ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES. 1. Introduction

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1 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO Abstract. We prove -linear analogues of the classical restriction an Kakeya conjectures in R. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely relate to heat flow. We conclue by giving some applications to the corresponing variablecoefficient problems an the so-calle joints problem, as well as presenting some n-linear analogues for n <. 1. Introuction For 2, let U be a compact neighbourhoo of the origin in R 1 an Σ : U R be a smooth parametrisation of a ( 1)-imensional submanifol S of R (for instance, S coul be a small portion of the unit sphere S 1 ). To Σ we associate the extension operator E, given by Eg(ξ) := g(x)e iξ Σ(x) x, U where g L 1 (U) an ξ R. This operator is sometimes referre to as the ajoint restriction operator since its ajoint E is given by E f = f Σ, where enotes the -imensional Fourier transform. It was observe by E. M. Stein in the late 1960 s that if the submanifol parametrise by Σ has everywhere non-vanishing gaussian curvature, then non-trivial L p (U) L q (R ) estimates for E may be obtaine. The classical restriction conjecture concerns the full range of exponents p an q for which such bouns hol. Conjecture 1.1 (Linear Restriction). If S has everywhere non vanishing gaussian curvature, q > 2 1 an p 1 +1 q, then there exists a constant 0 < C < epening only on an Σ such that for all g L p (U). Eg L q (R ) C g Lp (U) See for example [30] for a iscussion of the progress mae on this problem, the rich variety of techniques that have evelope in its wake, an the connection to Date: April Mathematics Subject Classification. 42B10. The first author was supporte by EPSRC Postoctoral Fellowship GR/S27009/02, the secon by EC project HARP, an the thir by a grant from the Packar Founation. 1

2 2 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO other problems in harmonic analysis, partial ifferential equations, an geometric analysis. In particular the restriction problem is intimately connecte to the Kakeya problem, which we shall iscuss later in this introuction. In recent years certain bilinear analogues of the restriction problem have come to light in natural ways from a number of sources (see for example [10], [19], [23], [31], [35], [28], [29], [32], [20], [14] concerning the well-poseness theory of non-linear ispersive equations, an applications to a variety of central problems in harmonic an geometric analysis). More specifically, given two such smooth mappings Σ 1 an Σ 2, with associate extension operators E 1 an E 2, one may ask for which values of the exponents p an q, the bilinear operator (g 1, g 2 ) E 1 g 1 E 2 g 2 may be boune from L p L p to L q/2. The essential point here is that if the submanifols parametrise by Σ 1 an Σ 2 are assume to be transversal (up to translations), then one can expect the range of such exponents to broaen; again see [31]. However, one of the more puzzling features of such bilinear problems is that, in three imensions an above, they seem to somewhat confuse the role playe by the curvature of the associate submanifols. For example, it is known that the bilinear restriction theories for the cone an paraboloi are almost ientical, whereas the linear theories for these surfaces are not (see [30] for further iscussion of this). Moreover, simple heuristics suggest that the optimal k-linear restriction theory requires at least k nonvanishing principal curvatures, but that further curvature assumptions have no further effect. In imensions it thus seems particularly natural to consier a -linear set-up, as one then oes not expect to require any curvature conitions. For each 1 j let Σ j : U j R be such a smooth mapping an let E j be the associate extension operator. Our analogue of the bilinear transversality conition will essentially amount to requiring that the normals to the submanifols parametrise by the Σ j s span at all points of the parameter space. In orer to express this in an appropriately uniform manner let A, ν > 0 be given, an for each 1 j let Y j be the ( 1)-form Y j (x) := 1 k=1 x k Σ j (x) for all x U j ; by uality we can view Y j as a vector fiel on U j. We will not impose any curvature conitions (in particular, we permit the vector fiels Y j to be constant), but we will impose the transversality (or spanning ) conition ( ) et Y 1 (x (1) ),..., Y (x () ) ν, (1) for all x (1) U 1,..., x () U, along with the smoothness conition Σ j C 2 (U j) A for all 1 j. (2) Remark 1.2. If U j is sufficiently small then E j g j = Ĝ j σ j, where G j : Σ j (U j ) C is the normalise lift of g j, given by G j (Σ j (x)) = Y j (x) 1 g j (x), an σ j is the inuce Lebesgue measure on Σ j (U j ).

3 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 3 By testing on the stanar examples that generate the original linear restriction conjecture (characteristic functions of small balls in R 1 see [27]) we are le to the following conjecture 1. Conjecture 1.3 (Multilinear Restriction). Suppose that (1) an (2) hol, q 2 1 an p 1 q. Then there exists a constant C, epening only on A, ν,, an U 1,..., U, for which L E j g j C g j L p (U j) (3) q/ (R ) for all g 1 L p (U 1 ),..., g L p (U ). Remark 1.4. Using a partition of unity an an appropriate affine transformation we may assume that ν 1 an that for each 1 j, Σ j (U j ) is containe in a sufficiently small neighbourhoo of the j th stanar basis vector e j R. Remark 1.5. By multilinear interpolation (see for example [6]) an Höler s inequality, Conjecture 1.3 may be reuce to the enpoint case p = 2, q = 2 1 ; i.e. the L 2 estimate L E j g j C g j L2 (U 2/( 1) (R ) j). (4) We emphasise that at this -linear level, the optimal estimate is on L 2, rather than L 2 1. It shoul also be pointe out that the conjecture range of exponents p an q is inepenent of any aitional curvature assumptions that one might make on the submanifols parametrise by the Σ j s. This is very much in contrast with similar claims at lower levels of multilinearity. It is instructive to observe that if the mappings Σ j are linear, then by an application of Plancherel s theorem, the conjecture inequality (4) (for an appropriate constant C) is equivalent to the classical Loomis Whitney inequality [22]. This elementary inequality states that if π j : R R 1 is given by π j (x) := (x 1,..., x j 1, x j+1,..., x ), then f 1 (π 1 (x)) f (π (x)) x f 1 1 f 1 (5) R for all f j L 1 (R 1 ). One may therefore view the multilinear restriction conjecture as a certain (rather oscillatory) generalisation of the Loomis Whitney inequality. The nature of this generalisation is clarifie in Section 2. Remark 1.6. Conjecture 1.3 in two imensions is elementary an classical, an is implicit in arguments of C. Fefferman an Sjölin. In three imensions this (trilinear) problem was consiere in [5] (an previously in [3]), where some partial results on the sharp line p = 1 q were obtaine. It is a well-known fact that the linear restriction conjecture implies the so-calle (linear) Kakeya conjecture. This conjecture takes several forms. One particularly simple one is the assertion that any (Borel) set in R n which contains a unit line 1 Strictly speaking, this is a multilinear extension or multilinear ajoint restriction conjecture rather than a multilinear restriction conjecture, but the use of the term restriction is well establishe in the literature.

