Restriction estimates for hyperboloids in higher dimensions via bilinear estimates
aa r X i v : . [ m a t h . C A ] M a r RESTRICTION ESTIMATES FOR HYPERBOLOIDS INHIGHER DIMENSIONS VIA BILINEAR ESTIMATES
ALEX BARRON
Abstract.
Let H be a ( d − R d and let Ef be the Fourier extension operator associated to H , with f supported in B d − (0 , k Ef k L p ( B (0 ,R )) ≤ C ǫ R ǫ k f k L p for all p ≥ d +2) d whenever d ≥ m + 1, where m is the minimumbetween the number of positive and negative principal curvatures of H .Bilinear restriction estimates for H proved by S. Lee and Vargas play animportant role in our argument. In this paper we study estimates for the operator Ef ( x, t ) = Z R d − f ( ξ ) e πi ( x · ξ + t ( ξ + ... + ξ d − m − − ξ d − m − ξ d − m +1 − ... − ξ d − )) dξ, supp( f ) ⊂ B d − (0 , . This is the extension operator associated to the hyperboloid H = { ξ ∈ R d : ξ d = ξ + ξ + ... + ξ d − m − − ξ d − m − ... − ξ d − } . We let M denote the ( d − × ( d −
1) diagonal matrix with M ii = 1 if i ≤ d − − m and M ii = − i > d − − m . Then the phase of Ef has theform x · ξ + t ( M ξ · ξ ) , ξ ∈ R d − . We can assume that m ≤ d − since otherwise we can replace t by − t .Note that m is the minimum between the number of positive and negativeprincipal curvatures of H .We will prove the following. Theorem 1.
Fix d ≥ . Suppose f is supported in B d − (0 , and fix R ≥ and ǫ > . If m ≤ d − and p ≥ d +2) d then (1) k Ef k L p ( B R ) ≤ C ǫ R ǫ k f k L p . By Tao’s ǫ -removal argument ([14]) the theorem holds for p > d +2) d withno loss of R ǫ . In the case d = 3 , m = 1 Theorem 1 was proved independently by Vargas([16]) and S. Lee ([11]) using a bilinear method. This was later improved byCho and J. Lee, who adapted the polynomial partitioning method developedby Guth in [7] to show that (1) holds for p > .
25 ([5]). In [12] Stovall provesendpoint cases when d = 3 that do not follow from arguments in [11] and[16]. See also the paper [10] by Kim. For other recent progress on restriction estimates for more complicated hyperbolic surfaces in dimension 3 see [4]and the references therein.When d ≥ d ≥ d is even then the estimate in Theorem 1 follows from their more gen-eral estimates for H¨ormander-type operators (see Remark 3.2 at the end ofSection 3 below).The main goal for this paper is to prove Theorem 1 via the bilinear esti-mates proved by S. Lee and Vargas, stated precisely in Theorem 2 in Section2 below. Our argument will follow a broad-narrow scheme adapted from [3],[6], [8]. This broad-narrow analysis allows us to use the estimates of S. Leeand Vargas except in certain exceptional cases which we analyze in Section2. The main idea is the following: if τ and τ are two caps in the supportof f and we do not have a favorable estimate for Ef τ Ef τ , then τ and τ must be arranged in a neighborhood of a cone-like surface C m . If we canfind no pairs ( τ , τ ) for which bilinear estimates apply then the geometryof C m forces the caps to in fact be contained in a neighborhood of an m -dimensional affine space; we can then treat this scenario using a ‘narrow’decoupling argument and induction on scales, at least when m ≤ d −
1. Inthe special case where d is odd and m = d − our method breaks down sincethe induction no longer closes. Note however that we always have m ≤ d − .We review some basic tools that we will use frequently in Section 1. InSection 2 we discuss some more history and background surrounding bilinearrestriction estimates. The key lemma describing how bilinear estimates for Ef can fail is then proved in Section 2.3. Finally, in Section 3 we carry outthe broad-narrow argument to complete the proof of Theorem 1. Notation.
