LECTURE 3: MULTI-LINEAR RESTRICTION THEORY

Transcription

1 LECTURE 3: MULTI-LINEAR RESTRICTION THEORY JONATHAN HICKMAN AND MARCO VITTURI 1. Bilinear restriction in R 2 Confident of the utility of the decoupling inequality, we now turn to discussing the prerequisite theory required for the proof of Bourgain and Demeter s theorem. The central theme of our analysis will be the multi-linear approach and in these lectures we will investigate the multi-linear restriction theorem of Bennett, Carbery and Tao [1]. To motivate this result, recall when n 2 the restriction conjecture asks us to prove }pgdσqq} Lq pr 2 q À }g} Lp pp 1 q (1) for q ą 4 and p 1 ď q{3. We have seen one approach to proving these estimates via local estimates, the square function conjecture and the Kakeya conjecture; when n 2 both the square function and the Kakeya conjectures are known to hold so this leads to a complete (unconditional) proof of (1) in this case. However, we will now present an alternative, and arguably more direct, argument which relies on recasting the problem in the bilinear setting. By real interpolation, to prove (1) it suffices to show for all Borel sets E Ď P 1 the restricted strong-type estimate }pχ E dσqq} Lq pr 2 q À E 1{p holds whenever q ą 4 and p 1 q{3. Fixing a set E and p, q satisfying these conditions, we observe by squaring this is equivalent to showing }pχ E dσqqpχ E dσqq} L q{2 pr 2 q À E 2{p (2) and hence we have reduced the problem to studying bilinear expressions pg 1, g 2 q ÞÑ pg j dσqq where the g j are functions on P 1. Now, at first sight this may appear more complicated than the original linear problem, but it turns out there are a number of features present in the bilinear setting which make the analysis significantly easier. In particular, whereas the curvature of P 1 was crucial in the linear world, it essentially does not play a rôle in the bilinear theory (provided n 2). Instead, the key geometric consideration is transversality. Definition. For 1 ď j ď n let S j be a compact hypersurfaces in R n and ν j : S j Ñ S n 1 its associated Gauss map. 1 We say the family is ω-transversal for some ω ě 0 if detpν 1 px 1 q... ν n px n qq ě ω whenever x j P S j for 1 ď j ď n. For n 2, the bilinear restriction theorem considers restriction/extension to pairs of transverse curves. x. 1 That is, νj continuously maps a point x P S j to a choice of unit normal vector ν j pxq to S j at 1

2 2 JONATHAN HICKMAN AND MARCO VITTURI Theorem 1. Let S 1, S 2 be a pair of compact, ω-transverse curves in the plane to which we associate smooth densities σ 1, σ 2, respectively. 2 The bilinear restriction estimate pg j dσ j qq À ω Op1q }g j } Lp ps L q{2 pr 2 q jq. (3) holds whenever q ě 4 and p 1 ď q{2 for any pair of L 2 -functions g 1, g 2 on S 1, S 2, respectively. Note that there are no curvature assumptions on S 1, S 2 and, indeed, we may even take the curves to be a pair of lines, provided their direction vectors are suitably separated. It is easy to show the trivial estimate pg j dσ j qq ď }g j } L 1 ps jq L8 pr 2 q holds and interpolation therefore reduces the problem to establishing (3) when q 4 and p 2. Remark. In view of what is to come, it is useful to highlight that the dependence of the constant on the transversality parameter is polynomial in (3) and also in other estimates below. Note that the range of exponents for this bilinear inequality is larger than that of the linear problem. Furthermore, by testing the inequalities against variants of the Knapp example, the range of exponents can be seen to be sharp (see, for instance, [8] for details). We will prove this estimate for S j given by transverse caps lying on the parabola; this is precisely the case of interest. Lemma 2. The bilinear restriction estimate pg j dσqq À ω 1{2 }g j } L2 pp 1 q. L2 pr 2 q for any pair of L 2 -functions g 1, g 2 which are supported on ω-transverse caps on P 1. Proof. Let I 1 and I 2 Ď r 1, 1s be intervals which parameterise the supporting caps of g 1 and g 2, respectively; the transversality hypothesis implies these intervals are Opω 1 q-separated. Observe, pg j dσqqpxq g j pt j, t 2 jqe 2πipx 1pt 1`t 2 q`x 2 pt 2 1`t2 2 qq dt 1 dt 2 I ij I 1 D g j pt j puq, t 2 jpuqq t 1 puq t puq 1 2 e 2πix u du, where we have applied the change of variables given by u 1 : t 1 ` t 2 and u 2 : t 2 1 ` t The latter expression is the Fourier transform of a bivariate function and 2 That is, σj is the product of a non-negative smooth function and arc-length measure on S j. 3 To see this change of variables is valid on I1 ˆ I 2, note if s j, t j P I j for j 1, 2 satisfy s 1 ` s 2 t 1 ` t 2 and s 2 1 ` s 2 2 t 2 1 ` t 2 2, then it follows from the formula 2ab pa ` bq 2 pa 2 ` b 2 q that s 1 s 2 t 1 t 2. Consequently, by comparing coefficients we see that ś 2 pz t jq and ś 2 pz s jq define the same polynomial (here z is a single complex variable) and hence the t j equal the s j, up to permutation. The separation of the intervals now implies t j s j for j 1, 2.

3 l 2 DECOUPLING INEQUALITY 3 so, by Plancherel, we have ij pg j dσqq L 2 pr 2 q 2 1 D I 2 g j pt j puq, t 2 jpuqq 2 t 1 t 2 2 du I 1 g j pt j, t 2 jq 2 t 1 t 1 2 dt 1 dt 2. The result now follows from the bound t 1 t 2 Á ω 1, which is a consequence of the separation hypothesis. Remark. This argument can easily be generalised to prove n-linear restriction estimates for ω 1 -transverse pieces of the moment curve t ÞÑ pt, t 2,..., t n q. Here the Jacobian arising from the change of variables is a (scalar multiple of a) Vandermonde determinant and one can use the same argument as of the footnote to prove the injectivity of the mapping, now invoking the Newton-Girard formulae. To apply these bilinear inequalities to estimate (2) we proceed by partitioning the parabola into disjoint pieces and considering pairs which are transverse. 4 In particular, for k P N 0 let Q Ď r 1, 1s be a dyadic interval of length 2 k and define the cap S SpQq : tpt, t 2 q : t P Qu. Let S k denote the collection of all such caps S, which forms a partition of P 1. If S 1, S 2 P S k are non-adjacent, then the geometry of the parabola ensures they are 2 k -transverse. For almost every pair of distinct points ξ 1, ξ 2 P P 1 there exists a least k P N such that if S j is the unique cap 5 in S k which contains ξ j for j 1, 2, then S 1 and S 2 are not (equal or) adjacent. It is then clear that the parents of these caps are adjacent. Hence, if we define the relation S 1 S 2 for S 1, S 2 P S k then it follows ðñ S 1, S 2 are not adjacent but have adjacent parents, pp 1 q 2 8ď k 1 ď S 1 ˆ S 2 S 1,S 2 PS k S 1 S 2 where the the union is disjoint, all up to a null set. By the triangle inequality and (2), it therefore suffices to show 8 k 1 S 1,S 2PS k S 1 S 2 pχ E dσ S1qqpχ E dσ S2qq À L q{2 pr 2 q E 2{p. (4) To estimate the summands, note that we have a trivial L 8 -bound pχ E dσ S1qqpχ E dσ S2qq À L8 pr 2 q S 1,S 2PS k S 1 S 2 S 1,S 2PS k S 1 S 2 E X S 1 E X S 2. On the other hand, we can use the transversality and Lemma 2 to deduce an L 2 - bound. First we apply the observation that, as S 1, S 2 runs over all pairs in S k satisfying S 1 S 2, the pχ E dσ S1 qqpχ E dσ S2 qqare almost-orthogonal. This is due 4 Here we again see the non-vanishing curvature condition is crucial to the linear problem: the fact that there will be many transverse pairs is only possible since P 1 satisfies this property. 5 Uniqueness holds provided the ξj are not dyadic numbers.