4 4 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO segment in every irection must have full Hausorff (an thus Minkowski) imension. Here we shall consier a more quantitative version of the conjecture, which is stronger than the one just escribe. For 0 < δ 1 we efine a δ-tube to be any rectangular box T in R with 1 sies of length δ an one sie of length 1; observe that such tubes have volume T δ 1. Let T be an arbitrary collection of such δ-tubes whose orientations form a δ-separate set of points on S 1. We use #T to enote the carinality of T, an χ T to enote the inicator function of T (thus χ T (x) = 1 when x T an χ T (x) = 0 otherwise). Conjecture 1.7 (Linear Kakeya). Let T an δ be as above. For each 1 < q there is a constant C, inepenent of δ an the collection T, such that Lq χ T Cδ ( 1)/q (#T) 1 1 q( 1). (6) (R T T ) The proof that Conjecture 1.1 implies Conjecture 1.7 follows a stanar Raemacherfunction argument going back implicitly to [15] an [2]. The enpoint q = 1 of (6) can be seen to be false (unless one places an aitional logarithmic factor in δ on the right-han sie), either by consiering a collection of tubes passing through the origin, or by Besicovitch set examples. See [37] for a etaile account of these facts. By a straightforwar aaptation 2 of the techniques in the linear situation, the multilinear restriction conjecture can be seen to imply a corresponing multilinear Kakeya-type conjecture. Suppose T 1,..., T are families of δ-tubes in R. We allow the tubes within a single family T j to be parallel. However, we assume that for each 1 j, the tubes in T j have long sies pointing in irections belonging to some sufficiently small fixe neighbourhoo of the j th stanar basis vector e j in S 1. It will be convenient to refer to such a family of tubes as being transversal. (The vectors e 1,..., e may be replace by any fixe linearly inepenent set of vectors in R here, as affine invariance consierations reveal.) 1 Conjecture 1.8 (Multilinear Kakeya). Let T 1,..., T an δ be as above. If q then there exists a constant C, inepenent of δ an the families of tubes T 1,..., T, such that ( T j T j χ Tj ) Lq/(R) C (δ /q #T j ). (7) Remark 1.9. Since the case q = is trivially true, the above conjecture is equivalent via Höler s inequality to the enpoint case q = 1. In contrast to the linear setting, there is no obvious counterexample prohibiting this claim holing at the enpoint q = 1, an inee in the = 2 case it is easy to verify this enpoint estimate. Remark By contrast with similar statements at lower levels of multilinearity, each family T j is permitte to contain parallel tubes, an even arbitrary repetitions 2 In the multilinear setting one consiers wave-packet ecompositions of Ej g j for general g j L 2 (U j ), an places ranom ± signs on each summan.

5 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 5 of tubes. By scaling an a limiting argument we thus see that the conjecture inequality reuces to the superficially stronger ( ) ) χ Tj µ Tj Lq/(R) C (δ /q µ Tj (8) T j T j T j T j for all finite measures µ Tj (T j T j, 1 j ) on R. Remark The ecision to formulate Conjecture 1.8 in terms of δ δ 1 tubes is largely for historical reasons. However, just by scaling, it is easily seen that (7) is equivalent to the inequality ( T j T j χ Tj ) Lq/(R) C (# T j ), (9) where the collections T j consist of tubes of with 1 an arbitrary (possibly infinite) length. (Of course we continue to impose the appropriate transversality conition on the families T 1,..., T here.) Remark As may be expecte given Remark 1.5, the special case of the conjecture inequality (or rather the equivalent form (8) with q = 1, an an appropriate constant C) where all of the tubes in each family T j are parallel, is easily seen to be equivalent to the Loomis Whitney inequality. We may therefore also view the multilinear Kakeya conjecture as a generalisation of the Loomis Whitney inequality. The geometric nature of this generalisation is of course much more transparent than that of Conjecture 1.3. In particular, one may fin it enlightening to reformulate (8) (with q = 1 ) as an l1 vector-value version of (5). Remark As mentione earlier, the linear Kakeya conjecture implies something about the imension of sets which contain a unit line segment in every irection. The multilinear Kakeya conjecture oes not have a similarly simple geometric implication, however there is a connection in a similar spirit between this conjecture an the joints problem; see Section 7. Remarkably, at this -linear level it turns out that the restriction an Kakeya conjectures are essentially equivalent. This equivalence, which is the subject of Section 2, follows from multilinearising a well-known inuction-on-scales argument of Bourgain [8] (see also [31] for this argument in the bilinear setting). Once we have this equivalence we may of course focus our attention on Conjecture 1.8, the analysis of which is the main innovation of this paper. The general iea behin our approach to this conjecture is sufficiently simple to warrant iscussion here in the introuction. First let us observe that if each T j T j is centre at the origin (for all 1 j ), then the left an right han sies of the conjecture inequality (7) are trivially comparable. This observation leas to the suggestion that such configurations of tubes might actually be extremal for the left han sie of (7). Question Is it reasonable to expect a quantity such as ( T j T j χ Tj ) Lq/(R)

6 6 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO to be monotone increasing for q slie to the origin? 1 as the constituent tubes simultaneously For reasons both analytic an algebraic, in pursuing this iea it seems natural to replace the rough characteristic functions of tubes by gaussians (of the form e π A(x v),(x v) for appropriate positive efinite matrices A an vectors v R ) aapte to them. As we shall see in Sections 3 an 4, with this gaussian reformulation the answer to the above question is, to all intents an purposes, yes 1 for q >. In Section 3 we illustrate this by giving a new proof of the Loomis Whitney inequality, which we then are able to perturb in Section 4. As a corollary of our perturbe result in Section 4 we obtain the multilinear Kakeya conjecture up to the enpoint, an a weak form of the multilinear restriction conjecture. More precisely, our main results are as follows. Theorem 1.15 (Near-optimal multilinear Kakeya). If 1 < q then there exists a constant C, inepenent of δ an the transversal families of tubes T 1,..., T, such that ( ) Lq/(R) C (δ /q #T j ). T j T j χ Tj Furthermore, for each ɛ > 0 there is a similarly uniform constant C for which ( T j T j χ Tj ) L1/( 1)(B(0,1)) Cδ ɛ (δ 1 #T j ). 2 1 Theorem 1.16 (Near-optimal multilinear restriction). For each ɛ > 0, q an p 1 q, there exists a constant C, epening only on A, ν, ɛ,, p an q, for which L E j g j q/ (B(0,R)) CRɛ g j Lp (U j) for all g j L p (U j ), 1 j, an all R 1. In Section 2 we show that Theorem 1.16 follows from Theorem In Section 4 we prove Theorem 1.15, an in Section 5 we iscuss the applications of our techniques to lower orers of multilinearity, an to more general multilinear k-plane transforms. In Section 6 we erive the natural variable coefficient extensions of our results using further bootstrapping arguments closely relate to those of Bourgain. Finally, in Section 7 we give an application of our results to a variant of the classical joints problem consiere in [12], [25] an [16]. Remark The monotonicity approach that we take here arose from an attempt to evise a continuous an more efficient version of an existing inuction-on-scales

