A bilinear Rubio de Francia inequality for arbitrary rectangles

Let $\mathscr{R}$ be a collection of disjoint dyadic rectangles $R$ with sides parallel to the axes, let $\pi_R$ denote the non-smooth bilinear projection onto $R$ \[ \pi_R (f,g)(x):=\iint \mathbf{1}_{R}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2\pi i (\xi + \eta) x} d\xi d\eta \] and let $r>2$. We show that the bilinear Rubio de Francia operator associated to $\mathscr{R}$ given by \[ f,g \mapsto \Big(\sum_{R\in\mathscr{R}} |\pi_{R} (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded whenever $1/p + 1/q = 1/s$, $r'<p,q<r$. This extends from squares to rectangles a previous result by the same authors, and as a corollary extends in the same way a previous result from Benea and the first author for smooth projections, albeit in a reduced range.


Introduction
In this paper we present an improvement over the results of Benea and the first author in [2] and the results of [7] by the same authors. The setup is as follows. Let D denote the standard dyadic grid, that is the collection of intervals of the form r2 k n, 2 k pn`1qs for n, k P Z. Let then R Ă DˆD be an arbitrary collection of disjoint rectangles R " R 1ˆR2 with R 1 , R 2 dyadic, and let π R denote the non-smooth bilinear frequency projection on R, that is π R pf, gqpxq :"¨p f pξqp gpηq1 R pξ, ηqe 2πipξ`ηqx dξdη.
We will consider the bilinear operators associated to the collection R given by T r R pf, gqpxq :"´ÿ RPR |π R pf, gqpxq| r¯1 {r for r ą 2.
Before stating our results we provide some context. Linear variants of the operator T r R are known in the literature as Rubio de Francia operators. Indeed, given a collection of arbitrary disjoint intervals I " tI n , n P Zu, Rubio de Francia proved in [19] that the operator {r is L p Ñ L p bounded for r 1 ă p ă 8 when r ě 2, this last condition being necessary (a consequence of Khinchin's inequality). Notice that when r " 2 then RdF 2 I is also bounded at the endpoint r 1 " 2, this endpoint being just a consequence of Plancherel's identity, but when r ą 2 the condition r 1 ă p is sharp (see [11]). Rubio de Francia's result is a generalization of the well known Littlewood-Paley inequalities (I n " r2 n , 2 n`1 s), extending an earlier result of Carleson [9] and (independently) Cordoba [10] for the collection I " trn, n`1s, n P Zu; as such, it can be thought of as a statement about orthogonality for arbitrary frequency intervals.
The result has also been extended and reproved by different means in a number of papers, see [8,12,21,22,14,3]. Bilinear operators of square-function type associated to collections of subsets of the frequency plane x R 2 have previously been considered as well. Perhaps the first such operator to have appeared in the literature is the square function where χ is a smooth bump function supported in r´1{2, 1{2s. Here the collection of subsets of x R 2 is evidently given by the strips S n :" tpξ, ηq s.t. |ξ´η´n| ă 1{2u for n P Z, and the bilinear frequency projections are given by a smooth bilinear multiplier. This square function was introduced by Lacey in [13], in which he proved it is L pˆLq Þ Ñ L 2 bounded for 2 ď p, q ď 8 satisfying 1{p`1{q " 1{2. This range was later extended in [15], [6] to show that the operator is L pˆLq Þ Ñ L s bounded for 1{p`1{q " 1{s, 1 ď s ď 2 and 2 ď p, q, ď 8, this last condition being sharp. The operator admits a non-smooth variant given by where the collection of subsets is again given by the strips S n but now the bilinear frequency projections are given by a non-smooth multiplier -specifically, each projection is essentially a modulated Bilinear Hilbert Transform. The non-smoothness makes the operator inherently harder to bound. The first author proved in [5] that the operator is L pˆLq Ñ L s bounded for 2 ă p, q ă 8, 1 ă s ă 2 and 1{p`1{q " 1{s. It should be remarked that the geometric regularity of the strips (in particular the fact that they have all the same width and separation from their neighbours) is fundamental to the proof, the case of arbitrary disjoint strips being an interesting open problem. Another example of a bilinear square-function associated to subsets of x R 2 can be found in [4], where the authors considered the operator f, g Þ Ñ´ÿ nPZˇä năξăηăan`1 p f pξqp gpηqe 2πipξ`ηqx dξdηˇˇr¯1 {r for r ě 1, where pa n q nPZ is a strictly increasing subsequence of reals (a smoother version of this operator was also originally considered in [1]). They can be thought of as bilinear Rubio de Francia operators for iterated Fourier integrals, and arise naturally in the study of the stability of solutions to AKNS systems of differential equations (see [16]). Here we can see that the subsets of the frequency space x R 2 consist of disjoint right triangles whose hypotenuses are aligned along the ξ " η diagonal, and the bilinear multipliers are non-smooth. In [4] it is proven that when r ě 2 the operator is L pˆLq Ñ L s bounded in the same range in which the Bilinear Hilbert Transform is bounded, and when 1 ď r ă 2 the operator is still bounded but in a range depending on r. The final example of a bilinear operator of the above kind we provide is the closest to the operator T r R . Consider a collection Ω of disjoint squares ω with sides parallel to the axes in x R 2 . Let χ denote a smooth bump function supported in r´1{2, 1{2sr´1 {2, 1{2s and let χ ω be the rescaling of χ with support ω, namely χ ω pξ, ηq " χ´ξ´c pω 1 q |ω 1 | , η´cpω 2 q |ω 2 |¯, cpIq denoting the center of the interval I. For r ě 2 fixed, one can associate to the collection Ω the bilinear Rubio de Francia operator f, g Þ Ñ S r Ω pf, gqpxq :"´ÿ ωPΩˇ¨p f pξqp gpηqχ ω pξ, ηqe 2πipξ`ηqx dξdηˇˇr¯1 {r .
Thus the sets are now squares with sides parallel to the axes and the frequency projections are smooth. In [2] by Benea and the first author the following is proven.
). Let r ą 2 and let Ω be a collection of disjoint squares. Let p, q, s be such that 1 p`1 q " 1 s and r 1 ă p, q ă 8, Then the operator S r Ω is L pˆLq Ñ L s bounded with constant independent of Ω, that is for every f P L p , g P L q the inequality The condition p, q ą r 1 is necessary, as can be seen by considering the collection of squares given by Ω " trn, n`1sˆrm, m`1s s.t. m, n P Zu (indeed, in this case the bilinear operator factorizes into linear smooth Rubio de Francia operators and the necessity follows from the linear case as by [11]). The condition 1{p`1{q " 1{s is equally necessary, as can be seen from a simple rescaling argument. In [7] we extended the above result for squares to the case of non-smooth frequency projections, albeit in a smaller range. 7]). Let r ą 2 and let Ω Ă DˆD be a collection of disjoint squares with dyadic sides. Let p, q, s be such that Then the operator T r Ω is L pˆLq Ñ L s bounded with constant independent of Ω, that is for every f P L p , g P L q the inequality First of all, notice that the condition r 1 {2 ă s ă r{2 is redundant in the above statement: it's a consequence of the other two conditions on p, q, s. Secondly, the larger range in Theorem 1.1 was obtained by means of a localisation argument (originally introduced in [4]) which is unavailable in the non-smooth case.
