a new type ofvariant appears. An example of such a variant is given by(9.1) F j,K = 1 K X ( L j ) ( lj ) = K | L j | | K | (cid:16) X K K | K | M h σ j i K , D (cid:0) h f j , h L j ⊗ h K ih σ j i − K , (cid:1) (cid:17) , where f j : R d × R d × R d → C . The goal is to estimate P K k F j,K k L ( σ j ) . Here we aredenoting K , = K × K the original tri-parameter rectangle being K = K × K × K ,and for brevity we only write h σ j i K , instead of h σ j i K , , , . Comparing with (6.16), the key difference is that in (6.16) the measure of the maximalfunction depended only on K . Here, it depends also on K , and therefore we havemaximal functions with respect to different measures inside the norms. We will use thefollowing lemma and the appearing new type of extrapolation trick to overcome this.9.2. Lemma.
Let µ ∈ A ∞ ( R d × R d ) be a bi-parameter weight. Let D = D × D be a gridof bi-parameter dyadic rectangles in R d × R d . Suppose that for each m ∈ Z and K ∈ D wehave a function f m,K : R d → C . Then, for all p, s, t ∈ (1 , ∞ ) , the estimate (cid:13)(cid:13)(cid:13)(cid:16) X m (cid:16) X K K M h µ i K D ( f m,K ) t (cid:17) st (cid:17) s (cid:13)(cid:13)(cid:13) L p ( wµ ) . (cid:13)(cid:13)(cid:13)(cid:16) X m (cid:16) X K K | f m,K | t (cid:17) st (cid:17) s (cid:13)(cid:13)(cid:13) L p ( wµ ) holds for all w ∈ A p ( µ ) .Proof. By extrapolation, see the discussion around (8.4), it suffices to take a function f : R d → C and show that (cid:13)(cid:13) K M h µ i K D f (cid:13)(cid:13) L q ( wµ ) . k K f k L q ( wµ ) for some q ∈ (1 , ∞ ) and for all w ∈ A q ( µ ) . We fix some w ∈ A q ( µ ) . The above estimatecan be rewritten as (cid:13)(cid:13) M h µ i K D f (cid:13)(cid:13) L q ( h wµ i K ) | K | q . k f k L q ( h wµ i K ) | K | q . We have the identity(9.3) h wµ i K ( x ) = h wµ i K ( x ) h µ i K ( x ) h µ i K ( x ) = h w ( · , x ) i µ ( · ,x ) K h µ i K ( x ) . Define v ( x ) = h w ( · , x ) i µ ( · ,x ) K . We show that v ∈ A q ( h µ i K ) . Let I be a cube in R d .First, we have that ˆ I v h µ i K = ˆ I h w ( · , x ) i µ ( · ,x ) K h µ ( · , x ) i K d x = ˆ K × I wµ | K | − . Therefore, h v i h µ i K I = h w i µK × I . Hölder’s inequality gives that (cid:0) h w ( · , x ) i µ ( · ,x ) K (cid:1) − q − ≤ (cid:10) w ( · , x ) − q − (cid:11) µ ( · ,x ) K , which shows that ˆ I v − q − h µ i K ≤ ˆ I (cid:10) w ( · , x ) − q − (cid:11) µ ( · ,x ) K h µ ( · , x ) i K d x = ˆ K × I w − q − µ | K | − . Thus, we have that (cid:0) h v − q − i h µ i K I (cid:1) q − ≤ (cid:0) h w − q − i µK × I (cid:1) q − . These estimates yield that [ v ] A q ( h µ i K ) ≤ [ w ] A q ( µ ) .Recall the identity (9.3). Since h µ i K ∈ A ∞ , we have that (cid:13)(cid:13) M h µ i K D f (cid:13)(cid:13) L q ( h wµ i K ) = (cid:13)(cid:13) M h µ i K D f (cid:13)(cid:13) L q ( v h µ i K ) . k f k L q ( v h µ i K )) = k f k L q ( h wµ i K )) , where we used Lemma 8.2. This concludes the proof. (cid:3) ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 33
We now show how to estimate (9.1). First, we have that k F j,K k L ( σ j ) is less than l j d s ′ multiplied by (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) X ( L j ) ( lj ) = K (cid:20) | L j | | K | (cid:16) X K K | K | M h σ j i K , D (cid:0) h f j , h L j ⊗ h K ih σ j i − K , (cid:1) (cid:17) (cid:21) s (cid:21) s (cid:13)(cid:13)(cid:13)(cid:13) L ( h σ j i K ) | K | . The exponent s ∈ (1 , ∞ ) is chosen small enough so that we get a suitable dependence onthe complexity through l j d s ′ , see the corresponding step in the bi-parameter case. Since h σ j i K ∈ A ∞ ( R d × R d ) , we can use Lemma 9.2 to have that the last term is dominatedby (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) X ( L j ) ( lj ) = K (cid:20) | L j | | K | (cid:16) X K K | K | (cid:12)(cid:12) h f j , h L j ⊗ h K ih σ j i − K , (cid:12)(cid:12) (cid:17) (cid:21) s (cid:21) s (cid:13)(cid:13)(cid:13)(cid:13) L ( h σ j i K ) | K | . After these key steps it only remains to use Proposition 5.8 twice in a very similar way asin the bi-parameter proof. We are done.10. A
PPLICATIONS
Mixed-norm estimates.
With our main result, Theorem 1.1, and extrapolation,Theorem 3.12, the following result becomes immediate.10.1.
Theorem.