We will write A . B if there is some constant c > A ≤ cB .If A . B and B . A we also write A ∼ B . Our uniform constants may alsovary from line-to-line, which is allowed since they will remain independentof R .Let B r be a ball of radius r in R d and let B r − denote a ball centered atthe origin in R d of radius r − . We let w B r be a smooth weight adapted to B r in the following sense: w B r ( x, t ) decays rapidly for ( x, t ) / ∈ B r , and d w B r is supported in a fixed dilate of B r − . Note that we can construct w B r bytaking a bump function w adapted to the unit ball such that | w ( x ) | . | x | ) d and then applying a suitable affine transformation. YPERBOLOID RESTRICTION 3 If S is a ball or rectangle in R d − we let f S = f · φ S , where φ S is asmooth bump function supported in a small dilate of S with φ S ( ξ ) = 1when ξ ∈ S . If M is a smooth manifold and ρ > N ρ ( M )denote the ρ -neighborhood of M . Acknowledgments.
This paper benefited from several helpful conversa-tions with M. Burak Erdo˘gan and Terence Harris.1.
Basic tools
In this section we review some basic tools we will use throughout theproof of Theorem 1. Below we will always assume that the support of f iscontained in B d − (0 , Wave packet decomposition and parabolic rescaling.
We firstrecall the standard wave packet decomposition for Ef (see for example [5],[8], [11], or [16]). Fix ρ ∈ (0 ,
1) and suppose { τ } is a collection of finitely-overlapping balls of radius ρ that cover the support of f . We will refer tothese τ as ρ -caps . Using a partition of unity we may decompose f = P τ f τ ,with f τ supported in a small dilate of τ . Then Ef = P τ Ef τ . We let G ( τ ) := ( ξ , ..., ξ d − m − , − ξ d − m , ..., − ξ d − , − | ( ξ , ..., ξ d − m − , − ξ d − m , ..., − ξ d − , − | when ξ is the center of τ , so G ( τ ) is the unit normal direction to H abovethe center of τ . If T τ is any tube in R d of dimensions ρ − × ... × ρ − × ρ − with long direction G ( τ ) then Ef τ is essentially constant on T τ .We also recall that Ef is invariant under parabolic rescalings in the fol-lowing way. Proposition 1.1.
Fix
R > and let B R = B d (0 , R ) . Also fix ρ ∈ (0 , with ρ − < R. Then for any ρ -cap τ one can find a function g supported in B d − (0 , such that k g k L p = ρ − d − p k f τ k L p and k Ef τ k L p ( B R ) ≤ ρ ( d − − d +1 p k Eg k L p ( B ρR ) . To prove the proposition one can use modulation invariance of Ef τ to reduceto the case where τ is centered at the origin, and then rescale ( x, t ) → ( ρ − ¯ x, ρ − ¯ t ) . The operator Ef has other scaling symmetries that differ from parabolicrescaling, although we will make no use of these symmetries in our argu-ments. Note, however, that the proof of Theorem 1 in the case d = 3 dueto S. Lee and Vargas ([11], [16]) does exploit these extra symmetries. Thesame is also true of the Bourgain-Guth proof of the case d = 3 (see Remark3.2 below), along with the improved estimate when d = 3 due to Cho andJ. Lee in [5]. ALEX BARRON
Flat decoupling and induction on scales.