7 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 7 argument 3 introuce by Wolff (an inepenently by the thir author). This inuctive argument allows one to euce linear (an multilinear) Kakeya estimates for families of δ-tubes from corresponing ones for families of δ-tubes. However, unfortunately there are inefficiencies present which prevent one from keeping the constants in the inequalities uner control from one iteration to the next. Our esire to minimise these inefficiencies le to the introuction of the formulations in terms of gaussians aapte to tubes (rather than rough characteristic functions of tubes). The suggestion that one might then procee by an inuction-on-scales argument, incurring constant factors of at most 1 at each scale, is then tantamount to a certain monotonicity property. We shoul emphasise, however, that this reasoning serve mainly as philosophical motivation, an that there are important ifferences between the arguments presente here an the aforementione inuction arguments. (Curiously however, one of the most natural seeming formulations of monotonicity fails at the enpoint q = /( 1) when 3 see Proposition 4.6.) We also remark that closely relate monotonicity arguments for gaussians are effective in analysing the Brascamp-Lieb inequalities [11], [4]; see Remark 3.4 below. Remark There is perhaps some hope that variants of the techniques that we introuce here may lea to progress on the original linear form of the Kakeya conjecture. For 3 it seems unlikely that our multilinear estimates (in their current forms) may simply be reassemble in orer to achieve this. However, our multilinear results o suggest (in some non-rigorous sense) that if there were some counterexamples to either the linear restriction or Kakeya conjectures, then they woul have to be somewhat non-transverse (or plany, in the terminology of [18]). Issues of this nature arise in our application to joints problems in Section 7. Remark The -linear transversality conition (1) has also turne out to be ecisive when estimating spherical averages of certain multilinear extension operators (g 1,..., g ) E 1 g 1 E g of the type consiere here. See [1] for further etails. Notation. For non-negative quantities X an Y, we will use the statement X Y to enote the existence of a constant C for which X CY. The epenence of this constant on various parameters will epen on the context, an will be clarifie where appropriate. Acknowlegement. We woul like to thank Jim Wright for many helpful iscussions on a variety of techniques touche on in this paper. 2. Multilinear Restriction Multilinear Kakeya It will be convenient to introuce some notation. For α 0, q 2 1 an p 1 q, we use R (p p q; α) 3 This inuction-on-scales argument is closely relate to that of Bourgain, an plays an important role in our applications in Section 6.

8 8 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO to enote the estimate L E j g j CR α q/ (B(0,R)) g j Lp (U j) for some constant C, epening only on A, ν, α,, p an q, for all g j L p (U j ), 1 j, an all R 1. Similarly, for 1 q, we use to enote the estimate ( T j T j χ Tj K (1 1 q; α) ) Cδ q/ α (δ /q #T j ) (10) for some constant C, epening only on α, an q, for all transversal collections of families of δ-tubes in R, an all 0 < δ 1. We again note that (10) is equivalent by stanar ensity arguments (in suitable weak topologies) to the superficially stronger ( ) χ Tj µ Tj Lq/(R) ) (δ Cδ α /q µ Tj T j T j T j T j for all finite measures µ Tj (T j T j, 1 j ) on R. With this notation, Theorem 1.15 is equivalent to the statements K (1 1 q; 0) for all 1 < q, an K (1 1 1 ; ɛ) for all ɛ > 0; Theorem 1.16 is equivalent to R ( ; ɛ) for all ɛ > 0. As we have alreay iscusse, a stanar Raemacher-function argument allows one to euce the multilinear Kakeya conjecture from the multilinear restriction conjecture. In the localise setting this of course continues to be true; i.e. for any α 0, (11) R ( ; α) = K (1 1 1 ; 2α). (12) Multilinearising a well-known bootstrapping argument of Bourgain [8] (again see [31] for this argument in the bilinear setting) we shall obtain the following reverse mechanism. Proposition 2.1. For all α, ɛ 0 an 2 1 q, R (2 2 q; α) + K (1 1 q 2 ; ɛ) = R (2 2 q; α 2 + ɛ 4 ). Using elementary estimates we may easily verify R ( ; α) for some large positive value of α. For example, noting that B(0, R) = c R for some constant c, we have that L E j g j c R ( 1)/2 2/( 1) (B(0,R)) E j g j c R ( 1)/2 g j L1 (U j),

9 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES ; ( 1) which by the Cauchy Schwarz inequality yiels R (2 2 2 ). In the presence of appropriately favourable Kakeya estimates, this value of α may then be reuce by a repeate application of the above proposition. In particular, Proposition 2.1, along with implication (12), easily allows one to euce the equivalence R ( ; ɛ) ɛ > 0 K (1 1 1 ; ɛ) ɛ > 0. Arguing in much the same way allows us to reuce the proof of Theorem 1.16 to Theorem 1.15, as claime in the introuction. The proof we give of Proposition 2.1 is very similar to that of Lemma 4.4 of [31], an on a technical level is slightly more straightforwar. We begin by stating a lemma which, given Remark 1.2 an the control of Y j implicit in (1) an (2), is a stanar manifestation of the uncertainty principle (see [13] for the origins of this, an Proposition 4.3 of [31] for a proof in the bilinear case which immeiately generalises to the multilinear case). Lemma 2.2. R (2 2 q; α) is true if an only if L f j q/ (B(0,R)) CRα /2 f j 2 (13) for all R 1 an functions f j supporte on A R j := Σ j(u j ) + O(R 1 ), 1 j. We now turn to the proof of the proposition, where the implicit constants in the notation will epen on at most A, ν,, p, α an ɛ. From the above lemma it suffices to show that L f j Rα/2+ɛ/4 /2 f j L q/ (B(0,R)) 2 (A R j ) for all f j supporte in A R j, 1 j. To this en we let φ be a real-value bump function aapte to B(0, C), such that its Fourier transform is non-negative on the unit ball. For each R 1 an x R let φ x (ξ) := e 2πix ξ R /2 φ(r 1/2 ξ). R 1/2 Observe that φ x R 1/2 is an L 1 -normalise moulate bump function aapte to B(0, C/R 1/2 ), whose Fourier transform is non-negative an boune below on B(x, R 1/2 ), uniformly in x. From the hypothesis R (2 2 q; α) an Lemma 2.2 we have φ x f L R 1/2 j Rα/2 /4 q/ (B(x,R 1/2 )) f j φ x R 1/2 L 2 (A R j ) (14) for all x. Averaging this over x B(0, R) we obtain L f j Rα/2 /4( R /2 q/ (B(0,R)) B(0,R) ( q/(2)x ) /q. f j φ x R 2 1/2 L 2 (R )) Now for each 1 j we cover A R j by a bounely overlapping collection of iscs {ρ j } of iameter R 1/2, an set f j,ρj := χ ρj f j. Since (for each j) the supports of (15)