It should be remarked that the proofs of Theorems 1.1 and 1.2 both rely essentially on the sets in Ω being squares.
In this paper we will extend the result above to the case where the collection consists of arbitrary dyadic rectangles instead, again taking the frequency projections to be non-smooth. Precisely, our main result can be stated as follows.
Then the operator T r R is L pˆLq Ñ L s bounded with constant independent of R, that is for every f P L p , g P L q the inequality

holds.
A standard argument shows that we have as a corollary that the same result holds for smooth frequency projections instead, but this time with the added benefit of allowing arbitrary disjoint rectangles, that is with sides not necessarily dyadic in any way (but still parallel to the axes, of course). Corollary 1.5. Let r ą 2 and let R be a collection of disjoint arbitrary rectangles in x R 2 with sides parallel to the axes. Let p, q, s be such that and such that r 1 ă p, q ă r, r 1 {2 ă s ă r{2.
Then the operator S r R is L pˆLq Ñ L s bounded with constant independent of R, that is for every f P L p , g P L q the inequality › › ›´ÿ The proof of the corollary is elementary but has nevertheless been included in Appendix A for completeness. The study of the boundedness of T r R pf, gq is reduced by duality to the study of the boundedness of the trilinear form which can then be further reduced to the study of the boundedness of the trilinear form where h " th R u RPR is a vector valued function, specifically taking values in ℓ r 1 pRq.
Observe moreover that the frequency projection π R pf, gq factorizes as π R1 f¨π R2 g, where R " R 1ˆR2 and π I denotes the multiplier given by y π I f " 1 I p f . We will prove the boundedness result stated below for the trilinear form Λ r R ; Theorem 1.4 then follows from the above remarks. Theorem 1.6. Let r ą 2 and let R Ă DˆD be a collection of disjoint dyadic rectangles in x R 2 . Let p, q, s be such that 1 p`1 q " 1 s and such that r 1 ă p, q ă r, r 1 {2 ă s ă r{2.
Then for every f P L p , g P L q and h P L s 1 pℓ r 1 q we have Let ecpRq denote the eccentricity of the rectangle R, defined as ecpRq :" We split the collection R into two subcollections, according to whether the eccentricity is high or low: of course, the rectangles of eccentricity 1 are just squares and as such we could remove them from the collection since they give rise to a bounded operator, by Theorem 1.2. We then split the trilinear form into Λ r R " Λ r R high`Λ r R low . We will prove Theorem 1.6 separately for R high , R low , using a time-frequency analysis of the trilinear form, and since the proof will be symmetric in the two cases we will concentrate exclusively on R high (see however Remark 7.4 for further details on this symmetry).
Let us briefly explain the structure of the proof by analogy with the linear case of Rubio de Francia operators RdF r I , where I " tI n , n P Zu is a given collection of disjoint intervals. Recall that the inequality }RdF 2 I f } L 2 ď }f } L 2 is a trivial consequence of Plancherel's inequality; it can be rephrased in a vector-valued flavour as }pπ In f q nPZ } L 2 pℓ 2 q ď }f } L 2 . On the other hand, observe that pointwise |π In f pxq| ď 2C f pxq independently of n, where C denotes the Carleson operator and therefore we have by the Carleson-Hunt theorem that }pπ In f q nPZ } L p pℓ 8 q À p }f } L p for any 1 ă p ă 8. By complex interpolation it follows immediately that }RdF r I f } L p À r,p }f } L p for r 1 ă p ă r. As stated before, the operator is also bounded in the range p ě r ą 2, but this is a consequence of the boundedness of RdF 2 I in p ě 2, and the proof of this fact requires methods other than simple interpolation. In the proof of Theorem 1.4 we have adopted a similar approach. Indeed, when r " 8 the bilinear operator becomes and this is easily seen to be bounded pointwise by C f¨C g, in parallel to the linear case, which in turn yields the full range of boundedness for T 8 R . Now, the natural bilinear analogue of the L 2 Ñ L 2 estimate would be L 2ˆL2 Ñ L 1 , and thanks to an interpolation result of Silva from [20] (see Lemma 8.6) Theorem 1.6 would follow from such an estimate for r " 2. However, there is no equivalent of Plancherel's theorem in the bilinear world, and this estimate is not straightforward. We currently do not know if such an estimate holds, the r " 2 case having resisted treatment in general even in the simpler case of squares and smooth frequency projections. As a replacement, we will prove preliminary L pˆLq Ñ L s estimates for T r R for each r ą 2, in a range that gets arbitrarily close to L 2ˆL2 Ñ L 1 as r Ñ 2. This will be enough to allow us to run the interpolation argument to conclude boundedness in the range stated in Theorem 1.4.
Remark 1.7. The overall proof structure explained above is the same the authors have adopted in the proof of Theorem 1.2 in [7], which in turn was inspired by the previous work for smooth squares as in [2] and drew insights from [1,3], in particular the idea of using non-local operators in the stopping-time arguments. The main difference here lies in the fact that we now have to deal with the simultaneous presence of two distinct scales in each π R pf, gq term, namely those given by |R 1 | and |R 2 |, which breaks the symmetry in how the contributions of f and g are treated. See particularly sections §2, §3 for how this is achieved.
The rest of the paper is structured as follows. In section §2, under the hypothesis that all rectangles have high eccentricity, we will reduce the problem to that of bounding a discretized trilinear sum with additional structure of certain (shifted) averages of f, g, h. We will introduce a notion of time-frequency tiles adapted to the particular situation. Sections §3, §4, §5 and §6 develop the time-frequency analysis tools needed to prove the main result. In particular, in section §3 we will introduce some structured collections of tiles referred to as n-columns 1 , where n denotes a shifting parameter, for which we will be able to bound the discretized trilinear sum explicitely. This process will give rise to sizes for f and h, but importantly not for g. Indeed, we will not try to optimize our estimates in g, and rather prove in section §7 some weaker estimates for the discretized trilinear sum in a partial range contained in p P p2, rq and q " r. In section §8 we will improve these estimates by interpolating the partial results with the case r " 8, thus concluding the proof of Theorem 1.4.
Acknowledgements. Both authors are supported by ERC project FAnFArE no. 637510. The authors are also very grateful to Cristina Benea for many useful comments/discussions and in particular for having shared with us a preprint of [3].