Decoupling allows us toseparate the contribution from different wave packets Ef τ . This is useful inthe ‘narrow case’ below when we cannot use bilinear restriction estimates.The strongest possible decoupling result for the hyperboloid has been provedby Bourgain and Demeter ([2]), though we will not need to use their theorem.Instead it will suffice to use the following more elementary ‘flat decoupling’result, which follows easily from orthogonality considerations. Proposition 1.2 (Flat Decoupling) . Suppose T is a collection of finitely-overlapping ρ -caps τ with ρ − < R . Then k X τ ∈T Ef τ k L p ( B R ) ≤ C ( T ) − p (cid:0) X τ ∈T k Ef τ k L p ( w BR ) (cid:1) , where w B R is a smooth weight adapted to B R .Proof. The case p = ∞ is just the Cauchy-Schwartz inequality, and when p = 2 the proposition follows from Plancharel’s theorem. The remainingcases follow by interpolation. (cid:3) Finally we recall that if R is small enough then Theorem 1 follows directlyfrom H¨older’s inequality. We can therefore assume by induction that Theo-rem 1 is true at scale ρR whenever ρ ≪
1. For technical reasons related tothe decoupling result in Proposition 1.2 we also remark that we can assumeby induction that the following weighted estimate holds: for any ǫ > k Ef k L p ( w BρR ) ≤ C ǫ ( ρR ) ǫ k f k L p , where w B ρR is a smooth weight adapted to B ρR . Bilinear restriction estimates for H In this section we will review some known bilinear estimates and provea lemma that characterizes what happens if these bilinear estimates fail.The following estimate was proved by S. Lee in dimension d ≥ Theorem 2 ([11], [16]) . Suppose f and f are supported in open sets τ and τ of diameter ∼ . If (2) inf ξ, ¯ ξ ∈ τ η, ¯ η ∈ τ | M ( ξ − η ) · ( ¯ ξ − ¯ η ) | ≥ c > then (3) k| Ef Ef | k L p ( B R ) ≤ C ǫ R ǫ k f k L k f k L whenever p ≥ d +2) d . If (2) fails then (3) can fail as well. We will need to use a version of Theorem 2 adapted to K − -caps for aparameter K such that 1 ≪ K ≪ R. The following is a consequence of Theorem 2.
YPERBOLOID RESTRICTION 5
Theorem 3.
Suppose f and f are supported in K − -caps τ and τ , re-spectively, whose centers are O ( K − ) -separated. Let A be a constant. If (4) inf ξ, ¯ ξ ∈ τ η, ¯ η ∈ τ | M ( ξ − η ) · ( ¯ ξ − ¯ η ) | ≥ AK − then (5) k| Ef Ef | k L p ( B R ) ≤ C A K O (1) k f k L k f k L whenever p ≥ d +2) d . If (4) fails then (5) can fail as well. We say that two K − -caps τ , τ are strongly separated if (4) holds.Since it is not immediately obvious from scaling that Theorem 2 impliesTheorem 3, we will prove the implication below in Section 2.2.2.1. Some background.
Bilinear restriction estimates in the full rangegiven in Theorem 2 were first proved by Wolff in the case of the cone [17].Wolff’s methods were later adapted by Tao in the case of the paraboloid [13],and then by Vargas and S. Lee independently in the case of hyperboloids.In the case of the cone and paraboloid the transversality condition (4) ismuch simpler.There is an argument due to Tao, Vargas, and Vega ([15]) that allows oneto deduce linear restriction estimates from bilinear restriction estimates forelliptic surfaces (e.g. paraboloids), and indeed linear restriction estimatesare obtained as corollaries of the main results in [13] and [17]. The main ideaof the argument from [15] is that any two points will belong to a unique pairof dyadic cubes that are separated by a distance proportional to their scale;one can then use this observation to efficiently decompose | E parab. f | as asum of terms to which bilinear estimates apply (after a parabolic rescaling).For hyperboloids this argument requires different ideas since the strongertransversality condition (4) is more complicated.In the special case d = 3 , m = 1 one can apply a simple change variablesand instead consider the extension operator associated to the surface { ξ ∈ R : ξ = ξ ξ , | ξ | ≤ } . Then (4) is equivalent to the following two-parameter separation condition:(6) | ξ − η | ' | ξ − η | ' ξ ∈ τ , η ∈ τ . Vargas and S. Lee were able to use this observation to almost recover thebilinear-to-linear reduction from [15], up to certain endpoint cases whichwere later proved by Stovall [12]. All of these arguments rely on the fact that(6) facilitates a two-parameter decomposition of frequency space analogousto the decomposition used in [15]. When d ≥ d = 3 then (6) can only fail if the caps are arrangedin a neighborhood of an axis-parallel line, but when d ≥ C = { ξ ∈ R d − : ξ + ... + ξ d − m − = ξ d − m + ... + ξ d − } . ALEX BARRON
After we deduce Theorem 3 we will analyze what can happen in theexceptional case where (4) fails for all pairs of caps in the support of f . Wewill see that failure of (4) for every pair of caps forces f to be supportednear an affine space of dimension m . We will then be able to use decouplingand induction to prove Theorem 1 in the ‘narrow’ cases where we cannotuse Theorem 3.As mentioned in the introduction, our methods do not work when d =3 , m = 1. In this case Theorem 1 is still true and follows from arguments byS. Lee, Stovall, or Vargas ([11], [12], [16]). Of course when d = 3 Theorem1 also follows from the stronger restriction estimate due to Cho and J. Lee[5].2.2. Proof that Theorem 2 implies Theorem 3.