10 10 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO the functions f j,ρj φ x have boune overlap, it suffices to show that R 1/2 L f j R α/2 /4( ( R /2 q/ (B(0,R)) B(0,R) ρ j f j,ρj φ x R 1/2 2 L 2 (R ) ) q/(2)x ) /q. The function φ x R 1/2 is rapily ecreasing away from B(x, R 1/2 ), an so by Plancherel s theorem, the left han sie of (16) is boune by CR α/2 /4( R /2 B(0,R) ( ρ j f j,ρj 2 L 2 (B(x,R 1/2 )) ) q/(2)x ) /q, since the portions of φ x on translates of B(x, R 1/2 ) can be hanle by translation R 1/2 symmetry. For each ρ j let ψ ρj be a Schwartz function which is comparable to 1 on ρ j an whose Fourier transform satisfies ψ ρj (x + y) R (+1)/2 χ ρ j (x) for all x, y R with y R 1/2, where ρ j enotes an O(R) O(R1/2 ) O(R 1/2 )-tube, centre at the origin, an with long sie pointing in the irection normal to the isc ρ j ; the implicit constants here epening only on A, ν an. We point out that this is where we use the full C 2 (U j ) control given by conition (2). If we efine f j,ρj := f j,ρj /ψ ρj, then f j,ρj an f j,ρj are pointwise comparable, an furthermore by Jensen s inequality, f j,ρj (x + y) 2 = f j,ρj ψ ρj (x + y) 2 R (+1)/2 f j,ρj 2 χ ρ j (x) whenever x R an y R 1/2. Integrating this in y we conclue f j,ρj 2 L 2 (B(x,R 1/2 )) R 1/2 f j,ρj 2 χ ρ j (x), an hence by rescaling the hypothesis K(1 1 q 2 ; ɛ) (in its equivalent form (11)) we obtain L f j q/ (B(0,R)) R α/2 /4( R /2 R α/2+ɛ/4 /2 R α/2+ɛ/4 /2 B(0,R) ( ( ) q/(2)x ) /q R 1/2 f j,ρj 2 χ ρ j (x) ρ j f j,ρj 2 L 2 (A R j ) ρ j f j L2 (A R j ). In the last two lines we have use Plancherel s theorem, isjointness, an the pointwise comparability of f j,ρj an f j,ρj. This completes the proof of Proposition 2.1. ) 1/2 (16)

11 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES The Loomis Whitney case As we have alreay iscusse, the Loomis Whitney case of Conjecture 1.8 correspons to the situation where each family T j consists of translates of a fixe tube with irection e j ; the transverality hypothesis allows us to assume (after a linear change of variables) that e 1,..., e is the stanar orthonormal basis of R. The gaussian reformulation of this inequality (or rather the equivalent inequality (9)) allue to in the introuction is now ( 0 j ( vj),( vj) ) L C #V j, (17) 1/( 1) (R ) v j V j e π A where for each 1 j, A 0 j is the orthogonal projection to the jth coorinate hyperplane {x R : x j = 0}, an V j is an arbitrary finite subset of R. The matrix A 0 j, which is just the iagonal matrix whose iagonal entries are all 1 except for the j th entry, which is zero, we refer to as the j th Loomis Whitney matrix. We will actually consier a rather more general n-linear setup where there are n istinct matrices A j, which are not necessarily commuting, an no relation between n an is assume. We aopt the notation that A p B if B A is positive semi-efinite, an A > p B if B A is positive efinite. The observations that if A p B, then D AD p D BD for any matrix D, an also if A p B > p 0 then B 1 p A 1 > p 0 (as can be seen by comparing the norms Ax, x 1/2 an Bx, x 1/2 on R an then using uality) will be useful at the en of the proof of the next proposition. Proposition 3.1. Let, n 1 an A 1,..., A n be positive semi-efinite real symmetric matrices. Let µ 1,..., µ n be finite compactly supporte positive Borel measures on R. For t 0, x R, an 1 j n let f j (t, x) enote the non-negative quantity f j (t, x) := e π Aj(x vjt),(x vjt) µ j (v j ). R Then if n ker A j = {0} 4 an p = (p 1,..., p n ) (0, ) n is such that the quantity is non-increasing in time. A 1,..., A n p p 1 A p n A n, (18) Q p (t) := R f j (t, x) pj x Corollary 3.2. Uner the conitions of Theorem 3.1, ( 1 f j (1, x) pj x f j (0, x) pj x = et A R where A := p 1 A p n A n. R ) 1 2 µ j pj, 4 Note that this is equivalent to the non-singularity of p1 A p na n. See [4] for further analysis of the conition (18).

12 12 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO Remark 3.3. The Loomis Whitney case (17) (with C = 1) now follows from Corollary 3.2 on setting n =, A j = A 0 j, p j = 1/( 1) for all j, an the measures µ j to be arbitrary sums of Dirac masses. Remark 3.4. Variants of Proposition 3.1 an Corollary 3.2, by the current authors an M. Christ [4], have recently lea to new proofs of the funamental theorem of Lieb [21] concerning the exhaustion by gaussians of the Brascamp Lieb inequalities (of which the Loomis Whitney inequality is an important special case). Although our proof of Proposition 3.1 is rather less irect than the one given in [4] (which is closely relate to the heat-flow approach of [11]), it oes seem to len itself much better to the perturbe situation, as we will iscover in the next section. Remark 3.5. In the statement of Proposition 3.1, we interpret t as the time variable so that the v j s are the velocities with which the gaussians slie to the origin. The proof we give of Proposition 3.1 is rather unusual. We begin by consiering the integer exponent case when p = (p 1,..., p n ) N n. By multiplying out the (p j ) th powers in the expression for Q p (t), an using Fubini s theorem, we may obtain an explicit formula for the time erivative Q p(t). With some careful algebraic an combinatorial manipulation, we are then able to rewrite this expression in a way that makes sense for p N n, an is manifestly non-positive whenever A 1,..., A n p p 1 A p n A n. Finally, we appeal to an extrapolation lemma (see the appenix) to conclue that the formula must in fact also hol for p (0, ) n. It may also be interesting to consier this approach in the light of the notion of a fractional cartesian prouct of a set (see [7]). Proof of Proposition 3.1. We begin by consiering the case when p N n. Then the quantity Q p (t) efine in this proposition can be expane as R (R ) p 1 e π n p j k=1 Aj(x v j,kt),(x v j,k t) (R ) pn On completing the square we fin that p j k=1 µ j (v j,k )x. p j A j (x v j,k t), (x v j,k t) = A (x vt), (x vt) + δt 2, k=1 where A := n p ja j is a positive efinite matrix, v := A 1 n A pj j k=1 v j,k is the weighte average velocity, an δ is the weighte variance of the velocity, δ := p j A j v j,k, v j,k A v, v. (19) k=1 Using translation invariance in x, we thus have Q p(t) = 2πt R (R ) p 1 (R ) pn δ p j k=1 e π Aj(x v j,kt),(x v j,k t) µ j (v j,k )x.