Reduction to a discretized model sum
Recall that we are considering the collection R high of rectangles of eccentricity bigger than 1. We start by performing a standard discretization procedure on the trilinear form Λ r R high . We have (using Radon duality, with dσ the induced Lebesgue measure on the plane 1 These play the analogous rôle played by trees, in classical time-frequency analysis terminology.
where we have denoted R 3 :" 2p´R 1´R2 q and χ R3 is a C 8 bump function adapted to R 3 and identically equal to 1 on´R 1´R2 . Notice that |R 3 | " |R 2 | by the assumption that R is of high eccentricity. Now, since the functions f˚q 1 Rj are morally roughly constant in modulus at scale |R j |´1, we do the following change of variables with respect to the smallest time-scale involved, the one given by |R 2 |´1 (indeed, we are assuming where we have gone back to writing π ω f for f˚q 1 ω in the last line.
Remark 2.1. Observe that |π R2 g| is morally roughly constant at scale |R 2 |´1, thus morally roughly constant on the intervals However, |π R1 f | is morally roughly constant at the (generally) larger scale |R 1 |´1. The presence of two different simultaneous scales is the major source of difficulty in the analysis and it is here that the difference with the proof of Theorem 1.2 will rely.
In light of the above remark, we want to take advantage of the fact that |π R1 f | is morally roughly constant at a larger scale in space to reduce to a trilinear form with additional structure. In order to explain this reduction properly we will make use of two types of tiles (mirroring the existence of two simultaneous scales in each π R pf, gq), the first of which we introduce now.

Definition 2.2.
A small tile ρ is a pair of the form pω 2ˆIρ , ω 3ˆIρ q where I ρ , ω 2 are dyadic intervals such that there exists a rectangle R P R with ω 2 " R 2 , ω 3 " R 3 (recall R 3 :" 2p´R 1´R2 q) and |ω 2 ||I ρ | " 1 (hence |ω 3 ||I ρ | " 1 as well; ω 3 is in general not dyadic). Given a small tile ρ we denote by Rpρq this unique rectangle R. The collection of all small tiles is denoted by S.
Given R P R high and n P Z there exists a unique small tile ρ " ρpR, nq such that Rpρq " R and I ρ " |R 2 |´1rn, n`1s, and viceversa. Because of this we can rewrite the trilinear form as by Hölder's inequality this is controlled by ÿ Now we will use the fact that |π R1 f | is roughly constant on many I ρ 's. Let ρ be fixed and let n be such that I ρ " |R 2 |´1rn, n`1s. There exists a unique k P Z such that n " k¨ecpRq`ℓ with 0 ď ℓ ă ecpRq, and thus we can associate uniquely to ρ the interval I " |R 1 |´1rk, k`1s. This is a dyadic interval of length |R 1 |´1 and it has the property that I ρ Ă I.
Given the interval ω let χ ω denote a smooth function such that χ ω pξq " 1 for ξ P ω and χ ω pξq " 0 for ξ R 2ω. By definition of the frequency projections we then have q χ R1˚p π R1 f q " π R1 f , and thus we can bound for every N ą 0 dy.
Therefore we see that we can bound (by Minkowski's inequality) where I is as above (in particular I Ą I ρ ) and that is I n denotes the interval I shifted by n times its length.
With this in mind, we now define the second type of tiles.
Definition 2.3. Given a rectangle R P R and a dyadic interval I, we define the (possibly empty) collection of small tiles S n R,I to be S n R,I :" tρ " pω 2ˆIρ , ω 3ˆIρ q small tile : ω 2 " R 2 , ω 3 " R 3 , I ρ Ď I n u. A super tile P is a pair of the form pR 1ˆI , S n R,I q where R P R and I is a dyadic interval such that |R 1 ||I| " 1 (notice that the pair pR, Iq completely determines the super tile). Given P as above, we let P 1 :" R 1ˆI and call it simply a tile, I P :" I, RpP q :" R (and analogously R j pP q :" R j for j P t1, 2, 3u) and S n P :" S n R,I . Finally, if P is a collection of super tiles, we let RpPq :" tRpP q : P P Pu. Now fix a parameter r 0 such that r ą r 0 ą 2. This will remain fixed throughout sections §3 - §7 (ultimately we will take r 0 to depend on r; one could take for example r 0 " pr`2q{2). Using the fact that ffl I |π R1 f | ď`ffl I |π R1 f | r0˘1 {r0 and the above remarks we can finally bound where Λ n P denote the shifted trilinear forms To bound Λ r R high pf, g, hq it will therefore suffice to bound the trilinear forms Λ n P pf, g, hq in the same range with a constant that is at most polynomial in n (indeed, we will be able to show it is at most logarithmic). In the following sections we will consider n fixed and study the trilinear forms Λ n P . Remark 2.4. It might seem that we are introducing a gratuitous inefficiency in our argument by replacing the L 1 -average ffl I |π R1 f | with the larger`ffl I |π R1 f | r0˘1 {r0 , but this will not be the case. This is due to our use of interpolation in the proof of Theorem 1.6: for the argument to carry through we will only need to prove boundedness of Λ r R high in any range sufficiently close to the L 2ˆL2 Ñ L 1 estimate, for all given r ą 2 (see Section 8). Using L 1 -averages would not enlarge the range obtained through interpolation, at least not with our argument.

n-Columns and sizes
We will now introduce some special collections of super tiles for which we will be able to bound the (shifted) trilinear form explicitely. In the process, we will come up with quantities that will be used as sizes in order to perform a time-frequency analysis of the general trilinear form.
First we define an order relation on the super tiles.
Definition 3.1. Let P, P 1 be super tiles and let n be fixed. Then we say that P ă n P 1 if We say that P ĺ n P 1 if P ă n P 1 or P " P 1 .
Definition 3.2. An n-column C with top a super tile P top is a collection of super tiles such that for every P P C P ĺ n P top .
We denote ToppCq " P top .
Remark 3.3. Notice the rôle of n in the above definitions. The difference between the above n-columns and the columns originally defined in [2] resides in the fact that there is now the shifting parameter n to be taken into account. Thus, unlike in [2], the tiles P 1 for P P C, that is the sets R 1 pP qˆI P , do not form an overlapping tree -unless n " 0. See figure 1 for further clarification.
The following lemma regarding the structure of an n-column is essentially obvious once one unpacks the definitions; it will find its application in the discussion that follows.
Lemma 3.4. Let C be a column with top P top . Then for every super tile P P C we have @ρ P S n P , I ρ Ă I n Ptop . Moreover, if R, R 1 P RpCq are distinct, then we have R 2 X R 1 2 " H. In other words, the small tiles belonging to an n-column agree with the timefrequency portrait of a column as in [2]. The second property in the lemma is due to the disjointness of the rectangles in R. See figure 1 for a pictorial representation of an n-column. Let then C be an n-column and consider the trilinear form Λ n C pf, g, hq. First of all, we fix a super tile P in C and look at the inner sum in the small tiles ρ in (4): we have by Hölder inequality ÿ For the term in g, by Lemma 3.4 we can bound by Hölder's inequality ÿ Figure 1. A composite pictorial representation of an n-column C for n " 3.