Let e j denote thestandard basis vectors in R d − . Let τ and τ be two K − -caps for which(4) holds. After translation we can assume that τ is centered at the origin.We may assume that dist( τ , τ ) & K − since otherwise the desired resultfollows easily by rescaling frequency space by K .Since τ is centered at the origin the condition (4) is invariant underlinear transformations of the form U = U ′ ⊕ U ′′ , where U ′ is a rotation in ξ , ..., ξ d − m − that fixes ξ d − m , ..., ξ d − , and U ′′ is a rotation in ξ d − m , ..., ξ d − that fixes ξ , ..., ξ d − m − . We can therefore assume that τ is centered at apoint of the form ξ ∗ = ( ξ , , ..., , ξ d − m , ξ d − m +1 , ..., ξ d − m )with | ξ − ξ d − m − ... − ξ d − | ≥ cK − . Let us first assume that ξ − ξ d − m − ... − ξ d − ≥ cK − . Since we are also assuming | ξ ∗ | ≥ cK − it follows that(7) ξ ≥ cK − . Now let S be the linear transformation such that Sξ ∗ = e Se j = e j j = 2 , ..., d − . One checks using (7) that k S k ∼ | ξ | . K . In particular the first column of S is(1 /ξ , , ..., , − ξ d − m /ξ , ..., − ξ d − /ξ )while the other columns are e , ..., e d − . Now suppose η, ¯ η ∈ τ . Since weare assuming that τ is centered at the origin we then have | M ( Sξ ∗ − Sη ) · ( Sξ ∗ − S ¯ η ) | = | M e · e + O ( K − ) | & . (8) YPERBOLOID RESTRICTION 7
Since changing ξ ∗ to any other ξ ∈ τ in (8) only introduces an error of O ( K − ) it follows that the caps Sτ , Sτ satisfy the condition (2), and so(5) follows from (3) after rescaling f , f (which is allowed since we can lose K O (1) in the bilinear estimate).In the case where − ξ + ξ d − m + ... + ξ d − ≥ cK − we apply another transformation U = U ′ ⊕ U ′′ to map ξ ∗ to U ξ ∗ = ( ˜ ξ , ˜ ξ , ..., ˜ ξ d − m − , ˜ ξ d − m , , ..., . Then since U does not change the norm of either ( ξ d − m , ξ d − m +1 , ..., ξ d − ) or( ξ , , ...,
0) it follows that˜ ξ d − m − ˜ ξ − ... − ˜ ξ d − m − ≥ cK − , and so we repeat the previous argument with ˜ ξ d − m playing the role of ξ .2.3. Failure of bilinear estimates.
We now prove that if the bilinearestimates in Theorem 3 fail then the caps τ must be localized near an m -dimensional affine space. We first prove some geometric lemmas that willlead us in this direction, with the main result of the section being Lemma 2.3below. Given ξ = ( ξ , ..., ξ d − ) ∈ R d − , we will write ξ ′ = ( ξ , ξ , ..., ξ d − m − )and also ξ ′′ = ( ξ d − m , ..., ξ d − ) . Lemma 2.1.
Let C denote the surface C = { ξ ∈ R d − : ξ + ... + ξ d − m − = ξ d − m + ... + ξ d − , | ξ | < } and let C r = { ξ ∈ B d − (0 ,
2) : | ξ · M ξ | ≤ r } . Suppose ξ, η ∈ C cK − . Let T ξ denote the subspace T ξ = { ω : ω · M ξ = 0 } . If ξ − η ∈ C CK − then η is in an O ( K − ) neighborhood of T ξ .Proof. Since ξ, η ∈ C cK − and ξ − η ∈ C CK − we have d − m − X i =1 ( ξ i − η i ) = d − X i = d − m ( ξ i − η i ) + O ( K − )= d − m − X i =1 ( ξ i + η i ) − d − X i = d − m ξ i η i + O ( K − ) . Multiplying this identity out we obtain ξ ′ · η ′ = d − X i = d − m ξ i η i + O ( K − ) = ξ ′′ · η ′′ + O ( K − ) . As a consequence η · M ξ = O ( K − ) , which proves the lemma. (cid:3) ALEX BARRON
Lemma 2.2.