13 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 13 If for each j we let v j be v j regare as a ranom variable associate to the probability measure e π Aj(x vjt),(x vjt) µ j (v j ), f j (t, x) an let v j,1,..., v j,pj be p j inepenent samples of these ranom variables (with the v j,k being inepenent in both j an k), then we can write the above as Q p(t) = 2πt E(δ) R f j (t, x) pj x where δ is now consiere a function of the v j,k, an E() enotes probabilistic expectation. By linearity of expectation we have E(δ) = p j E( A j v j,k, v j,k ) E( A v, v ), (20) k=1 (where of course v is v regare as a ranom variable). By symmetry, the first term on the right-han sie is n p je( A j v j, v j ). As for the secon term, by efinition of v we have E( A v, v ) = E( A 1 = p j A j v j,k, k=1 p j p j j =1 k=1 k =1 p j j =1 k =1 A j v j,k ) E( A 1 A j v j,k, A j v j,k ). When (j, k) (j, k ) we can factorise the expectation using inepenence an symmetry to obtain E( A v, v ) = p j E( A 1 A j v j, A j v j ) + p j A 1 A j E(v j ), A j E(v j ) 1 j,j n Combining these observations together, we obtain Q p(t) = 2πt G(p, t, x) R p j p j A 1 A j E(v j ), A j E(v j ). f j (t, x) pj x (21)

14 14 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO where G is the function { G(p, t, x) := p j E( (Aj A j A 1 A j )v j, v j ) (A j A j A 1 A j )E(v j ), E(v j ) } + = + p j A j E(v j ), E(v j ) j,j p j p j A 1 A j E(v j ), A j E(v j ) p j E( (A j A j A 1 A j )(v j E(v j )), (v j E(v j )) p j A j (E(v j ) E(v)), (E(v j ) E(v)). Note that G makes sense now not just for p N n, but for all p (0, ) n. On the other han by the chain rule we have ( n Q p(t) = 2π p k E ( A k v k, x tv k )) n f j (t, x) pj x. (22) R k=1 Now we observe that since the ajugate matrix aj(a ) := et(a )A 1 is polynomial in p, et(a )G(p, t, x) is also polynomial in p. Hence multiplying both (21) an (22) by et(a ) an using Lemma 8.2 of the appenix, (along with the hypothesis that A is non-singular), we may euce that (21) in fact hols for all p (0, ) n. Now, if we choose p so that A p A j hols for all 1 j n, the inner prouct (A j A j A 1 A j ), is positive semi-efinite, an hence G(p, t, x) is manifestly non-negative for all t, x. This proves Proposition 3.1. Remark 3.6. The secon formula for G(p, t, x) above may be re-expresse as G(p, t, x) = p j E( (A j A j A 1 A j )(v j E(v j )), (v j E(v j )) + p j A j E(v j ), E(v j ) A E(v), E(v). While in this formulation it is not obvious at a glance that G(p, t, x) 0, we can make a centre of mass change in the preceeing argument to euce this. Inee, if we subtract v 0 = v 0 (t, x) from each v j,k in the efinition (19) of δ, then the value of δ remains unchange. If we now choose v 0 = E(v), the last term in our expression for G(p, t, x) vanishes, whence G is nonnegative as before. This type of Galilean invariance will also be use crucially in the next section to obtain a similar positivity. While this proof of Proposition 3.1 is a little more laboure than the one we have presente in [4] (see also [11]), it nevertheless paves the way for the arguments of the next section where the (constant) matrices A j are replace by ranom matrices A j an are thus subject to the expectation operator E. In that context, we are able to prove a suitable variant of the formula for G presente in this remark.

15 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 15 Remark 3.7. If n p ja j > A l for all l, we can expect there to be room for a stronger estimate to hol. This will also play an important role in the next section where we will use it to hanle error terms arising in our analysis. 4. The perturbe Loomis-Whitney case In this section we prove a perturbe version of Proposition 3.1 of the previous section. Theorem 1.15 will then follow as a special case. Although Theorem 1.15 is our main goal, working at this increase level of generality has the avantage of proviing more general (multilinear) k-plane transform estimates at all levels of multilinearity. We shall iscuss these further applications briefly in the next section. If A is a real symmetric matrix, we use A to enote the operator norm of A (one coul also use other norms here, such as the Hilbert-Schmit norm, as we are not tracking the epenence of constants on ). Proposition 4.1. Let, n 1, ε > 0 an M 1,..., M n be positive semi-efinite real symmetric matrices. In aition suppose that n ker M j = {0} an p = (p 1,..., p n ) (0, ) n is such that p 1 M p n M n > p M j (23) for all 1 j n. For each 1 j n let Ω j be a collection of pairs (A j, v j ), where v j R, A j p 0, A 1/2 j M 1/2 j ε, an let µ j be a finite compactly supporte positive Borel measure on Ω j. For t 0, x R, an 1 j n let f j (t, x) enote the non-negative quantity f j (t, x) := e π Aj(x vjt),(x vjt) µ j (A j, v j ). Ω j Then if ε is sufficiently small epening on p an the M j s, we have the approximate monotonicity formula R f j (t, x) pj x (1 + O(ε)) R f j (0, x) pj x for all t 0, where we use O(X) to enote a quantity boune by CX for some constant C > 0 epening only on p an the M j s. Corollary 4.2. If p is such that (23) hols, then if ɛ is small enough epening on p an the M j s, we have the inequality R where M = p 1 M p n M n. ( ) f j (1, x) pj x (1 + O(ɛ)) et M µ j pj,