Quadrant I represents the frequency plane, quadrant II represents the timefrequency picture of g, quadrant IV represents the (rotated) time-frequency picture of f . The column consists of supertiles tP top , P 1 , . . . , P 5 u and has top P top , whose time interval is Itop. The dashed rectangles in quadrant I are the frequency rectangles: RpCq " tR, R 1 , R 2 u. The rectangles in quadrant II are small tiles (each has area 1). The thicker rectangles in quadrant II represent the union of all small tiles in S n P for P " P top , P 1 , P 3 . The rectangles in quadrant IV are the tiles P top 1 , P 1 1 , . . . , P 5 1 as labeled (each has area 1). Observe that RpP top q " R, RpP 1 q " RpP 2 q " R 1 , RpP 3 q " RpP 4 q " RpP 5 q " R 2 . Notice also that for any j " 1, . . . , 5 we have I n because r ą 2.
For the term in h instead, we see that we have for any small tile ρ and any large where Φ I denotes a rapidly decaying function concentrated in I and equal to 1 there. In particular, Φ I decays like p1`distpz, Iq{|I|q´N for some large N . Therefore, summing over ρ P S n P , we have ÿ With these preliminary estimates at hand, we proceed to estimate Λ n C pf, g, hq. Write I top in place of I ToppCq for shortness. We take the supremum in the terms in f and use Hölder's inequality with exponents pr, r 1 q once again, this time in the super tiles, thus obtaining We then define sizes for f and h according to the above estimate.
Definition 3.5 (Sizes). Let P be a collection of super tiles, and let f P L 1 loc pRq, h P L 1 loc pℓ r 1 q. Then we define Notice that, strictly speaking, Size n f does not depend on the shifting parameter n. However, the associated energy will and thus we keep the index n as a reminder. We can then summarise the above discussion in the following proposition.
Proposition 3.6. Let C be an n-column. Then we can bound As explained in the overview of the proof, we will not introduce a size for g. Control of the contribution of g will rather be achieved through the following lemma. Proof. By the definition of n-column, we have that I n P Ă I n ToppCq for all P P C, and moreover if P, P 1 P C are such that R 2 pP q ‰ R 2 pP 1 q we necessarily have R 2 pP qXR 2 pP 1 q " H (see Lemma 3.4). In other words, the "shifted tiles" R 2 pP qˆI n P are all disjoint and contained in the strip RˆI n ToppCq . If we then rewrite we see that we can bound the integrand pointwise by pVar r C gpxqq r , and we are done.
Remark 3.8. The above lemma relies crucially on the structure of the n-columns to hold; thus we can see that the lemma provides motivation for our definition.
Finally, we conclude this section with the observation that the sizes introduced are controlled by maximal averages, which will allow us to perform the timefrequency analysis. Lemma 3.9. Let P be a collection of super tiles. Then we have where C is the Carleson operator.

Lemma 3.10. Let P be a collection of super tiles. Then we have
Size n h pPq À sup P PP´1 |I P |ˆ} Proof. The claim follows quickly from the definition of an n-column and the rapid decay of the functions Φ I n P .

Energies
In this section we will introduce the remaining quantities that we need in order to enable the time-frequency analysis of Λ n P . It will be necessary to introduce a special notion of disjointness for n-columns (quite similar to that in [2]) in order to ensure good control of the quantities we are going to introduce, which is as follows.
Definition 4.1. Let C be a collection of n-columns. We say that the n-columns in C are mutually disjoint if i) for every C, C 1 P C we have C X C 1 " H (that is, the n-columns are disjoint as sets of super tiles); ii) any two tops are not comparable under the ĺ n relation, or equivalently the sets R 1 pToppCqqˆI n ToppCq for C P C are pairwise disjoint. Notice the presence of the shifting parameter n in the above definition. It is not the tiles ToppCq 1 that are assumed to be disjoint sets in time-frequency, but rather their shifted versions. With this notion of mutual disjointness we can now define the energies. where the inner supremum ranges over all collections C of mutually disjoint n-columns C such that for each C P C and each P P Ć where the inner supremum ranges over all collections C of mutually disjoint n-columns C such that for each C P Ć 1 Remark 4.4. Observe that Energy n f is an L r0 quantity, while Energy n h is an L r 1 quantity. The fact that r 1 ă 2 ă r 0 will be fundamental in summing up the energies coming from a decomposition of P into n-columns (see Proposition 6.1).
We now show how to control the energies in terms of L p norms of the functions f, h. We start with the latter. Lemma 4.5. Let P be a collection of super tiles and let h P L r 1 pℓ r 1 q. Then we have Energy n h pPq À }h} L r 1 pℓ r 1 q . Proof. This lemma is essentially the same as Proposition 2.10 in [7]. Let n P Z and C be a collection of mutually disjoint n-columns such that the pair realizes the supremum in the definition of Energy n h within a factor of 2, that is Energy n h pPq r 1 " 2 r 1 n ÿ CPC |I ToppCq |.
Then the right hand side is dominated by ÿ Observe that as P ranges over the super tiles such that RpP q " R, the intervals I n P are all disjoint; as the n-columns are mutually disjoint we then have that ÿ CPC ÿ P PC: RpP q"R Φ I n P À 1 and the conclusion follows.
Next we will show how to control Energy n f in terms of }f } L r 0 . Here the presence of the shift parameter will produce an unavoidable logarithmic loss in n in the inequality, which will however be more than acceptable for our purposes. Lemma 4.6. Let P be a collection of super tiles and let f P L r0 pRq. Then where log`pxq " logp2`|x|q.
Proof. Let n P Z and C be a collection (which we can assume to be finite) of ncolumns such that they realize the supremum in the definition of Energy n f within a factor of 2, that is We'd like to argue by pointwise control by a Variational Carleson operator as in Lemma 3.7, but we can't do so at this stage because, by the nature of the definition of mutually disjoint n-columns, the tops ToppCq 1 are not disjoint in general. However, we will show that up to introducing a loss of log`pnq in the estimate, we can reduce to the disjoint case. To do so it is convenient to introduce a second ordering on the super tiles (distinct from the previously defined order ă n ).
Definition 4.7. Let P, P 1 be super tiles. We say that P ă P 1 if Thus, the difference between the ă n order relation and the newly introduced ă is that the former involves the shifting parameter n, as the subscript suggests, while the latter doesn't. Now, let P tops " tToppCq s.t. C P Cu and let P max tops denote the subcollection of super tiles that are maximal with respect to the ordering ă just introduced. Notice that the tiles in P tops are all distinct by assumption, which has the following consequence: if one fixes P 0 P P max tops then trivially ÿ P PPtops :P ăP0, |IP |Án´1|IP 0 | |I P | À log`pnq|I P0 |.