4 4 JONATHAN HICKMAN AND MARCO VITTURI to the fact that the Fourier supports of these functions are essentially disjoint. In particular, we have the estimate 1{2 2 pχ E dσ S1qqpχ E dσ S2qq À }pχ E dσ S1qqpχ E dσ S2qq} L 2 pr 2 L q 2 pr 2 q S 1,S 2PS k S 1 S 2 S 1,S 2PS k S 1 S 2 Now, if S 1 S 2, then we can use the 2 k -transversality of these caps to show }pχ E dσ S1 qqpχ E dσ S2 qq} L2 pr 2 q À 2 k{2 E X S 1 1{2 E X S 2 1{2 and from this it follows pχ E dσ S1qqpχ E dσ S2qq S 1,S 2PS k S 1 S 2 À 2 k{2 L 2 pr 2 q S 1,S 2PS k S 1 S 2 E X S 1 E X S 2 1{2. By Hölder s inequality and the preceding estimates, we can bound the left-hand side of (2) by 8 1 2{q. 2 2k{q E X S 1 E X S 2 (5) k 1 S 1,S 2PS k S 1 S 2 We can easily bound the sum in the above expression in two ways (using the fact for any fixed S 2 P S k there are only Op1q caps S 1 P S k for which S 1 S 2 ) to give E X S 1 E X S 2 À E mint2 k, E u. S 1,S 2PS k S 1 S 2 Using this estimate it is easy to deduce (5) is Op E 2{q q, as required. 2. Multi-linear restriction and Kakeya One can easily prove bilinear restriction estimates in R 2 and such estimates can be effectively applied to study the linear problem. In general dimensions n ě 2 it is natural to consider n-linear restriction estimates; that is, inequalities of the form pg j dσ j qq À }g j } Lp ps L q{n jq (6) pr n q for hypersurfaces S 1,..., S n endowed with smooth measures σ 1,..., σ n, respectively. As in the n 2 case, one hopes that curvature conditions will no longer be relevant in the multi-linear setting and the key consideration will be the more tractable notion of transversality. One may test (6) against variants of the Knapp example to ascertain necessary conditions on the exponents p, q and formulate a reasonable multi-linear version of the restriction conjecture. Remarkably, the resulting conjecture was almost completely resolved by Bennett, Carbery and Tao. In [1] the aforementioned authors succeeded in proving local estimates with sub-polynomial loss in the constants, viz. Theorem 3 (Multi-linear restriction, general case [1]). Let S 1,..., S n be a family of ω-transverse hypersurfaces in R n to which we associate smooth densities σ 1,..., σ n, respectively. The local estimate pg j dσ j qq Æ ω Op1q }g j } Lp ps j q L q{n pb R q holds for q ě 2n{pn 1q and p 1 ď pn 1qq{n.

5 l 2 DECOUPLING INEQUALITY 5 Replacing Æ with À would give a sharp result and so we see the theorem is within ϵ of completely resolving the conjecture in all dimensions. This is quite remarkable, given the dearth of sharp results for the linear problem! By a simple adaptation of the analysis of the first lecture, one can show that Theorem 3 implies a multi-linear variant of the Kakeya conjecture. In analogy with the restriction theory, multi-linearising the Kakeya problem replaces the condition that a single collection of rectangles contains many directions (which loosely corresponds to the non-vanishing curvature hypothesis in restriction theory 6 ) with a transversality condition on n collections of rectangles (corresponding, of course, to the above transversality condition on n hypersurfaces). Definition. Let T 1,..., T n each denote a collection of R 1{2 ˆ ˆ R 1{2 ˆ R rectangles. The collections are ω-transverse if detpνpt 1 q... νpt n qq ě ω whenever T j P T j for 1 ď j ď n, where νpt q denotes the unit vector in which a rectangle T is orientated. Theorem 4 (Multi-linear Kakeya [1]). Let T 1,..., T n each denote a collection of R 1{2 ˆ ˆ R 1{2 ˆ R rectangles and suppose these collections are ω-transverse. Then the estimate L R pn 1q{2 #T j holds. T j PT j χ Tj 1{pn 1q avg pb R q Æ ω Op1q Remark. The R pn 1q{2 factors appearing on the right-hand side of the inequality correspond to the reciprocal of the measure of the cross sections of the T P T j. The estimate is insensitive to the lengths of the rectangles and in fact remains true if the T j have width R 1{2 and infinite length. One clear indication of the relative simplicity of restriction in the present setting is the surprising fact that the multi-linear Kakeya theorem is equivalent to the multilinear restriction theorem! Indeed, Theorem 3 was proved by first establishing the above multi-linear Kakeya estimate, which turns out to be vastly more tractable than its linear counterpart. The proof of this implication will be discussed below. The original proof of Theorem 4 was based on an elegant heat-flow argument. Since then at least three other proofs have appeared and, in particular, a strengthened version of the theorem without the ϵ-loss in R was shown to hold by Guth [5] (see [3] for a subsequent simplification of Guth s argument). It is certainly worth noting that recently Guth [6] gave a remarkably short and accessible proof of Theorem 4 (with ϵ-loss). The details of these proofs will not be discussed here and the interested reader is directed to the above references. We will, however, make a few remarks about the relationship between the multi-linear Kakeya inequality and the classical Loomis-Whitney inequality (Theorem 5). Note that the extreme case of Theorem 4 when the collections of rectangles are 1- transverse corresponds (by Hadamard s inequality) precisely to the situation when all the rectangles in T j are orientated in the same direction ν j and the vectors ν 1,..., ν n form an orthonormal basis. By applying a rotation one may assume that ν j e j are the natural basis vectors on R n. Under these hypotheses, (7) is now a consequence of the following. (7) 6 Note, if a hypersurface has non-vanishing Gaussinan curvature, then the normals to the hypersurface point in many directions.