16 16 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO To prove the corollary it is enough to show that there is a c > 0 (epening only on p an the M j s) such that n f j (0, x) pj e π (1 cɛ)m x,x µ j pj (24) for all x R. To this en we first observe that for any λ > 0 we have the ientity ( ) pj f j (0, x) pj = e π Ajx,x µ j (A j, v j ) Ω j ( = e π (1 λɛ n ) pj k=1 p k)m x,x e π (Aj Mj+λɛM )x,x µ j (A j, v j ). Ω j Now since M > p 0, there exists a constant c > 0, such that M p c I, where I enotes the ientity matrix. Using this an (26) below, we may choose λ > 0 (epening only on p an the M j s) such that A j M j + λɛm is positive efinite, an hence e π (Aj Mj+λɛM )x,x 1 for all x. Setting c = λ p k completes the proof of (24), an hence the corollary. Remark 4.3. When the irections of the tubes in T j are sufficiently close to e j, Theorem 1.15 follows from Corollary 4.2 on setting n =, M j = A 0 j, p j = p > 1 1 an the measures µ j to be appropriate sums of Dirac masses. By affine invariance an the triangle inequality one can hanle any linearly inepenent irection sets, although of course the constants may now get significantly larger than 1. Proof of Proposition 4.1. From the estimate A 1/2 j M 1/2 j ε we have an thus A 1/2 j = M 1/2 j + O(ε) (25) A j = M j + O(ε). (26) Again, we begin by consiering the case when p is an integer. Let Q p enote the quantity Q p (t) := f j (t, x) pj x; we can expan this as Q p (t) =... R Ω p 1 1 Ω pn n R e π n p j k=1 A j,k(x v j,k t),(x v j,k t) p j k=1 µ j (A j,k, v j,k )x. It turns out that this quantity will not be easy for us to stuy irectly (mainly because of the quantity A 1 which will appear in the erivative of Q p ). Instea, we shall consier the moifie quantity Q p (t) efine by Q p (t) := et(a )e π p n p j j k=1 A j,k(x v j,k t),(x v j,k t) µ j (A j,k, v j,k )x R Ω p 1 1 Ω pn n k=1 (27)

17 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 17 where A := n pj k=1 A j,k is a positive efinite matrix: the point is that the eterminant et(a ) will eventually be use to convert A 1 into a quantity which is a polynomial in the A j,k (the ajugate or cofactor matrix of A ); i.e. et(a )A 1 = aj(a ). Let us now see why the weight et(a ) is mostly harmless. From (26) we have A = M + O(ε), where M = p 1 M p n M n, an so in particular we have et(a ) = et(m ) + O(ε). Thus for ε small enough, et(a ) is comparable to the positive constant et(m ). Since all the terms in Q p (t) an Q p (t) are non-negative, we have thus establishe the relation Q p (t) = (1 + O(ε)) et(m ) 1 Qp (t) (28) for integer p 1,..., p n. However, for applications we nee this relation for non-integer p 1,... p n. There is an obvious ifficulty in oing so, namely that Q p (t) is not even efine for p N n. However this can be fixe by performing some manipulations (similar to ones consiere previously) to rewrite Q p (t) as an expression which makes sense for arbitrary p (0, ) n. To this en, for each j we let (A j, v j ) be ranom variables (as before) associate to the probability measure e π Aj(x vjt),(x vjt) µ j (A j, v j ) f j (t, x) an let (A j,1, v j,1 ),..., (A j,pj, v j,pj ) be p j inepenent samples of these ranom variables (with the (A j,k, v j,k ) being inepenent of (A j,k, v j,k ) when (j, k) (j, k )). Then we can rewrite (27) as Q p (t) = E(et(A )) f j (t, x) pj x, R where A := n pj k=1 A j,k. If we write R j,k := M j A j,k then the ranom variables R j,k are O(ε) (by (26)). Observe that for fixe j, all the R j,k have the same istribution as some fixe ranom variable R j, an we can write et(a ) = et(m ) + P ((R j,k ) 1 j n;1 k pj ) where P is a polynomial in the coefficients of the R j,k which has no constant term (i.e. P (0) = 0). Thus we have Q p (t) = et(m )Q p (t) + E(P (R j,k ) 1 j n;1 k pj ) f j (t, x) pj x. R Now by taking avantage of inepenence an symmetry of the ranom variables R j,k, we can write the expression E(P (R j,k )) as a polynomial combination of p an of the (tensor-value) moments E(R m j ) for some finite number m = 1,..., M of m (with M epening only on n, an of course the M j s); thus we have Q p (t) = et(m )Q p (t) + P (p, E(R m j ) 1 j n;1 m M ) f j (t, x) pj x R (29)

18 18 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO for some polynomial P epening only on n, an the Mj s. Since P has no constant term it is easy to see that P (p, ) also has no constant term, i.e. P (p, 0) = 0 (this can also be seen by consiering the ε = 0 case). We remark that this polynomial can be compute explicitly using the Lagrange interpolation formula, although we make no use of this here. The right-han sie of (29) makes sense for any p 1,..., p n > 0, not necessarily integers, an so we shall aopt it as our efinition of Qp in general. Since R j = O(ε) an P has no constant term we observe that P (p, E(R m j ) 1 j n;1 m M ) = O(ε), whence we obtain (28) for all p 1,..., p n > 0 (not just the integers), though of course the implicit constants in the O notation will certainly epen (polynomially) on p. In orer to prove the proposition, it thus suffices by (28) to show that the quantity Q p (t) is non-ecreasing in time for sufficiently small ε. Again, we begin by working with p N n, so that we may use (27). Now we ifferentiate Q p (t). As before, we can complete the square an write p j A j,k (x v j,k t), (x v j,k t) = A (x vt), (x vt) + δt 2 k=1 where v := A 1 n pj k=1 A j,kv j,k is the weighte average velocity, an δ is the weighte variance δ := p j A j,k v j,k, v j,k A v, v. (30) k=1 Arguing as in the previous section, we thus have Q p(t) = 2πt... R Ω p 1 1 Ω pn n et(a )δ p j k=1 Recalling the ranom variables A j,k, v j,k, we thus have Q p(t) = 2πt E(et(A )δ) R e π A j,k(x v j,k t),(x v j,k t) µ j (A j,k, v j,k ) x. f j (t, x) pj x (31) where δ is now consiere a function of the A j,k an v j,k. Let us rewrite δ slightly by inserting the efinition of v an using the self-ajointness of A 1, to obtain δ = p j A j,k v j,k, v j,k A 1 k=1 p j A j,k v j,k, k=1 p j j =1 k =1 A j,k v j,k. We now take avantage of a certain Galilean invariance of the problem. We introuce an arbitrary (eterministic) vector fiel v 0 (t, x) which we are at liberty to select later, an observe that the above expression is unchange if we replace all