As for the tiles P P P tops such that P ă P 0 but |I P | ! n´1|I P0 |, observe that the latter implies that I n P Ă 3I P0 . Since the n-columns were assumed to be mutually disjoint, the shifted tiles R 1 pP qˆI n P are all disjoint, and therefore ÿ P PPtops :P ăP0, |IP |!n´1|IP 0 | |I n P | À |I P0 |.
We have thus shown that ÿ P PPtops Since we have by assumptioń IP |π R1pP q f | r0¯1 {r0 ě 2 n for all P P P tops , we are therefore able to bound Energy n f pPq r0 À log`pnqˆÿ P0PP max tops |π R1pP0q f | r0 1 IP 0 ; but by maximality the super tiles in P max tops are all disjoint (as tiles), and therefore we can bound pointwise (as in Lemma 3.7) ÿ P0PP max tops |π R1pP0q f pxq| r0 1 IP 0 pxq ď pVar r0 C f pxqq r0 .

Thus we have proven
Energy n f pPq r0 À log`pnqˆpVar r0 C f pxqq r0 dx, and the lemma follows (even with the smaller constant Oplog`pnq 1{r0 q) from the fact that r 0 ą 2 and hence Var r0 C is L r0 Ñ L r0 bounded (see [18]).
Remark 4.8. The argument in the above lemma motivates our choice of introducing the parameter r 0 ą 2. Indeed, the Variational Carleson operator Var q C is unbounded when q ď 2, as shown in [18].

Decomposition lemmas
Now that all the relevant quantities are in place, we will establish two decomposition lemmas that will allow us to partition every collection of super tiles P into structured subcollections which have controlled size and energy. The results in this section are classical and are based on simple stopping time arguments.
Lemma 5.1 (Decomposition lemma for f ). Let P be a collection of super tiles, f P L 1 loc pRq, and let n P Z be such that Size n f pPq ď 2´nEnergy n f pPq. Then we can decompose P into P high \ P low so that Size n f pP low q ď 1 2 2´nEnergy n f pPq, and P high can be partitioned into a collection C of mutually disjoint n-columns such that ÿ CPC |I ToppCq | À 2 r0n .
Proof. We can assume the collection P is finite, for simplicity. Let P stock be initialized to and let C be initialized to C :" H. We set right away P low :" PzP stock , which will not be changed throughout the algorithm. The size property is then immediate from the definition of (the initial state of) P stock . As for the organization of P high :" PzP low into n-columns, we proceed as follows. Let P max be the maximal super tile in P stock (with respect to ă n ) such that inf I n Pmax is minimum and sup R 1 pP max q is maximum (any other total order will do, of course). Then we let C be the maximal n-column (with respect to ă n ) in P stock with top P max , update C to be CYtCu and update P stock to be P stock z Ť P PC tP u. Repeat the process until P stock is empty and the algorithm stops. Then we see that by maximality the n-columns in C are mutually disjoint, and as for the bound on ř CPC |I C | notice that we have for each C and for each P P Ć ą 2´n´1Energy n f pPq and therefore just by definition of energy (and its monotonicity) À Energy n f pP high q ď Energy n f pPq, which proves the claim.
Next we have Lemma 5.2 (Decomposition lemma for h). Let P be a collection of super tiles, γ " 2 r0{r 1 , h P L 1 loc pℓ r 1 q, and let n P Z be such that Size n h pPq ď γ´nEnergy n h pPq. Then we can decompose P into P high \ P low so that Size n h pP low q ď γ´n´1Energy n h pPq, and P high can be partitioned into a collection C of mutually disjoint n-columns such that ÿ CPC |I ToppCq | À γ r 1 n " 2 r0n .
We have chosen to introduce the constant γ in order to make the global decomposition lemma below more readily apparent. The proof of the decomposition lemma for h is quite similar to the one for f and is thus omitted. Finally, we can combine the two decomposition lemmas above iteratively in order to produce a global decomposition of the collection P as follows.

Lemma 5.3. Let P be a collection of super tiles. Then there exists a decomposition P "
Ů n P n with the properties: i) Size n f pP n q À minp2´nEnergy n f pPq, Size n f pPqq, ii) Size n h pP n q À minp2´r 0n{r 1 Energy n h pPq, Size n h pPqq, iii) P n is organized into a collection C n of mutually disjoint n-columns, iv) ř CPCn |I C | À 2 r0n . Furthermore, the collection P n is empty if n is such that is largest, putting the resulting P high collection in the corresponding P n and updating P stock at the end of each step to be the collection pP stock q low resulting from the last application of a decomposition lemma. We omit the details.

Generic estimate
In this section we will combine the estimate for a given n-column in Proposition 3.6 with the global decomposition obtained in Section 5 in order to obtain a global estimate for the trilinear form Λ n P in terms of the sizes and energies of the collection P. More precisely, we obtain the following (recall that r ą r 0 ą 2). Then for every 0 ď θ 1 , θ 2 ď 1 such that θ 1`θ2 " 1 we have Proof. Apply the global decomposition lemma (Lemma 5.3) to the collection P, which yields the subcollections P n and their partitions C n into n-columns. For each such subcollection we then have by Proposition 3.6 and Lemma 5.3 that we can bound By Lemma 3.7 the term 1 |I ToppCq | ř P PC´I n P |π R2pP q g| r can be replaced by and therefore we obtain by property (iv) of Lemma 5.3 that |Λ n Pn pf, g, hq| À sup It thus suffices to show that upon summing in n we obtain a quantity that is bounded exactly by the right hand side of (5). This requires a tedious but easy case by case analysis.
Assume then that , the other cases being similar. We have to distinguish two situations: : in this case we have to bound the sum since 1´r 0 {r " σ this sum is readily evaluated to be bounded by : in this case we have ÿ n : S f pE f q´1ă2´nďS which we have already seen is acceptable in the previous case. This concludes the proof. 7. Preliminary result when p " r or q " r In this section we will prove a preliminary result that, when combined with the interpolation technique in Section §8 will yield Theorem 1.6. In particular, we will prove restricted weak type estimates for our trilinear forms with either p " r (for rectangles of low eccentricity) or q " r (for rectangles of high eccentricity).
We begin by stating rigorously the results that will be proven in this section. There are two such results, one for each of the collections R high , R low defined in Section §2. Let A be a measurable set. We say that a measurable set A 1 is a major subset of A if A 1 Ă A and |A 1 | ą 1 2 |A|. Proposition 7.1 (Rectangles of high eccentricity). Let r ą r 0 ą 2 and let R high be a collection of disjoint rectangles such that for every R P R high it holds that ecpRq ą 1. Let p be such that 1 100 1 r 0`9 9 100 1 r ď 1 p ď 1 r 0 and s be such that 1 p`1 r " 1 s .