6 6 JONATHAN HICKMAN AND MARCO VITTURI Theorem 5 (Loomis-Whitney inequality). For measurable f j : e K j Ñ R the inequality ź n f j π L j ď }f 1{pn 1q pr n q j } L1 pe K j q (8) is the orthogonal projection onto the co-ordinate hyper- holds where π j : R n Ñ e K j plane e K j. Note that if the T j are 1-transverse as above and QpT j q : T j X e K j for T j P T j, then each QpT j q is a cube of side-length R 1{2 and T j Ď π 1 j QpT j q. Defining f j : e K j Ñ R by f j : χ QpTj q, T jpt j the Kakeya estimate (7) now follows immediately from (8). The Loomis-Whitney inequality admits many different proofs; here, for completeness, we include a very simple argument based on Hölder s inequality. Proof (of Theorem 5). We induct on dimension, noting the n 2 case is trivial. For x P R n write x px 1, x n q P e K n ˆ R so that the left-hand side of (8) is given by n 1 ź f j π j px 1, x n q 1{pn 1q f n px 1 q 1{pn 1q dx 1 dx n n 1. R e K n By Hölder s inequality this expression is dominated by n 1 ź f j π j px 1, x n q 1{pn 2q dx 1 pn 2q{pn 1qdxn n 1}fn } L1 pe K n q. R e K n The proof is concluded by applying the hypothesised pn 1q-dimensional case of the Loomis-Whitney inequality together with the estimate n 1 ź R }f j p, x n q} 1{pn 1q n 1 n 1 ź L 1 pe K qdx j XeK n ď n }f j } L1 pe K j q, which is a simple consequence of the pn 1q-linear Hölder inequality and Fubini s theorem. It turns out that one can reduce the proof of the multi-linear Kakeya theorem to studying cases where the collections of rectangles are close to being 1-transverse in the sense that the directions of the rectangles belonging to T j lie in a small cap on S n 1 centred around e j. In this way one can think of Theorem 4 as a perturbed version of the Loomis-Whitney inequality. These observations, combined with a simple multi-scale analysis, are used to provide the aforementioned short proof of the multi-linear Kakeya theorem in [6]. The fact that the multi-linear Kakeya inequality is valid even when the collections T j contain many rectangles orientated in the same direction (something which is certainly not true in the linear case!) has a useful consequence. In particular, we can immediately deduce the following weighted version. Theorem 6 (Multi-linear Kakeya, weighted version [1]). Let T 1,..., T n each denote a collection of R 1{2 ˆ ˆ R 1{2 ˆ R rectangles and suppose these collections are ω-transverse. For each T j P T j let w Tj ě 0 be a weight and define the simple functions g j : w Tj χ Tj. T jpt j

7 l 2 DECOUPLING INEQUALITY 7 Then the estimate ź n holds. g j L 1{pn 1q avg pb R q Æ ω Op1q R pn 1q{2 T jpt j w Tj Proof. If w Tj P N for all T j P T j, then the result is a consequence of the original multi-linear Kakeya inequality by including repeats of the tubes in the collections. We can now prove the estimate for rational weights by re-scaling and then for real weights by continuity. 3. Multi-linear Kakeya ñ multi-linear restriction Assuming the multi-linear Kakeya theorem we now show how the multi-linear restriction inequality follows. For ease of notation, we will consider restriction to patches contained in the paraboloid, but the proof can easily be seen to carry through in the generality expressed above. First note that to prove Theorem 3 it suffices to consider the endpoint case where p 2 and 1 ď q ď 2n{pn 1q. Indeed, the remaining estimates then immediately follow from trivial inequalities and interpolation. By applying various uncertainty principle techniques one may rewrite the desired estimate in the following form. Theorem 7 (Multi-linear restriction). Let τ 1,..., τ n be a family of ω-transverse regions (see below) on N R 1pP n 1 q. If 1 ď q ď 2n{pn 1q, then ź n f j 1{n Lq pb R q Æ ω Op1q R n{2 (9) }f j } 1{n L 2 pr n q (10) holds whenever each f j is a Schwartz function with Fourier support in τ j. Here we work with a rather specific notion of a region. Definition. In the coming lectures we ll employ the following terminology. For R 1! ρ! 1 a ρ-region will be the intersection of N R 1pP 2 q with a ρ-ball whose centre lies on P 2. Often the precise value of ρ is unimportant. In this case, we say a set is a region if it is ρ-region for some value of ρ (belonging to the range stated above). A family of regions pτ j q 3 is ω-transverse if the surfaces pτ j X P 2 q 3 are ω-transverse. The proof of the equivalence of Theorem 3 (for patches τ j X P n 1 ) and Theorem 7 involves a simple adaptation of the arguments discussed in the first lecture to the multi-linear setting. The details are left to the reader. Typically we will be interested in the present formulation in applications. It is useful to further note that (10) is equivalent to an ostensibly stronger weighted variant, where the right-hand norm is localised to B R. Weighted estimates will be a common feature of our future analysis and at this juncture it is useful to introduce the following definition. Definition (Smooth localisation). Throughout this discourse w BR will denote rapidly decaying weights concentrated on the ball B R ; that is, w BR is a non-negative measurable function satisfying w BR pxq 1 for x P B R and w BR pxq À 1 ` x cpb N Rq R for some large N Op1q. The precise choice of w BR may vary from line to line or, indeed, within a single line.

8 8 JONATHAN HICKMAN AND MARCO VITTURI For various technical reasons it is preferable to work with this general class of weights rather than Schwartz functions. Definition (Smoothly localised norm). For 1 ď p ď 8, let } } Lp pw BR q and } } L p avg pw BR q denote the L p -norms defined with respect to the measures w BR pxqdx and B R 1 w BR pxqdx, respectively. It is easy to see Theorem 7 is equivalent to showing ź n f L j Æ ω Op1q q{n }f avgpb R q j } L 2 avg pw BR q whenever q and the f j satisfy the hypotheses of the theorem. This follows immediately by taking φ BR to be a Schwartz function which satisfies φ BR pxq Á 1 for x P B R and supp ˆφ BR Ď Bp0, R 1 q and applying (10) to the functions f j φ BR. In this new formulation the norms are localised on both sides of the inequality, which will facilitate an inductive procedure. The proof of Theorem 7 is an instance of what is known as an induction on scales argument. Here one attempts to compare an estimate at a choice of scale R with the same estimate at a smaller scale R 1{2. A recursive application of the analysis then reduces the problem to the small scale case which can be tackled using elementary methods. In particular, it turns out that the theorem is a simple consequence of the following proposition. Proposition 8 (Inductive step - Restriction). Let τ 1,..., τ n be ω-transverse regions, B R a fixed R-ball and B a covering of B R by R 1{2 -balls with bounded-overlap. For 1 ď q ď 2n{pn 1q the inequality }f j } 1{n L 2 avg pw B R 1{2 q l q avg pbq Æ ω Op1q R n{2 }f j } 1{n L 2 pr n q. (11) holds whenever each f j is a Schwartz function with Fourier support in τ j. Roughly, Proposition 8 can be thought of as controlling L 2 avgpw BR q norms by 1{2 L 2 avgpw BR q norms. Indeed, although the right-hand side of (11) involves global norms, in applications we will be able to convert the resulting global estimates into their local counterparts using the simple techniques we ve already encountered. Proposition 8 therefore provides a mechanism for comparing quantities relevant to Theorem 7 at the distinct scales R 1{2 and R. To see how Theorem 7 follows from Proposition 8, we first note that trivially Theorem 7 holds with a sub-optimal dependence on R 1 ; this provides a base case for our induction. Lemma 9 (Trivial multi-linear restriction). Let 1 ď p ď 8, r ď R and fix an R-ball B R and B r a covering of B R by r-balls with bounded-overlap. Then ź n f j 1{n L p avg pb R q À rn{2 }f j } 1{n L 2 avg pw Br q l p avg pb r q holds whenever each f j is a Schwartz function with Fourier support in τ j. Proof. Observe ź n f j 1{n L p avg pb R q À ź n f j 1{n L p avg pb r q ď ź n }f l p j } 1{n avg pb r q L 8 pb r q. l p avg pb r q