Let T be an n -linear m -parameter Calderón-Zygmund operator. Let < p ji ≤∞ , i = 1 , . . . , n , with p j = P i p ji > , j = 1 , . . . , m . Then we have that k T ( f , . . . , f n ) k L p ··· L pm . n Y i =1 k f i k L p i ··· L pmi . Remark.
We understand this as an a priori estimate with f i ∈ L ∞ c – this is only aconcern when some p ji is ∞ . In [32], which concerned the bilinear bi-parameter case with tensor form CZOs, we went to great lengths to check that this restriction can always beremoved. We do not want to get into such considerations here, and prefer this a prioriinterpretation at least when n ≥ . See also [34] for some previous results for bilinearbi-parameter CZOs that are not of tensor form, but where, compared to [32], the range ofexponents had some limitations in the ∞ cases. Lastly, see [11].The proof is immediate by extrapolating with tensor form weights. For the generalidea see [32, Theorem 4.5] – here the major simplification is that everything can be donewith extrapolation and the operator-valued analysis is not needed. This is because theweighted estimate, Theorem 1.1, is now with the genuinely multilinear weights unlikein [32, 34].10.B. Commutators.
We will state these applications in the bi-parameter case R d = R d × R d . The m -parameter versions are obvious. We define [ b, T ] k ( f , . . . , f n ) := bT ( f , . . . , f n ) − T ( f , . . . , f k − , bf k , f k +1 , . . . , f n ) . One can also define the iterated commutators as usual. We say that b ∈ bmo if k b k bmo = sup R | R | ˆ R | b − h b i R | < ∞ , where the supremum is over rectangles. Recall that given b ∈ bmo , we have(10.3) k b k bmo ∼ max (cid:0) ess sup x ∈ R d k b ( x , · ) k BMO( R d ) , ess sup x ∈ R d k b ( · , x ) k BMO( R d ) (cid:1) . See e.g. [26]. In the one-parameter case the following was proved in [9, Lemma 5.6].10.4.
Proposition.
Let ~p = ( p , . . . , p n ) with < p , . . . , p n < ∞ and p = P i p i < . Let ~w = ( w , . . . , w n ) ∈ A ~p . Then for any ≤ j ≤ n we have ~w b,z := ( w , . . . , w j e Re( bz ) , . . . , w n ) ∈ A ~p with [ ~w b,z ] A ~p . [ ~w ] A ~p provided that | z | ≤ ǫ max([ w p ] A ∞ , max i [ w − p ′ i i ] A ∞ ) k b k BMO , where ǫ depends on ~p and the dimension of the underlying space. If < p , . . . , p n < ∞ , then there holds that [ ~w ] A ~p ( R d ) < ∞ if and only if max (cid:0) ess sup x ∈ R d [ ~w ( x , · )] A ~p ( R d ) , ess sup x ∈ R d [ ~w ( · , x )] A ~p ( R d ) (cid:1) < ∞ . Moreover, we have that the above maximum satisfies max( · , · ) ≤ [ ~w ] A ~p ( R d ) . max( · , · ) γ where γ is allowed to depend on ~p and d . The first estimate follows from the Lebesguedifferentiation theorem. The second estimate can be proved by using Lemma 3.6 and thecorresponding linear statement, see (3.1). Using this, (10.3) and Proposition 10.4 gives abi-parameter version of Proposition 10.4 – the statement is obtained by replacing BMO with bmo , and the quantitative estimate is of the form [ w b,z ] A ~p . [ ~w ] γA ~p . Now, we haveeverything ready to prove the following commutator estimate.10.5. Theorem.
Suppose T is an n -linear bi-parameter CZO in R d = R d × R d , < p , . . . , p n ≤∞ and /p = P i /p i > . Suppose also that b ∈ bmo . Then for all ≤ k ≤ n we have thecommutator estimate k [ b, T ] k ( f , . . . , f n ) w k L p . k b k bmo n Y i =1 k f i w i k L pi for all n -linear bi-parameter weights ~w = ( w , . . . , w n ) ∈ A ~p . Analogous results hold for iteratedcommutators.Proof. We assume k b k bmo = 1 . It suffices to study [ b, T ] , and in fact we shall prove thefollowing principle. Once we have k T ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi for some ~p in the Banach range, then k [ b, T ] ( f , . . . , f n ) w k L p . n Y i =1 k f i w i k L pi . ENUINELY MULTILINEAR WEIGHTED ESTIMATES FOR SINGULAR INTEGRALS IN PRODUCT SPACES 35
In this principle the form of the n -linear operator plays no role ( T does not need to be aCZO). The iterated cases follow immediately from this principle and the full range thenfollows from extrapolation.Define T z ( f , . . . , f n ) = e zb T ( e − zb f , f , . . . , f n ) . Then, by the Cauchy integral theorem, we get for nice functions f , . . . , f n , that [ b, T ] ( f , . . . , f n ) = dd z T z ( f , . . . , f n ) (cid:12)(cid:12)(cid:12) z =0 = − πi ˆ | z | = δ T z ( f , . . . , f n ) z d z, δ > . Since p ≥ , by Minkowski’s inequality k [ b, T ] ( f , . . . , f n ) w k L p ≤ πδ ˆ | z | = δ k T z ( f , . . . , f n ) w k L p | d z | . We choose δ ∼ w p ] A ∞ , max i [ w − p ′ i i ] A ∞ ) . This allows to use the bi-parameter version of Proposition 10.4 to have that k T z ( f , . . . , f n ) w k L p = k T ( e − zb f , f , . . . , f n ) we Re( bz ) k L p . k e − zb f w e Re( bz ) k L p n Y i =2 k f i w i k L pi = n Y i =1 k f i w i k L pi . The claim follows. 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