Then, for any F, G, H finite measurable subsets of R and for any measurable functions f, g such that |f | ď 1 F , |g| ď 1 G , there exists a major subset H 1 Ă H such that for any measurable vector-valued function h " ph R q RPR high that satisfieś ÿ The choice of major subset may depend on r, r 0 , f, g, H but not on p, s.
Observe that the range of exponents is restricted to a subset of the line q " r.
Remark 7.2. The rather clumsy appearance of the above statement is due to some technicalities. Some comments are in order. Firstly, there is the fact that in our argument the major subset H 1 will end up depending on f, g rather than on the sets F, G only -the latter being the standard in typical time-frequency analysis arguments. This is because we will control the contributions of f and g by non-local non-positive averages (see Lemma 3.9 and Lemma 3.7) and thus we will not be allowed to replace f and g by 1 F , 1 G in the definition of the exceptional set that we are going to remove from H (see proof below). However, this will not be a problem for the interpolation argument in the quasi-Banach range (that is, when s ă 1), as will be clear from Lemma 8.6 and its proof. Secondly, the range of boundedness in the statement looks unnatural and arbitrary. This is due to the fact that, in order to perform interpolation between estimates for different values of r, we need the major subset to be independent of p, s. If we gave up this requirement, the methods employed in the proof would yield the more natural range 2 ă p ă r, but then we would not be able to ensure that the choice of major subset is indeed independent of p. In view of a more transparent proof, we have preferred the above formulation.
Similarly, we have Proposition 7.3 (Rectangles of low eccentricity). Let r ą r 0 ą 2 and let R low be a collection of disjoint rectangles such that for every R P R low it holds ecpRq ă 1.
Then, for any F, G, H finite measurable subsets of R and for any functions f, g such that there exists a major subset H 1 Ă H such that for any measurable vector-valued function h " ph R q RPR low that satisfieś ÿ RPR low it holds that |Λ r R low pf, g, hq| À q,r |F | 1{r |G| 1{q |H| 1{s 1 .
The choice of major subset H 1 may depend on r, r 0 , f, g, H but not on q, s.
The range of exponents this time is restricted to a subset of the line p " r instead. The same remarks as in 7.2 apply. Although the ranges in the two above propositions don't intersect, upon interpolation with the r " 8 case they will each yield the same range of estimates, allowing us to sum their two contributions. See next section for details. We now proceed with the proof. We will prove the first of the two propositions above and then comment in Remark 7.4 on the modifications one has to make to the argument in order to adapt it to the second one.
Proof of Prop. 7.1. The proof follows a standard argument originating from [17] (although implicitly present in previous work), together with ideas from [1,4], in particular the idea of using non-local operators for the stopping-time argument below. It will suffice to prove, for data as given in the statement, that for any collection of super tiles P it is Λ n P pf, g, hq À p,q,r rlog`pnqs Op1q |F | 1{p |G| 1{r |H| 1{s 1 for 1 100r0`9 9 100r ď 1 p ď 1 r0 and 1{p`1{r`1{s 1 " 1. Indeed, since the constant is only poly-logarithmic in the shifting parameter n and since by Section §2 the trilinear form Λ r R high is dominated by ÿ nPZ p1`|n|q´N Λ n P pf, g, hq for an arbitrarily large N ą 0, the above implies the same bounds for Λ r R high itself. Given sets F, G, H and functions f, g such that |f | ď 1 F , |g| ď 1 G , we set out to define an exceptional set E (independent of p, s but depending on f, g) which will be removed from H. It will be useful to introduce the following maximal operators: for any m P Z, define M m to be the shifted maximal function It is well known that the shifted maximal function M m is L p Ñ L p bounded for 1 ă p ď 8 and also L 1 Ñ L 1,8 bounded, with constant at most Oplog`pmqq (see for example §4.3.4 of [16]). Now we can define the exceptional set E to be with implicit constants to be chosen below, where xmy :" p1`|m| 2 q 1{2 . Define then the set H 1 :" HzE; we claim that H 1 is a major subset of H. Indeed, for a given m P Z we see by the L 1 Ñ L 1,8 boundedness of M´n`m thaťˇˇ!
x P R s.t. M´n`mppC f q 100r0 qpxq Á log`pm´nqxmy 2 |F | |H| )ˇˇÀ where in the last line we have used the fact that the Carleson operator C is L 100r0 Ñ L 100r0 bounded. Since this is summable in m, we see that for a suitable choice of implicit constants the contribution to |E| of these sets is at most ! |H|. Similarly, for a given m 1 P Z we see by the L 1 Ñ L 1,8 boundedness of M m 1 and by the L r Ñ L r boundedness of Var r C g thaťˇˇ!
which is again summable in m 1 and thus we conclude that |E| ! |H|, which proves the claim. Notice H 1 depends on f, g, r, r 0 and H but not on p. Now, we partition the collection P into P small :"tP P P s.t. I n P Ć Eu, P large :"PzP small , and will estimate the trilinear forms Λ n P small and Λ n P large separately. Notice the presence of the shifting parameter n in the definitions. We start with P small . Given P P P small we observe that since I n P Ć E there exists an x P I n P such that M´nppC f q 100r0 qpxq À log`pnq|F |{|H|, and thus a fortiori it must be that 1 but pI n P q´n " I P , and therefore we must have by Lemma 3.9 that (we are not keeping track of the optimal powers of log`pnq because they will be inconsequential to us). Similarly, we see that for a given P P P small there must be an x P I n P such that M 0 ppVar r C gq r qpxq À |G|{|H|, and therefore for all such P we have 1 |I P |ˆIn P pVar r C gq r dx À |G| |H| .