9 l 2 DECOUPLING INEQUALITY 9 Furthermore, for each B r P B r we have }f j } L 8 pb rq À r n{2 }f j } L 2 pw B r q whenever f j has Fourier support in τ j Ă Bp0, 10q, by an easily deduced local variant of the Bernstein inequality (see Lemma 10 below). We presently record the local Bernstein inequality applied in the previous argument, which will be of use in our future analysis. Lemma 10 (Local Bernstein). If f has frequency support in some B 1{R, then for any r ě R we have for all 1 ď p ď q ď 8. }f} L q avg pb r q À prrq np1{p 1{qq }f} L p avg pw B r q Proof. For any such f we have the global Bernstein inequality }f} Lq pr n q À R np1{p 1{qq }f} Lp pr n q. To obtain the local version, simply apply this to the function fψ Br where ψ Br is a Schwartz function adapted to B r so that supp ˆf ˆψ Br Ď B 2{R. For any R ě 1 let M q prq denote the smallest constant C for which ź n f j 1{n L q avg pb R q ď C }f j } 1{n L 2 avg pw B R q (12) holds for all f j Schwartz with Fourier support in τ j. With this notation, Theorem 7 can now be succinctly expressed as M q prq Æ ω Op1q (13) for all 1 ď q ď 2n{pn 1q. If, say, 1 ď R ď 100 then M q prq À 1 is an immediate consequence of Lemma 9. 7 The problem is therefore to prove (13) holds at very large scales R " 1. The idea is to use Proposition 8 to compare the problem at a large scale R with the situation at a smaller scale R 1{2 and, in particular, prove an inequality of the form M q prq Æ ω Op1q M q pr 1{2 q. Such an inequality can be applied recursively to essentially reduce the problem to the amenable small-scale case. Proof (of Theorem 7, assuming Proposition 8). By applying a simple transformation one may assume without loss of generality that the τ j are 1-transverse. Fix R ą 100 and 1 ď q ď 2n{pn 1q and observe f j 1{n L À f q j 1{n L q avg pb R q avg pb R 1{2 q l q avg pbq where B is as above. By definition, the right-hand side of the preceding expression is bounded by M q pr 1{2 q }f j } 1{n l L 2 avg pw B R 1{2 q. q avg pbq Now apply Proposition 8 to deduce f j 1{n L Æ M q qpr 1{2 qr n{2 avg pb R q }f j } 1{n L 2 pr n q. 7 Simply take r : R and BR : tb R u.

10 10 JONATHAN HICKMAN AND MARCO VITTURI We have global L 2 -norms appearing on the right-hand side which isn t quite what we want but this can easily be remedied. Indeed, for any R-ball let ψ B2R be a Schwartz function satisfying ψ B2R pxq 1 for all x P B 2R and supp ˆψ B2R Ď Bp0, p2rq 1 q. Replace R with 2R in the above argument and apply the resulting inequality to the functions f j ψ B2R (which have Fourier support in τ j ). From this one concludes and so f j 1{n L Æ M q qpp2rq 1{2 q avg pb R q M q prq ď C ϵ R ϵ{4 M q pp2rq 1{2 q }f j } 1{n L 2 avg pw B R q for every ϵ ą 0. Applying this recursively, for any m P N we have M q prq ď C m ϵ R ϵ{4p1`1{2` `1{2m 1q M q p2 p1{2` `1{2mq R 1{2m q ď C m ϵ R ϵ{2 M q p2 p1{2` `1{2mq R 1{2m q. Now choosing m : rlog 2 log 2 Rs, it follows 2R 1{2m ď 100 and so M q prq À plog 2 Rq 2 log 2 C ϵ R ϵ{2 À ϵ R ϵ. Hence M q prq Æ 1, as required. Thus, the proof of the multi-linear restriction theorem has been reduced to establishing the inductive step. We will first give a heuristic argument to prove Proposition 8, assuming the multi-linear Kakeya theorem holds. We will return to fill in the technical details in the next section. Proof (of Proposition 8 - heuristic version). It suffices to show (11) holds in the endpoint case q 2n{pn 1q. The presence of the L 2 -norms in the desired estimate allow for the application of orthogonality techniques. As in the earlier notes, let f θ denote the frequency restriction of a function f to an R 1{2 -slab θ Ă N R 1pP n 1 q. Since the f θ have disjoint Fourier support, }f} L2 pr n q }f} L p,r 1 pr n q whenever supp ˆf Ď N R 1pP n 1 q, by Plancherel s theorem. It will be useful to have a localised version of this inequality which holds for averages and we assume that for r ě R 1{2 the following estimate holds }f j } L 2 avg pw Br q }f j } L 2,R 1 avg pb r q. (14) This is not a rigorous statement and in truth we cannot guarantee perfect localisation to B r in the norms. For now it is convenient, however, to work with (14) and we will return to the issue of making this rigorous below. The uncertainty principle tells us that the f j,θ should be roughly constant on balls of radius R 1{2 and we will also assume this literally holds in the sense that for every B R 1{2 P B the exists a constant f j,θ pb R 1{2q such that f j,θ pxq f j,θ pb R 1{2q for all x P B R 1{2. (15) Once again, this is merely a heuristic and some work will be required to make formulate a rigorous version of this statement. However, for now we note (14) and (15) together imply ` }f j } L 2 avg pw BR 1{2 q À f j,θ pb R 1{2q 2 1{2

11 l 2 DECOUPLING INEQUALITY 11 so that }f j } 2{pn 1q L 2 avg pw B R 1{2 q À 1 B R 1{2 B R 1{2 ` Averaging over all the balls B R 1{2 P B yields }f j } 1{n l ` L 2 avg pw B R 1{2 q À 2n{pn 1q avg pbq f j,θ pxq 2 1{pn 1q dx. f j,θ 2 1{2n Lavg 1{pn 1q pb R q. (16) The uncertainty principle tells us f j,θ should be roughly constant on rectangles given by translates of the dual of (a suitable rectangle containing) θ. For a fixed j the function f j is Fourier supported in τ j and hence the f j,θ are only non-zero for the slabs θ which intersect this region. These observations lead to the heuristic f j,θ pxq 2 f j pt j q 2 χ Tj pxq (17) T jpt j where T 1,..., T n form a ω-transverse family of rectangles and each f j pt j q is a constant. We are now in position to apply the weighted form of the multi-linear Kakeya inequality (9) which implies ` f j pt j q 2 L χ Æ ω Op1q R pn 1q{2 Tj f 1{pn 1q j pt j q 2 avg pb T j PT R q j T j PT j ω Op1q R n f j pt j q 2 dx R n T j PT j ω Op1q R n f j,θ pxq 2 dx, L 1{pn 1q avg pb R q R n where we have invoked (17). By a second application of (17), we now have the following estimate ` 2 1{2n f j,θ Æ ω Op1q R n{2 }f j } 1{n. L 2,R 1 pr n q Combining this inequality with (16) and Plancherel s theorem immediately yields (11), as required. 4. The rigorous argument We now return to the argument sketched above in order to treat some of the points glossed over in the heuristics. The first issue is to formalise the local orthogonality principle from (14). Note it is important that the choice of scale at which we spatially localise is consistent with the uncertainty principle. Proposition 11 (Local orthogonality). For r ě R 1{2 the following inequalities hold: i) }f} L 2 avg pb r q À }f} L 2,R 1 avg pw B r q ii) }f} L 2 avg pw B r q À }f} L 2,R 1 avg pw B r q whenever supp ˆf Ď N R 1pP n 1 q. Proof. i) Let ψ B2r be a Schwartz function satisfying ψ B2r pxq Á 1 for x P B 2r with supp ˆψ B2r Ă Bp0, p2rq 1 q and observe }f} L 2 avg pb r q À B r 1{2 }fψ B2r } L2 pr n q B r 1{2 ˆf θ ˆψ L B2r 2 pr p. n q