Let V be a subspace of R d − and suppose that V ∩ B d − (0 , ⊂C cK − a , where a > . Then if K is sufficiently large we must have dim V ≤ m .Proof. Let { v , ..., v k } be an orthonormal basis for V . By hypothesis weknow that v i · M v i = O ( K − a )for each i . Also note that v i − v j ∈ V ∩ B d − (0 ,
2) and therefore v i − v j ∈C cK − a . Then from Lemma 2.1 we conclude that(9) v i · M v j = O ( K − a )for each pair i, j . Of course(10) v i · v j = 0 , i = j by hypothesis. Now let P ω denote the orthogonal projection
P ω = ( ω d − m , ..., ω d − ) ∈ R m . From (9) and (10) we conclude that(11)
P v i · P v j = O ( K − a ) if i = j, P v i · P v i = 12 + O ( K − a ) . But if K is large enough, depending only on a and the implicit constantsabove, then (11) implies that { P v , ..., P v k } are independent. Indeed, from(11) it follows that there is some α > | Angle(
P v i , P v j ) − π/ | ≤ αK − a , i = j || P v i | − / | ≤ αK − a which implies the claimed independence if K is large enough (depending onthe value of a and α ). Since { P v , ..., P v k } are all vectors in R m we musthave k ≤ m and so dim V ≤ m . (cid:3) The following lemma is the main result of this section.
Lemma 2.3.
Let { τ } be a collection of finitely-overlapping K − -caps in B d − (0 , with Ef = P τ Ef τ . If K is sufficiently large then one of thefollowing must occur.(i) There exists a uniform α > and an m -dimensional affine space V such that every τ is contained in an O ( K − α ) neighborhood of V .(ii) There are two K − -caps τ, τ ′ for which inf ξ, ¯ ξ ∈ τω, ¯ ω ∈ τ ′ | M ( ξ − ω ) · ( ¯ ξ − ¯ ω ) | ≥ AK − . Proof.
Suppose that (ii) fails and let τ , τ , ..., τ k be distinct caps in B d − (0 , f . We can assume we can find such caps with k ≥ Ef we can also assumethat τ is centered at the origin.Pick η i ∈ τ i for i = 1 , ..., k . Since (ii) fails for each pair of caps ( τ , τ i ) wesee that η i ∈ C cK − for each i , with the constant c depending only on d, A . YPERBOLOID RESTRICTION 9
Since (ii) also fails for each pair ( τ i , τ j ) when i = j we see that η i − η j ∈ C cK − as well. Then by Lemma 2.1 we conclude that(12) η i · M η j = O ( K − )for each i, j (including i = j ).Now fix k ′ ≤ d −
1. After possibly re-labeling, suppose that { η , η , ..., η k ′ } is a maximal subset of { η , η , ..., η k } such that(13) | η i | ≥ cK − d − + σ d − and | η ∧ η ∧ ... ∧ η k ′ | & K − + σ , where 0 < σ ≪ σ is independent of K . Let V = span { η , η , ..., η k ′ } . If ω ∈ V ∩ B d − (0 ,
2) with ω = P k ′ i =1 a i η i then by (13) we have | a i | . K − σ . It follows that ω · M ω = X i,j a i a j ( η i · M η j ) = O ( K − σ ) , and as a consequence V ∩ B d − (0 , ⊂ C cK − σ . But then Lemma 2.2 implies that k ′ = dim V ≤ m (provided K is sufficientlylarge). It follows that (i) must be true for some uniform α , for example α = d − + σ d − . Otherwise we could take k ′ > m in (13) , but we justsaw this is not possible. (cid:3) In the next section we will take K = R δ for some δ = δ ( ǫ ). We are allowedto assume that K ≥ C ǫ by induction, and therefore we will always be ableto assume K is large enough that Lemma 2.3 applies.3. The broad-narrow argument