Moreover, by Lemma 3.10 and the hypothesis that }h} ℓ r 1 ď 1 H 1 we have the trivial bound Size n h pPq À 1 (9) for any arbitrary collection of super tiles P. Recall that σ " pr´r 0 q{r. Combining (7), (8) and (9) with the generic estimate of Proposition 6.1 and the energy estimates of Lemmas 4.6, 4.5, we obtain after a little algebra that for any 0 ď θ 1 , θ 2 ď 1, θ 1`θ2 " 1, Since 1 r 0´9 9 100 σ r 0 " 1 100r 0`9 9 100r 0 , this yields precisely the desired estimates (6) as θ 1 ranges over r0, 1s. Now we are left with showing that (6) holds for Λ n P large as well. In order to do so, we decompose P large into Ů dPN P d where it then suffices to prove that the contribution of Λ n P d is summable in d and that the sum is bounded by the right hand side of (10). Let then d be fixed and observe that if P P P d then 2 d`1 I n P Ć E and 2 d I n P Ă E. This means that for some integer l such that |l| " 2 d the shifted interval pI n P q l " I n`l P is not contained in E. This implies, by definition of E, that for some x P I n`l P we have M´n´lpC f 100r0 qpxq À log`pn`lqxly 2 |F |{|H|, which in turn implies as before Similarly, given P P P d we see that there must be an integer l such that |l| " 2 d and the shifted interval pI n P q l " I n`l P contains a point x such that M´lppVar r C gq r qpxq À log`plqxly 2 |G|{|H|; this in turn implies the estimate Now, estimates (11), (12) are worse than the corresponding ones (7), (8); however, we now have an improved estimate for h. Indeed, if P P P d , we must have by definition of H 1 that distpI n P , H 1 q Á 2 d |I P |; then, by the rapid decay of the functions Φ I n P , for any large M ą 0 we can bound (again by Lemma 3.10) Combining estimates (11), (12) and (13) with Proposition 6.1 and the energy Lemmas 4.6, 4.5 as done before, we obtain for any 0 ď θ 1 , θ 2 ď 1, θ 1`θ2 " 1, that the estimate holds, where M 1 ą M {2, provided M was chosen sufficiently large. This is summable in d ą 0, and since ÿ dą0 ÿ l : |l|"2 d log`pn`lq Op1q d2´d M 1 À plog`pnqq Op1q we see that |Λ n P large pf, g, hq| is controlled by the right hand side of (10) as desired, thus concluding the proof.
Remark 7.4. We conclude the section with a comment on how Proposition 7.3 is proven. In the previous sections we have concentrated on the collection R high of rectangles with large eccentricity (i.e. |R 2 | ą |R 1 |). However, it is clear that the analysis is completely symmetric in the opposite case of R low , in which the rôles of functions f and g are simply swapped. In particular, one has to re-define super tiles reversing f and g and use n-rows instead of n-columns (adapting the definitions is a trivial exercise); then one performs a global decomposition of the collection of super tiles into uniformly controlled n-rows. All lemmas and propositions from previous sections then hold with f replaced by g and viceversa, without substantial changes.

Proof of the main theorem
In this section we will finally complete the proof of Theorem 1.6. This will be achieved by interpolating the results of Section §7 for generic r ą 2 with the trivial case of r " 8, as in the authors' related work [7].
First of all, we state (and prove) the r " 8 case of Theorem 1.6.
Theorem 8.1. Let R be a collection of disjoint rectangles. Then for all 1 ă p, q ă 8 and 1 p`1 q " 1 s it holds that for any f P L p , g P L q is dominated pointwise by C f¨C g. The result is then a trivial consequence of Hölder's inequality and the Carleson-Hunt theorem.
Remark 8.2. We record here for future reference that Theorem 8.1 implies the L pˆLrˆLs 1 pℓ 1 q Ñ R boundedness of the trilinear form Λ 8 R pf, g, hq as given in (2).
In order to combine this trivial result with Propositions 7.1, 7.3, we will need a multilinear interpolation argument for vector-valued operators. This argument is originally due to Silva, who introduced it in [20], and is based upon complex interpolation. We introduce the following definition that extends the usual notion of generalized restricted weak type inequalities to the vector-valued setting we need. Definition 8.3. Let Λ be a trilinear form, let α " pα 1 , α 2 , α 3 q be such that 0 ď α 1 , α 2 ď 1, α 3 ď 1, α 1`α2`α3 " 1 and let t ě 1. We say that Λ is of generalized restricted weak type pα; tq if for every measurable subsets F, G, H of R of finite measure and for every functions f, g such that One can also bound the constant of the interpolated inequality explicitely in terms of the data, but we will not be interested in doing so here.
Proof. The lemma is a particular case of a more general interpolation lemma of Silva (specifically Lemma 4.3 in [20]). We sketch the proof here for the reader's convenience. We argue by complex interpolation and assume t 0 , t 1 ă 8. Let F, G, H, f, g, H 1 , θ be given and let h be such that For z P C with Rez P r0, 1s define h z by h z k pxq :" |h k pxq| tpzq for every k, where tpzq :" p1´zq t θ r 0`z t θ r 1 .
Since the function Ψpzq :" Λpf, g, h z q is easily seen to be holomorphic in the open strip S " tz P C s.t. 0 ă Rez ă 1u, continuous in its closure and bounded, we can apply to it Hadamard's three-lines-lemma and conclude that since h θ`i0 " h we have as desired.
We are now ready to prove Theorem 1.6. It will be a straightforward consequence of the interpolation Lemma 8.6, but some extra care is needed to make sure all hypotheses are verified. The resulting proof is elementary but technical and painful to read; in order to remedy this fact, we have included in Figure 2 a picture that illustrates the proof geometrically.
Proof of Th. 1.6. First of all, as noticed in Remark 8.5, the trilinear form Λ 8 R is of generalized restricted weak type´´1 p0 , 1 q0 , 1 1¯for 1 ă p 0 , q 0 ă 8, 1{p 0`1 {q 01 {s 1 0 " 1 and for any collection R. Secondly, set r 0 " r 0 prq :" pr`2q{2 in the statement of Proposition 7.1 in order to remove the dependence on r 0 ; write t 1 in place of r 1 , so that r 0 prq " p3t 1´2 q{p2t 1´2 q. Then we have that the trilinear form Λ r R high is of generalized restricted weak type´´1 p1 , 1 t 1¯f or all 1 ă t 1 ă 2 and all p 1 , s 1 such that t 1´1 50p3t 1´2 q`9 9 100 1{q 0 1 0 1 r Figure 2. The interpolation argument in geometric form. The horizontal coordinate represents the exponent r, the other coordinates represent the (inverse) exponents 1{p-1{q. A point at coordinates pr, 1{p, 1{qq represents the L pˆLqˆLs 1 pℓ r 1 q Ñ R estimate for the trilinear form Λ r R , where 1{s " 1{p`1{q. In particular, the horizontal line through through the axis of the parallelepiped represents (dualized) L 2ˆL2 Ñ L 1 estimates. At r " 8 the range of boundedness for Λ 8 R is the whole r0, 1sˆr0, 1s (hatched) square in the 1{p-1{q