12 12 JONATHAN HICKMAN AND MARCO VITTURI Note each ˆf θ ˆψ B2r is supported in N R 1{2pθq. Moreover, the choice of scale r ensures this support is contained in the union of only Op1q slabs. Thus, the supp ˆf θ ˆψ B2r have bounded overlap and, by Cauchy-Schwarz, }f} L 2 avg pb rq À B 1 r ˆfθ ˆψ B2r 2 1{2 and taking w Br : ψ B2r 2 yields the desired estimate. ii) By the rapid decay of the weight, }f} 2 L 2 avg pw Br q À L 2 p p R n q kpz n p1 ` k q N }f} 2 L 2 avg pbr`krq where N is a large integer (depending on n only), and we can apply part i) to each of the }f} L 2 avg pb r`krq to deduce }f} 2 L 2 avg pw Br q À p1 ` k q N }f} 2. L 2,R 1 kpz n avg pw B r `krq The right-hand side is given by B 1 r f θ pxq 2` R n kpz n p1 ` k q N w Br`krpxq dx, and the expression in brackets is another weight adapted to B r. In addition, we wish to formalise the heuristic (14) that the f j,θ are constant on balls of radius R 1{2. This is achieved by introducing some non-negative ζ R 1{2 P S pr n q which satisfies: i) f j,θ f j,θ Mod ξθ ζ R 1{2; ii) ˆζ R 1{2pξq 1 for ξ P Bp0, R 1{2 q and iii) }ζ R 1{2} L1 pr n q À 1. Hölder s inequality then implies f j,θ pxq 2 À f j,θ 2 ζ R 1{2pxq. (18) By selecting ζ R 1{2 appropriately, one may assume in addition that the function is approximately constant at a certain scale. Definition. A function ζ is approximately constant at scale r ą 0 if ζpxq ζpyq À 1 whenever x y ď r. Lemma 12. If ζ is approximately constant at scale r and ζpxq 1 for x P Bp0, rq, then for any non-negative measurable function F it follows whenever x y À r. F ζ pxq F ζ pyq Proof. The approximately constant property implies F pzq sup ζpx ` w zq dz À F pzq R n wpbp0,rq R n and the result follows. inf wpbp0,rq ζpx ` w zq dz With these heuristic principles now formalised, we can proceed to give a rigorous proof of the inductive step for the multi-linear restriction.

13 l 2 DECOUPLING INEQUALITY 13 Proof (of Proposition 8 - rigorous version). Again, it suffices to assume q 2n{pn 1q. Applying the local orthogonality proposition, one may estimate the left-hand side of (11) thus }f j } 1{n }f j } L 2,R 1 (19) L 2,R 1 avg l À pw BR 1{2 q 2n{pn 1q avg pbq avg pw BR 1{2 q 1{n lavg 2{pn 1q pbq We now choose ζ R 1{2 to be approximately constant on R 1{2 -balls and, using (18), write }f j } 2 À 1 f L 2,R 1 j,θ 2 ζ avg pw BR 1{2 q B R 1{2 R 1{2pxqw BR pxq dx. 1{2 R n Now the locally-constant property is applied: define c j,θ pb R 1{2q 2 to be the value of f j,θ 2 ζ R 1{2 at the centre of the ball B R 1{2. By Lemma 12, it follows that }f j } 2{pn 1q À c L 2,R 1 j,θ pb R 1{2q 2 1{pn 1q pw BR 1{2 q avg c j,θ pb R 1{2q 2 1{pn 1q f j,θ 2 ζ R 1{2 Lavg 1{pn 1q pb R 1{2 q 1{pn 1q Lavg 1{pn 1q. pb R 1{2 q By averaging over B R 1{2 P B and using (19) and the rapid decay of the weights we conclude that }f j } 1{n l À f L 2,R 1 avg pw BR 1{2 q 2n{pn 1q j,θ 2 ζ R 1{2 1{2n avg pbq L 1{pn 1q avg pb R q, (20) which is the rigorous version of (16). Again we wish to envoke the locally-constant property in order to deduce the f j,θ are roughly constant on various rectangles, paving our way to the application of the multi-linear Kakeya inequality. For each R 1{2 -slab θ, let θ denote the R 1{2 ˆ... R 1{2 ˆ R rectangle which is dual to a R 1{2 ˆ... R 1{2 ˆ R 1 rectangle containing θ. Define ζ θ to be a Schwartz function which satisfies i) ζ R 1{2pxq À ζ θ pxq for all x P R n ; ii) ζ θ pxq 1 is approximately constant on tubes. Consequently, f j,θ 2 ζ θ pxq T PT j c j pt q 2 χ T pxq (21) where T j is a collection of rectangles and the c j pt q 2 are suitable constants. The Fourier support properties of the f j allow one to ensure that T 1,..., T n constitutes a ω-transverse family. We apply the weighted multi-linear Kakeya (9) as in the sketch above to deduce that ` c j pt q 2 L χ T 1{pn 1q avg T PT j pb R q Æ ω Op1q The L 1 -estimate for the ζ θ implies f j,θ 2 ζ θ pxq dx À R n R n f j,θ 2 ζ θ. L1 pr n q }f j,θ } 2 L 2 pr n q }f j} 2 L 2 pr n q (22)

14 14 JONATHAN HICKMAN AND MARCO VITTURI and consequently, combining this together with (20), (21) and (22), we have a rigorous proof of (11). 5. An introduction to the Bourgain-Guth method It is natural to ask what are the consequences of Theorem 3 for the linear problem. Unfortunately, the transversality hypotheses make it difficult to apply multilinear restriction results directly to obtain new linear estimates in dimensions n ą 2. After some years Bourgain and Guth [2] introduced certain techniques which allow one to use Theorem 3 to obtain improved partial results on the restriction conjecture in higher dimensions. The rudiments of the Bourgain-Guth method will be important for us in the following lecture and here we will discuss how one can apply this method to prove results for the linear problem in R 3 using the tri-linear restriction theorem. We recall from the first lecture that one formulation of the restriction problem in R 3 is to show that the local estimate }f} Lq pb R q Æ R 1 } ˆf} L8 pn R 1 pp 2 qq (23) holds for q ě 3 whenever f is a smooth function whose Fourier transform is supported in N R 1pP 2 q. Theorem 13. The restriction estimate (23) holds for q ě 3 ` 1{ This result was originally due to Tao [7] who gave a proof using bilinear restriction theory. We ll present a proof using the Bourgain-Guth method: the advantage here is that these techniques can be augmented with additional Kakeya / X-ray transform information to obtain improvements on the exponent. In particular, it was shown in [2] that (23) holds for q ě 56{ , although we will not prove this here. At the time this was the best known result for restriction to P 2, but the range q ě 13{ has now been obtained by Guth [4] by combining ideas from [2] with recently developed techniques used to study Kakeya-like problems (in particular, polynomial partitioning). In high dimensions the Bourgain-Guth method still yields the best known bounds for the linear restriction problem Setting up the induction. The proof of Theorem 13 will utilise another kind of induction on scale argument and for each R ě 1 we define R q prq : R sup }f} Lq pb R q : supp ˆf Ď N R 1pP 2 q and } ˆf} L8 pn R 1 pp 2 qq ď 1u so that the problem is to show R q prq Æ 1. (24) The inequality (24) trivially holds at low scales (1 ď R ď 100, say) since }f} Lq pb R q À R 3{q }f} L8 pr 3 q ď R 3{q } ˆf} L1 pn R 1 pp 2 qq À R 3{q 1 } ˆf} L8 pn R 1 pp 2 qq and so R q prq À R 3{q. Our task is to establish an inductive step to reduce the problem to this situation. Rather than comparing R q at well-separated scales R 1{2 and R as in the previous section, this time we will consider three much closer scales. In particular, we introduce the well-separated scales 1! K 1! K! R and are interested in comparing R q prq with quantities such as R q pr{kq and R q pr{k 1 q. Fix f P S pr n q with supp ˆf Ď N R 1pP 2 q and suppose that } ˆf} L 8 pn R 1 pp 2 qq 1. It will be convenient to also assume supp ˆf Ď Bp0, 1{2q, which we can do without any loss of generality.