19 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 19 the v j,k by v j,k v 0 : δ = p j A j,k (v j,k v 0 ), (v j,k v 0 ) k=1 A 1 p j A j,k (v j,k v 0 ), k=1 p j j =1 k =1 A j,k (v j,k v 0). If we multiply this quantity by et(a ) then we obtain a polynomial: et(a )δ = p j et(a ) A j,k (v j,k v 0 ), (v j,k v 0 ) k=1 aj(a ) p j A j,k (v j,k v 0 ), k=1 p j j =1 k =1 A j,k (v j,k v 0) where aj(a ) is the ajugate or cofactor matrix of A, which is a polynomial in the coefficients of A an is thus polynomial in the coefficients of the A j,k. To make this expression more elliptic, we introuce the matrices B j,k := A 1/2 j,k an the vectors w j,k := B j,k (v j,k v 0 ), an write (32) as et(a )δ = p j et(a ) w j,k 2 k=1 aj(a ) p j B j,k w j,k, k=1 p j j =1 k =1 (32) B j,k w j,k. (33) Note that A = n pj k=1 B2 j,k is a polynomial in the B j,k, so the right-han sie of (33) is a polynomial in the B j,k an w j,k. We can then take expectations, taking avantage of the inepenence an symmetry of the ranom variables (B j,k, w j,k ), an obtain a formula of the form E(et(A )δ) = S(p, (E((B j, w j ) m )) 1 j n;1 m M ) (34) for some polynomial S, an where the power of moments M epens only on (in fact it is 2 + 2, which is the egree of the polynomial in (33)). Here of course (B j, w j ) represents any ranom variable with the same istribution as the (B j,k, w j,k ). Also note that as the right-han sie of (33) is purely quaratic in the w j,k, the expression (34) must be purely quaratic in the w j. Just as before, one may of course use the Lagrange interpolation formula here to write own an explicit expression for S. Inserting this formula back into (31) we obtain Q p(t) = 2πt S(p, (E((B j, w j ) m )) 1 j n;1 m M ) R f j (t, x) pj x (35)

20 20 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO when p N n. However, ifferentiating with respect to t through the integral in (29) leas to an alternative expression for Q p(t) of the form Q p(t) = 2π K(p, t, x) R f j (t, x) pj x, vali for all p (0, ) n, where K is polynomial in p an of polynomial growth in x. Hence by Lemma 8.2 of the appenix, ientity (35) must also be true for arbitrary p (0, ) n. Thus to conclue the proof it will suffice to show that S(p, (E((B j, w j ) m )) 1 j n;1 m M ) 0 (36) for any p satisfying the hypotheses of the theorem, if ε is sufficiently small epening on p an the M j s. From (25) we know that B j = M 1/2 j + O(ε), an in particular we have B j = O(1). Since S is purely quaratic in the w j, we thus have S(p, (E((B j, w j ) m )) 1 j n;1 m M ) = S(p, (E((M 1/2 j, w j ) m )) 1 j n;1 m M ) O ( ε E( w j 2 ) ) (37), as we can use the Cauchy Schwarz inequality to control any cross terms such as E(w j ) 2 or E(w j ) E(w k ). Lemma 4.4. For arbitrary p 1,..., p n > 0 S(p, (E((M 1/2 j, w j ) m )) 1 j n;1 m M ) { n = et M p j E(w j ) 2 + M 1/2 p j E( (I M 1/2 j M 1 M 1/2 j )(w j E(w j )), (w j E(w j )) ) p j M 1/2 j E(w j ) 2} (38) Proof Since the esire ientity is polynomial in p 1,..., p n, it suffices to restrict our attention to p 1,..., p n N. From (34) an (33) (with the B j replace by M 1/2 j ) we have S(p, (E((M 1/2 j,w j ) m )) 1 j n;1 m M ) = E ( n et(m ) w j,k 2 aj(m ) p j M 1/2 j w j,k, k=1 p j k=1 p j j =1 k =1 M 1/2 j w j,k ).

21 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 21 Writing aj(m ) = et(m )(M ) 1, we thus have S(p, (E((M 1/2 j, w j ) m )) 1 j n;1 m M ) p j = et(m ) E( w j,k 2 ) k=1 et(m )E( M 1 p j M 1/2 j w j,k, k=1 which by inepenence an symmetry we can rewrite as et(m ) p j E( w j 2 ) et(m ) et(m ) 2 et(m ) p j E( M 1 M 1/2 j w j, M 1/2 j w j ) p j j =1 k =1 p j (p j 1) M 1 M 1/2 j E(w j ), M 1/2 j E(w j ) 1 j<j n p j p j M 1 M 1/2 j E(w j ), M 1/2 j E(w j ). Now we argue as in the previous section an write p j E( w j 2 ) p j E( M 1 M 1/2 j w j, M 1/2 j w j ) an Hence = E( (I M 1/2 j M 1 M 1/2 j )w j, w j ) = (I M 1/2 j M 1 M 1/2 j )E(w j ), E(w j ) p j E( (I M 1/2 j M 1 M 1/2 j )w j, w j ), + E( (I M 1/2 j M 1 M 1/2 j )(w j E(w j )), (w j E(w j )) ). S(p, (E((M 1/2 j, w j ) m )) 1 j n;1 m M ) { n = et(m ) p j (I M 1/2 j M 1 M 1/2 j )E(w j ), E(w j ) + 2 M 1/2 j w j,k ), p j E( (I M 1/2 j M 1 M 1/2 j )(w j E(w j )), (w j E(w j )) p j (p j 1) M 1 M 1/2 j E(w j ), M 1/2 j E(w j ) 1 j<j n p j p j M 1 M 1/2 j E(w j ), M 1/2 j E(w j ) }.

22 22 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO We can rearrange the right-han sie as { n et(m ) p j E(w j ) p j E( (I M 1/2 j M 1 M 1/2 j )(w j E(w j )), (w j E(w j )) p 2 j M 1 M 1/2 j E(w j ), M 1/2 j E(w j ) 1 j<j n which can be rearrange further as { n et(m ) p j E(w j ) 2 + M 1/2 p j p j M 1 M 1/2 j E(w j ), M 1/2 j E(w j ) } p j E( (I M 1/2 j M 1 M 1/2 j )(w j E(w j )), (w j E(w j )) ) p j M 1/2 j E(w j ) 2}. This conclues the proof of the lemma. The last term in (38) has an unfavourable sign. However, we can at last select our vector fiel v 0 in orer to remove this term. More precisely, we efine v 0 by the formula v 0 := (E ( n p j M 1/2 j A 1/2 j ) ) 1 n p j E(M 1/2 j A 1/2 j v j ), which is well efine for ε sufficiently small since A j = M j + O(ε) an thus n p jm 1/2 j A 1/2 j is close to the invertible matrix M. With this choice of v 0 we compute that p j M 1/2 j E(w j ) = 0 an so the last term in (38) vanishes. Recalling (37), (38), an using the fact that there exists a positive constant c epening only on p an the M j s, such that I M 1/2 j M 1 M 1/2 j p ci for all 1 j n, we see that it suffices to show that p j E(w j ) 2 + c p j E( w j E(w j ) 2 ) O ( ε E( w j 2 ) ) 0. However we have w j 2 2 w j E(w j ) E(w j ) 2, an so the claim now follows if ε is sufficiently small epening on p an the M j s.