plane, as given by Theorem 8.1. At exponent r 1 intermediate between 2 and 8, Proposition 7.1 gives for Λ r 1 R low the range of boundedness represented by the thickened interval above, as per label. The other thickened interval represents instead the range of boundedness for Λ r 1 R high given by Proposition 7.3. Interpolation between exponents r 1 and 8 can be interpreted geometrically in the above picture as taking the convex hull of the corresponding endpoint ranges. This gives then for an intermediate exponent r 2 a range of boundedness (for Λ r 2 R low ) represented by the darker shaded area at r " r 2 , a subset of the r 1 2 ă p, q ă r 2 range represented by the lighter shaded area. As r 1 tends to 2, the darker shaded area fills the lighter one. Interpolation with the other thickened segment fills the same area in the limit. and 1{p 1`1 {t 1 1`1 {s 1 1 " 1. Fix then some other r ą 2. We claim that for any p, q such that r 1 ă p, q ă r we can choose p 0 , q 0 , p 1 , t 1 , θ so that where 1 ă p 0 , q 0 ă 8, t 1 and p 1 satisfy condition (14), and 0 ă θ ă 1. By the above discussion, a direct application of Lemma 8.6 will then show that Λ r R high is of generalized restricted weak type´´1 p , 1 q , 1 s 1¯; r 1¯. Notice that the major subset can be taken to be the one given by the value t 1 , since for t 0 " 1 the major subset is the set itself. That we can find such quantities as above follows from the limiting case obtained by taking t 1 " p 1 " 2, that is assuming we have the estimate``1 2 , 1 2 , 0˘; 1˘(notice that for such a choice of parameters we don't know if the trilinear form is of generalized restricted weak type, but at the moment we are only concerned with showing the existence of solutions to system (15)). Indeed, in this case it's easy to solve the above system of equations and inequalities: one has θ 2 " 1 r and therefore 1 p0 "´1 p´1 r¯´1´2 r¯´1 and 1 q0 "´1 q´1 r¯´1´2 r¯´1 . In the general case of t 1 ă 2 (for which we know the trilinear form is of generalized restricted weak type), by choosing t 1 sufficiently close to 2 we can solve the system by a perturbation of the solution for the limiting case (by continuity), because the range where Λ r R high is of generalized restricted weak type according to Proposition 7.1 is arbitrarily close to the``1 2 , 1 2 , 0˘; 1˘estimate. Finally, observe that if Λ r R high is of generalized restricted weak type´´1 p , 1 q , 1 s 1¯; r 1t hen r Λ r R high (as given by (1)) is of generalized restricted weak type´1 p , 1 q , 1 s 1¯i n the classical sense, and therefore we can conclude the strong L pˆLqˆLs 1 Ñ R boundedness of r Λ r R high by classical multilinear interpolation. This is equivalent to the L pˆLqˆLs 1 pℓ r 1 q Ñ R boundedness of the trilinear form Λ r R high , of course.
The argument for Λ r R low is essentially identical, and one can easily verify that it yields the same range of estimates. Thus we can conclude estimate (3) by summing up the separate contributions given by collections R high , R low . and such that r 1 ă p, q ă r, r 1 {2 ă s ă r{2.
Then for every f P L p , g P L q and h P L s 1 pℓ r 1 q we have Λ r R pf, g, hq À r,p,q }f } L p }g} L q }h} L s 1 pℓ r 1 q .
Proof. We need only observe that if the rectangles are 3-separated, then for each rectangle R P R there exists a rectangle r R with R Ă r R Ă 3R that belongs to one of the dyadic (shifted) grids DˆD 1 , where D, D 1 are either of the collections 3 D 0 :" tr2 k ℓ, 2 k pℓ`1qs : k, ℓ P Zu, D 1 :" t " 2 k`ℓ`p´1 q k 3˘, 2 k`ℓ`1`p´1 q k 3˘‰ : k, ℓ P Zu, : k, ℓ P Zu.
Then one repeats the proof of Theorem 1.6 for each such product of grids using rectangles r R. The details are left to the reader. Now we claim that Corollary 1.5 follows for arbitrary disjoint rectangles if it holds for, say, 12-separated collections of rectangles. Indeed, let W denote a Whitney decomposition of r´1{2, 1{2s chosen so that I P W ñ 16I Ă r´1{2, 1{2s. Thus we have a decomposition of the square r´1{2, 1{2s 2 given by the rectangles in WˆW, where elements can be indexed with pj, kq P N 2 in such a way that the element corresponding to pj, kq has horizontal side of length À δ j and vertical side of length À δ k for some absolute constant 0 ă δ ă 1. We consider then a smooth partition of unity tφ j,k u associated to the decomposition WˆW, that is 1 r´1{2,1{2s 2 " ř j,k φ j,k , which we can take so that each φ j,k pξ, ηq factorizes as a tensor product φ j b φ k of functions of one variable and so that each φ j,k is supported in the 4{3-enlargement of the rectangle of WˆW indexed by pj, kq. For a given rectangle R, we denote by Ψ R the affine map that sends R to r´1{2, 1{2s 2 preserving orientation (thus χ R " χ˝Ψ R ); if the rectangle is the 4{3-enlargement of the one in WˆW indexed by pj, kq we write Ψ jk for this map. We decompose χ using the partition of unity above: we let c jk :" }χφ jk } 8 and χ jk :" c´1 jk pχ¨φ jk q˝Ψ´1 jk , so that χ " ÿ j,k c jk χ jk˝Ψjk and each χ jk is L 8 normalized. It's easy to see that c jk À minpγ j , γ k q for some 0 ă γ ă 1. Moreover, each χ jk is adapted to r´1{2, 1{2s 2 , in the sense that χ jk is supported in r´1{2, 1{2s 2 and }B α ξ B β η χ jk } L 8 À α,β 1. Using this decomposition of χ we decompose the trilinear form Ξ r χ,R as follows. Notice that pχ¨φ jk q˝Ψ R " c jk χ jk˝p Ψ jk˝ΨR q, so if we define the rectangle R jk to be R jk " pΨ jk˝ΨR q´1pr´1{2, 1{2s 2 q we can see that t3{4R jk u jk is a Whitney decomposition of R which is an affine image of WˆW. We define the collections R jk to be tR jk s.t. R P Ru. Therefore we can write Ξ r χ,R " ÿ j,k c jk Ξ r χ jk ,R jk , and the boundedness of Ξ r χ,R follows from the (uniform in j, k) boundedness of the Ξ r χ jk ,R jk thanks to the summability of c jk . It's easy to verify from the construction above that the rectangles in R jk are indeed 12-separated, and hence it suffices to prove boundedness for 12-separated collections. Now, fix χ and take R to be 12-separated. We can assume that χpξ, ηq " χ 1 pξqχ 2 pηq because we can always reduce to this case by a windowed Fourier series expansion 4 (the coefficients will be summable thanks to the fact that χ is smooth). Letting φ j pxq :" χ 1 j pxq for j " 1, 2 we can write Ξ r χ,R pf, g, hq We want to use Fubini to take the integration in dζdθ out, so for ζ, θ fixed and R P R we let R ζ,θ :" R X prΨ´1 R1 pζq,`8qˆrΨ´1 R2 pθq,`8qq and let R ζ,θ :" tR ζ,θ s. |φ 1 pζq||φ 2 pθq||Λ r R ζ,θ pf, g, hq|dζdθ.
To conclude, notice that R ζ,θ is certainly a 3-separated collection of rectangles and therefore the variant of Theorem 1.6 above applies, giving a bound independent of ζ, θ. Since φ 1 , φ 2 P L 1 , integrating in dζdθ concludes the argument.