15 l 2 DECOUPLING INEQUALITY 15 Cover N R 1pP 2 q with a family of finitely-overlapping K 1 1 -regions tαu, noting such a family must consist of OpK1q 2 regions. Let f α denote a frequency restriction of f to some subset of α so that f α:k 1 1 reg. f α. Further divide each α into OppK{K 1 q 2 q regions tτu, each of diameter K 1 and define f τ in an analogous way. It follows that f α for each K 1 1 -region α. τ:k 1 reg. τăα Remark. It is important to bear in mind that regions such as α and τ are quite different from the slabs θ. In particular, by the uncertainty heuristics R 1 is the scale at which we see the finest possible detail on the frequency side. Consequently, any slab θ can essentially be thought of as a rectangle. However, at scale K 1 1, for instance, we still see the curvature of the paraboloid and each α is thought of as the R 1 -neighbourhood of some small paraboloid (it possesses curvature ). We will borrow the following terminology from [4]. Definition. For 0 ă λ ă 1 and x P B R we say x is λ-broad for f if f τ max α:k 1 1 reg. f α pxq ď λ fpxq. We define the λ-broad part of f to be the function Br λ f which agrees with f on the set of broad points and vanishes everywhere else. Hence, if x is not λ-broad, then fpxq ď λ 1` and therefore, by the triangle inequality, }f} L q pb R q ď }Br λ f} L q pb R q ` λ 1` α:k 1 1 reg. f α pxq q α:k 1 1{q 1 reg. }f α } q L q pb R q 1{q. (25) The broad contribution will be the difficult part to estimate: the remainder, which will refer to as the narrow contribution, can be dealt with using a simple scaling argument, shown below. The advantage of considering broad points is that it facilitates the following bilinear-reduction. Lemma 14. There exists some c ą 0 such that whenever 0 ă λ ă c the following holds. For each x P B R there exists K 1 1 -regions αx 1, α2, x depending on x such that 1) distpα1, x α2q x Á K 1 1 ; 2) Br λ fpxq À K1 2 mint f α x 1 pxq, f α x 1 pxq u. Proof. Suppose x is λ-broad for some 0 ă λ ă 1. Choose the αj x so that they satisfy f α x 1 pxq max f α pxq and f α x α:k 1 1 reg. 2 pxq max α:k 1 1 reg. distpα,α x qą10k f α pxq. Clearly fpxq À K 2 1 f α x j pxq for j 1 and it remains to show the same holds for j 2. Observe that fpxq À f α x 1 pxq ` K 2 1 f α x 2 pxq ď λ fpxq ` K 2 1 f α x 2 pxq,

16 16 JONATHAN HICKMAN AND MARCO VITTURI where the latter inequality is due to the λ-broad hypothesis. estimate holds provided λ is sufficiently small. Thus, the desired Henceforth we assume λ is sufficiently small so that we have the pointwise bound Br λ fpxq À K 2 1 f α x 1 pxq 1{2 f α x 2 pxq 1{2. Remark. In the n 2 case α1, x α2 x form a K 1{2 1 -transverse pair and so one can begin to apply the bilinear restriction theory 8 to obtain good estimates for the broad contribution (provided K 1 is chosen to be very tiny compared with R). This, combined with estimates for the narrow term described below, lead to a slightly different proof of the restriction conjecture for P 1 which can be thought of as a baby version of the Bourgain-Guth argument Estimating the narrow contribution. We will estimate the }f α } L q pb R q appearing in (25) individually and sum; since there are only OpK1q 2 regions τ (and K 1! R) the summation will not be costly. As remarked earlier, each α can be thought of as a neighbourhood of a miniature version of the whole paraboloid. Crucially, however, such a neighbourhood is relatively fat since its width to thickness ratio is 1{K 1 : 1{R, compared with 1 : 1{R for the whole of N R 1pP 2 q. This fattening can be effectively exploited by applying a transformation which sends τ to N K1{RpP 2 q. Lemma 15 (Parabolic rescaling). Let R 1! ρ ď 1. If g has frequency support on a ρ-region, then }g} Lq pb R q À ρ p4{q 2q R q pρrqr 1 }ĝ} L8 pn R 1 pp 2 qq. Since the number of K 1 1 -regions is OpK2 1q, applying the rescaling in the present situation yields q 1{q λ 1` }f α } L q pb R q À λ 1 K 6{q 2 1 R q pr{k qr 1 1, (26) α:k 1 1 reg. which controls the narrow term. Proof (of Lemma 15). Let ξ 0 denote the centre of the ρ-region supporting ĝ and A: R 3 Ñ R 3 an isometry such that A 1 ξ 0 0 and suppose A 1 maps the tangent plane to P 2 at ξ 0 to the horizontal plane te 3 0u (which is the tangent plane to P 2 at 0). In particular, we choose A so that A 1 pp 2 q pu, u 2 q : u P R 2 and u ` πpξ 0 q P r 1, 1s 2(, where π : R 3 Ñ R 2 denotes the orthogonal projection onto the plane te 3 0u. Provided R 1! ρ, it follows supp ĝ A Ď N R 1pP 2 q X Bp0, 2ρq. By applying a simple change of variables, gpx 1, x 3 q ĝpξ, ξ 2 ` ζqe 2πipx1.ξ 1`x 3.p ξ 1 2`ζqq dξ 1 dζ ζ ă1{r 16ρ 4 16ρ 4 ξ 1 ă4ρ ζ ă1{ρ 2 R ζ ă1{ρ 2 R ξ 1 ă1 ĝpρξ, ρ 2 p ξ 2 ` ζqqe 2πipρx1.ξ 1`ρ 2 x 3.p ξ 1 2`ζqq dξ 1 dζ `ĝρ P 2 dσ ζ qρ pxqdζ ζ where P 2 ζ : P 2 ` p0, ζq is endowed with the obvious measure σ ζ and F ρ px 1, x 3 q : F pρx 1, ρ 2 x 3 q for any F : R 3 Ñ C. 8 There is a slight issue here as the regions depend on x, but this can be easily remedied.