23 ON THE MULTILINEAR RESTRICTION AND KAKEYA CONJECTURES 23 When each p j is an integer, the quantity Q p (t) is monotonic without further hypotheses on the matrices A j since δ efine in (30) is manifestly non-negative. This in particular inclues the two imensional bilinear Kakeya situation n = = 2 an p 1 = p 2 = 1. (While we o not in this paper establish monotonicity of Q p (t) uner the hypotheses of Proposition 4.1, neither o we rule it out.) The trilinear enpoint Kakeya situation in three imensions correspons to n = = 3, p 1 = p 2 = p 3 = 1/2 an the matrices A j each having rank 2. Interestingly, in this case Q p (t) is not monotonic ecreasing, as the following lemma will emonstrate. Lemma 4.5. Let 3. For j = 1, 2,..., suppose A j is a nonnegative efinite matrix of rank ( 1) such that ker A 1, ker A 2,... ker A span. Suppose furthermore that 1 1 (A 1 + A A ) A 1, A 2,..., A. Then A is uniquely etermine by A 1, A 2,... A 1. Proof Accoring to [4, Proposition 3.6], uner the hypotheses of the lemma there exists an invertible matrix D an rank ( 1) projections P j such that A j = D P j D for all j an 1 1 (P 1 + P P ) = I. We claim that for j k, ker P j an ker P k are orthogonal. To see this let x j be a unit vector in ker P j an note that x 1 = 1 1 (P 1 + P P )x 1 = 1 1 (P P )x 1. So 1 1 (P 2 + +P )x 1, x 1 = 1. On the other han P j x 1, x 1 1 for all j since P j is a projection. Thus P j x 1, x 1 = 1, an x 1 = P j x 1 for all j 1. Similarly P j x i = x i for all i j. Hence for i j, x i, x j = P j x i, x j = x i, P j x j = 0. Thus the kernels of the P j s are mutually orthogonal. By choosing a suitable orthonormal basis (an possibly changing D) we may assume therefore that P j = A 0 j, the jth Loomis Whitney matrix. Now let us suppose that D is another invertible matrix an that Ãj = D A 0 D. j Then the statement A i = Ãi is the same as A 0 i = R A 0 i R with R = DD 1 ; i.e. R leaves both ker A 0 i an its orthogonal complement ima0 i invariant an acts as an isometry on the latter. If now A 1 = Ã1,..., A 1 = Ã 1, this means that R acts as an isometry on every coorinate hyperplane except possibly {x = 0}, an the stanar basis vectors e 1,..., e 1 are eigenvectors of R. This forces the matrix of R with respect to the stanar basis to be iagonal with entries ±1 an thus A = Ã too. Proposition 4.6. For j = 1, 2,..., let W j be a set of nonnegative efinite rank ( 1) matrices such that for all A j W j, ker A 1, ker A 2,..., ker A span. Let µ j be a positive Borel measure on W j R. Let p = (1/( 1),..., 1/( 1)). If Q p (1) Q p (0) for all such positive measures µ j, then either = 2 or each W j is a singleton, an 1 1 (A 1 + A A ) A 1,..., A. Remark 4.7. In the latter case 3 of Proposition 4.6 the family {A j } is an affine image of the Loomis Whitney matrices {A 0 j }, i.e. for some invertible D, Ã j = D A 0 D, j as the proof of Lemma 4.5 shows. Proof Let µ # j be arbitrary positive finite Borel measures on R, an let A j W j. With f j (t, x) := e π Aj(x vjt),(x vjt) µ # j (v j) R

24 24 JONATHAN BENNETT, ANTHONY CARBERY, AND TERENCE TAO an Q # p (t) := f 1 (t, x) 1/( 1) f (t, x) 1/( 1) x, R setting µ j := µ # j δ 0(A j ) we now have Q # p (1) Q # p (0). By [4, Proposition 3.6] this forces 1 1 (A 1 + A A ) A 1,..., A. When 3, Lemma 4.5 shows that any 1 of the A j s etermine the remaining one. Thus no W j may contain more than one point. Remark 4.8. The same argument applies if instea of Q p we consier Q p. Consequently the crucial single-signeness of S in (36) fails in this setting at the enpoint p = (1/( 1),..., 1/( 1)). 5. Lower levels of multilinearity As we remarke in the introuction, if n < then the conjecture exponents for n- linear restriction type problems epen on the curvature properties of submanifols in question; with flatter surfaces expecte to enjoy fewer restriction estimates. Our methos here cannot easily take avantage of such curvature hypotheses, though, an we will instea aress the issue of establishing n-linear restriction estimates in R which assume only transversality properties rather than curvature properties. In such a case we can establish quite sharp estimates. A similar situation occurs when consiering n-linear Kakeya estimates in R. The analogue of curvature woul be some sort of irection separation conition (or perhaps a mixe Lebesgue norm conition) on the tubes in a given family. Here we will only consier estimates in which the tubes are counte by carinality rather than in mixe norms, an transversality is assume rather than irection separation. We begin by iscussing the n-linear Kakeya situation. Fix 3 n (the case n = 1 turns out to be voi, an the n = 2 case stanar, see e.g. [31]). Suppose T 1,..., T n are families of δ-tubes in R. Suppose further that for each 1 j n, the tubes in T j have long sies pointing in irections belonging to some sufficiently small fixe neighbourhoo of the jth stanar basis vector e j in S 1. (Again, the vectors e 1,..., e n may be replace by any fixe set of n linearly inepenent vectors in R here, as affine invariance consierations reveal.) n Theorem 5.1. If n 1 < q then there exists a constant C, inepenent of δ an the families of tubes T 1,..., T n, such that ( ) n Lq/n(R) C (δ /q #T j ). (39) T j T j χ Tj Remark 5.2. One can conjecture the same result to hol at the enpoint q = note for instance that the estimate is easily verifiable at this enpoint when n = 2. For q < n n 1 the estimate is false, as can be seen by choosing each family T j to be (essentially) a partition of a δ-neighbourhoo of the unit cube in span{e 1,..., e n } n n 1 ;