17 l 2 DECOUPLING INEQUALITY 17 Applying Minkowski s inequality and rescaling we deduce }g} Lp pb R q À ρ 4 `ĝ ρ P 2 ζ dσ Lq ζ qρ dζ pb R q ζ ă1{ρ 2 R ď ρ p4{q 4q ζ ă1{ρ 2 R `ĝ ρ P 2 ζ dσ ζ q Lq pb ρr q dζ where B ρr is an ρr-ball which contains tpρx 1, ρ 2 x 3 q : x P B R u. Now we use the equivalence of the various formulations of the local restriction inequality to deduce `ĝ ρ P 2 ζ dσ ζ q L À R q pb ρr q qprq}ĝ ρ P 2 ζ } L8 ppζ 2q ď R q prq}ĝ} L8 pn R 1 pp 2 qq and the desired estimate is immediate Estimating the broad contribution. The Bourgain-Guth method gives an effective way to estimate the broad term by roughly interpolating between the trilinear restriction estimate and 2-dimensional restriction inequalities. Presently we sketch some of the basic ideas of the proof. Consider the frequency decomposition at the medium scale K 1 ; the functions f τ are frequency localised to balls of radius K 1 and, consequently, each f τ should be roughly constant on balls of radius K. Assuming this is the case, if we fix one such ball B K, then there exist constants f τ pb K q such that f τ pxq f τ pb K q for all x P B K and so on B K we have a heuristic bound Br λ fpxq À f τ pb K q. τ:k 1 reg. Letting T B K denote the set of all K 1 -regions τ for which f τ pb K q is large there are (roughly speaking) two possible situations: 1) Transverse case: There exists a triple of transverse regions τ 1, τ 2, τ 3 P T B K. 2) Coplanar case: All the τ P T B K lie in a thin strip of the R 1 -neighbourhood of the paraboloid. In the transverse case we essentially have a bound 3ź Br λ fpxq À f τj pxq 1{3 on B K which can be estimated using the tri-linear restriction estimate. On the other hand, in the coplanar case the dominant regions are aligned along a (1-dimensional) parabola embedded in P 2 so that the situation is inherently lower-dimensional. We can therefore use techniques from the well-understood restriction theory for curves. In particular, we will apply a variant of the Fefferman-Córdoba inequality introduced in the first lecture; here our reduction to bounding the broad term Br λ f is useful as it will engender the use of bilinear rather than linear estimates. This sketch is somewhat lacking in detail - for instance, all the estimates described above are over small balls B K rather than the big ball B R - but it gives the essence of the central geometric dichotomy. To begin the rigorous argument, we interpret the locally-constant property as earlier in the notes. In particular, we estimate f τ pxq À f τ ζ K pxq : c τ pxq, (27) where ζ K ě 0 is an L 1 -normalised Schwartz function which is locally-constant at scale K. Given x P B R, define and also let c pxq : max c τ pxq τ:k 1 reg. T x : tτ : c τ pxq ą K 4 c pxqu;

18 18 JONATHAN HICKMAN AND MARCO VITTURI this is the set of regions which provide a large contribution to ř τ c τ pxq. To describe the geometric dichotomy we introduce some notation. Definition. We will let π : R 3 Ñ R 2 denote the orthogonal projection onto the plane te 3 0u R 2. Thus, if ξ P P 2, then it follows ξ pπpξq, πpξq 2 q. The centre of (the ball defining) a ρ-region τ will be denoted ξ τ and we write π τ : πpξ τ q. Consider the following situations: 1) Transverse case: There exist K 2 -transverse regions τ x 1, τ x 2, τ x 3 P T x (which depend on x). 2) Coplanar case: There exists a line lpxq Ă R 2 such that if distpπ τ, lpxqq ą 10 3 K 1, then c τ pxq ď K 4 c pxq. Lemma 16. For any x P B R either the transverse or the coplanar case holds. Proof. Let τ and τ satisfy c τ pxq c pxq and c τ pxq max c τ pxq. τ:distpτ,τ qą10 2K 1 Define lpxq to be the unique line through π τ and π τ. We may suppose that there exists some K 1 -region τ such that distpπ τ, lpxqq ą 10 3K 1 and c τ pxq ě K 4 c pxq; indeed, if not, then the coplanar case holds. Note that this immediately implies c τ pxq ě K 4 c pxq. We claim the triple τ, τ, τ form a K 2 -transverse triple and so the transverse case holds. This claim follows from the separation properties described above together with the following observation: if ξ P P 2, then the unit normal νpξq to P 2 at ξ is proportional to p2πpξq, 1q, and therefore for ξ 1, ξ 2, ξ 3 P P 2 we have πpξ 1 q 1 detpνpξ 1 q, νpξ 1 q, νpξ 1 qq πpξ 2 q πpξ 3 q 1 πpξ 1q πpξ 2 q πpξ 3 q πpξ 2 q, where the latter is proportional to the area of the triangle pπpξ 1 q, πpξ 2 q, πpξ 3 qq. Let E trans denotes the set of all x P B R for which the transverse case holds so that }Br λ f} Lq pb R q ď }Br λ f} Lq pe trans q ` }Br λ f} Lq pb R ze trans q. We will estimate the two terms separately. 1) Transverse case. For x P E trans it follows that 3ź c τ pxq ď c pxq ă K 4 c τ x j pxq 1{3 for every K 1 -region τ and, consequently, fpxq ď τ:k 1 reg. 3ź f τ pxq À K 6 c τ x j pxq 1{3. The regions τj x above depend on x; we remove this dependence by summing over all K 2 -transverse triples of regions. Thus, 3ź fpxq q À K 6q c τj pxq q{3 pτ j q 3 :K 2 trans. and we note that, by Hölder s inequality, 3ź 3ź c τj pxq q{3 ` fτj ζ K pxq q{3 À pr 3 q 3 3ź f τj px z j q q{3 Z K pzq dz

19 l 2 DECOUPLING INEQUALITY 19 where Z K pzq : ś 3 ζ Kpz j q for z pz 1, z 2, z 3 q P pr 3 q 3. Integrating over all E trans and applying Fubini s theorem, we deduce that }Br λ f} q L q pe trans q À K6q pτ j q 3 :K 2 trans. pr 3 q 3 3ź f τj p z j q q{3 L q{3 pb R q Z Kpzq dz Recall } ˆf τ } 8 ď 1 for every K 1 -region τ. It follows from Theorem 3 that for any K 2 -transverse triple pτ j q 3 one has ź 3 f τj p z j q L q{3 pb R q Æ KOp1q R 3 so that }Br λ f} Lq pe trans q Æ K Op1q R 1. 2) Coplanar case. We have reduced the problem to considering the contribution from the points which fail the transversal condition 1) and therefore satisfy the coplanar condition 2). It is in this part of the argument the we will use the reduction to broad points and locally-constant property of the functions c τ. Cover B R ze trans with OpK 3 q balls tb K u of radius K. Fix a ball B K in our cover and assume, without loss of generality, that B K X pb R ze trans q H. By the definition of the coplanar case and the local constant property there exists some line l with the property that if distpπ τ, lq ą 10 3 K 1, then c τ pxq À K 4 c pxq (28) holds uniformly for all x P B K. Thus, the significant contributions to fpxq are given by c τ pxq for τ aligned along the parabola lying above l; this can be seen as a dimensional reduction. For broad x we can apply the bilinear reduction described earlier (recall, it was remarked that this reduction is useful to analyse the n 2 case). In particular, from Lemma 14 it follows that Br λ fpxq À K1 2 f α pxq 1{2 f α 1pxq 1{2. (29) α,α 1 :K 1 1 reg. distpα,α 1 qák 1 1 Here the summation is over all pairs pα, α 1 q of K 1 1 -regions which satisfy distpα, α1 q Á K 1 1 ; we refer to such pairs as non-adjacent. Define Bil l pf α, f α 1qpxq : f τ pxq 1{2 f τ 1pxq 1{2 τăα π τ PLpB K q τ 1 Ăα 1 π τ 1 PLpB K q where LpB K q is the strip LpB K q : N K1{Kplq. Provided 10 3 ď K 1, it follows from (28) that f α pxq 1{2 f α 1pxq 1{2 À Bil l pf α, f α 1qpxq ` ` c τ pxq q 1{q τ:k 1 reg. (30) (it is for this reason we choose to have K 4 appearing in the definition of T x ). Note that the latter term is independent of the ball B K and we proceed to estimate

20 20 JONATHAN HICKMAN AND MARCO VITTURI its L q -norm over the large ball B R. Observe that ` c τ q 1{q L ` }c τ } q 1{q q L pb R q q pb R q τ:k 1 reg. τ:k 1 reg. À }f τ p zq} q L q pb R q ζ Kpzq dz R 3 τ:k 1 reg. 1{q À K 6{q 2 R q pr{kqr 1, (31) where the final inequality is due to the parabolic rescaling lemma. We now concentrate on the bilinear term in (30), noting that the line l appearing in this expression depends on the choice of ball B K. We wish to bound the L q pb K q- norm of Bil l pf α, f α 1q for a non-adjacent pair pα, α 1 q. Unfortunately, we don t have a satisfactory way to do this directly; we can, however, estimate the L 4 pb K q-norm using a variant of the Córdoba-Fefferman argument from lecture 1. Proposition 17 (Bilinear Córdoba/Fefferman inequality). If α, α 1 are non-adjacent K 1 1 -regions, then }Bil l pf α, f α 1q} L4 pb K q À K1 2 ` f τ 2 1{2` f τ 1 2 1{2 L 4 pw BK q. τăα π τ PLpB K q τ 1 Ăα 1 π τ 1 PLpB K q Proposition 17 is proved using the argument for the classical Córdoba/Fefferman square function estimate; the details can be found below. Presently we see how the proposition aids our analysis. Assume 2 ď q ď 4 and observe that q{4 Bil l pf α, f α 1qpxq q dx À K 3p1 q{4q Bil l pf α, f α 1qpxq 4 dx (32) B K B K by Hölder s inequality. Remark. This step in the proof is not efficient and the K 3p1 q{4q factor incurred here is costly. Proposition (17) now bounds the right-hand side of (32) by K 3p1 q{4q K 2q 1 }f τ f τ 1} 2 L 2 pw BK q q{4. τăα τ π τ PLpB K q 1 Ăα 1 π τ 1 PLpB K q To estimate the terms appearing on the right-hand side, we use the locally-constant property c τ pxq c τ pyq for all x, y P B K. (33) In particular, since OpK 1 Kq regions τ satisfy π τ P LpB K q, it follows from (33) and Hölder s inequality that q{4 q{2 }f τ f τ 1} 2 L 2 pb K q À K 3q{4 }c τ } 2 L 8 pb K q τ 1 Ăα 1 π τ 1 PLpB K q π τ PLpB K q ď K 3q{4 pk 1 Kq q{2 1 π τ PLpB K q À K 3p1 q{4q pk 1 Kq q{2 1 }c τ } q L 8 pb K q τ:k 1 reg. }f τ } q L q pb K q. Combining the above estimates and summing over all B K we obtain 1{q 1{2 1{q Bil l pf α, f α 1qpxq q dx À K 2 1 pk 1 Kq }f τ } q L q pb R q B R ze trans τ:k 1 reg. 1{q

21 l 2 DECOUPLING INEQUALITY 21 for any pair of non-adjacent K 1 1 regions. Applying parabolic rescaling, }f τ } Lq pb R q À K 4{q 2 R q pr{kqr 1 for each K 1 -region τ and therefore Bil l pf α, f α 1q À Lq pb R ze trans q KOp1q 1 K 5{q 3{2 R pr{kqr 1 q. α,α 1 :K 1 1 reg. distpα,α 1 qák 1 1 Combining this with (31), we conclude that }Br λ f} Lq pb R ze trans q À K Op1q 1 K 5{q 3{2 R q pr{kqr 1. To conclude this part of the argument, we present the proof of Proposition 17. As mentioned earlier, this is a minor modification of the classical argument described in lecture 1. Proof (of Proposition 17). Fixing α, α 1, pairs of non-adjacent K 1 1 -regions we estimate f τ pxq 2 f τ 1pxq 2 ϕ BK pxq 2 dx R 3 τăα π τ PLpB K q which, by Plancherel, is equal to pr 3 ˇˇ τăα τ π τ PLpB K q 1 Ăα 1 π τ 1 PLpB K q τ 1 Ăα 1 π τ 1 PLpB K q ˆf τ,x ˆ fτ 1,x ˆϕ BK pξqˇˇ2 dξ. Observe that supp ˆf τ ˆ fτ 1 ˆϕ BK Ď N 2K 1pτ 1 τq and so, by Cauchy-Schwarz, the above integral is dominated by ˆf τ ˆ fτ 1 ˆϕ BK pξq 2 χ N2K 1 pτ 1 τqpξq dξ. pr 3 τăα τ π τ PLpB K q 1 Ăα 1 π τ 1 PLpB K q τăα τ π τ PLpB K q 1 Ăα 1 π τ 1 PLpB K q From this we see that to prove the proposition, it suffices to show that for any ξ P R p3 the bound χ N2K 1 pτ 1 τqpξq #tpτ, τ 1 q : ξ P N 2K 1pτ 1 τqu À K1 8 (34) τăα τ π τ PLpB K q 1 Ăα 1 π τ 1 PLpB K q holds. To see this, suppose N 2K 1pτ 1 1 τ 1 q X N 2K 1pτ 1 2 τ 2 q H for K 1 -regions τ j Ă α, τj 1 Ă α1 satisfying π τj, π τ 1 j P LpB K q for j 1, 2. Thus, there exist y j P τ j and yj 1 P τ j 1 for j 1, 2 such that py 1 y 1 1q py 2 y 1 2q À K 1. Since the regions lie in a R 1 neighbourhood of P 2, one may assume without loss of generality that y j pπpy j q, πpy j q 2 q and y 1 j pπpy1 j q, πpy1 j q 2 q. It then follows that pπpy 1 q πpy 1 1qq pπpy 2 q πpy 1 2qq À K 1 ; p πpy 1 q 2 πpy 1 1q 2 q p πpy 2 q 2 πpy 1 2q 2 q À K 1. Recall that each of our regions lies above LpB K q. If v P S 1 ˆ t0u and b P R 2 ˆ t0u, b À 1 are such that l ttv ` b : t P Ru, then there exist t j, t 1 j P R, with t j À 1 and t 1 j À 1, such that πpy j q pb ` t j vq, πpy 1 jq pb ` t 1 jvq À K 1 {K (35)