We now prove Theorem 1 using a broad-narrow argument adapted from[3], [8], [6]. Fix ǫ > δ > δ ≪ ǫ and set K = R δ and K = K α , where α is as in part (i) of Lemma 2.3 (for example, α = d − − σ d − worksfor some small σ > T be a collection of finitely-overlapping K − -caps τ covering the support of f and use a partition of unity to decompose f = P τ f τ with f τ supported in (a small dilate of) τ . We also let { θ } be acollection of finitely-overlapping K − -caps covering the support of f . Then f = P θ f θ as well.On the spatial side we fix a collection Q of finitely-overlapping K -cubesthat cover B d (0 , R ). Given Q ∈ Q we define its significant set S p ( Q ) = { τ ∈ T : k Ef τ k L p ( Q ) ≥ T ) k Ef k L p ( Q ) } . Note that for τ / ∈ S p ( Q ) we have k X τ / ∈S p ( Q ) Ef τ k L p ( Q ) ≤ k Ef k L p ( Q ) , and so we will always be able to absorb these error terms into the left-handside of our estimates for k Ef k L p ( Q ) below.Now fix a uniform constant A > K -cube Q is narrow and write Q ∈ N if there is an ( m + 1)-dimensionalsubspace W such that Angle( G ( τ ) , W ) ≤ AK − for all τ ∈ S p ( Q ), where G ( τ ) is the unit normal to the surface H above thecenter of τ . If a cube Q is not narrow then we say it is broad and write Q ∈ B . We of course have k Ef k pL p ( B R ) ≤ X Q ∈N k Ef k pL p ( Q ) + X Q ∈B k Ef k pL p ( Q ) , and so it suffices to consider separately the cases when the broad and narrowterms dominate.3.1. The broad case.
We first consider the broad case. We will need touse the following lemma which is a consequence of Theorem 3 and the factthat Ef is essentially constant at scale one. Lemma 3.1.
Suppose f is supported in B d − (0 , . Let τ and τ be twostrongly separated K − -caps. Then X Q ∈B k Ef τ k p L p ( Q ) k Ef τ k p L p ( Q ) ≤ K O (1) k f k pL whenever p ≥ d +2) d . The proof of this lemma is contained in the proof of Proposition 3.1 in [6],though for completeness we include most of the argument.
Proof.
We define f i = e ix i · ξ + t i · ( Mξ · ξ ) f τ i for some choice of ( x i , t i ) ∈ R d . Let φ be a bump function on R d with b φ = 1 in B d (0 ,
2) and b φ supported in B d (0 , . Note that Ef i = Ef i ∗ φ for any modulation f i .Decompose Q as a union of lattice cubes of side-length . Then we mayfind ( x i , t i ) as above such that k Ef i ∗ φ k L ∞ ( Q ) is attained in the same latticecube C Q for i = 1 ,
2. Then k Ef τ k L p ( Q ) k Ef τ k L p ( Q ) ≤ K O (1) k Ef ∗ φ k L ∞ ( C Q ) k Ef ∗ φ k L ∞ ( C Q ) . We may pick our bump function φ so that φ decays rapidly outside B d (0 , w ∈ B d ( z, φ ( w ) . φ ( z ) for any z ∈ R d . YPERBOLOID RESTRICTION 11
Therefore k Ef ∗ φ k p L ∞ ( C Q ) k Ef ∗ φ k p L ∞ ( C Q ) . (cid:0) Z C Q Z R d Z R d | Ef ( z ) || Ef ( z ) | φ ( z − z ) φ ( z − z ) dz dz dz (cid:1) p = C (cid:0) Z R d Z R d Z C Q | Ef ( z − z ) || Ef ( z − z ) | φ ( z ) φ ( z ) dzdz dz (cid:1) p . By Minkowsi’s and H¨older’s inequalities we then have X Q ∈B k Ef ∗ φ k p L ∞ ( C Q ) k Ef ∗ φ k p L ∞ ( C Q ) . (cid:20) Z R d Z R d (cid:0) Z B R | Ef ( z − z ) | p | Ef ( z − z ) | p φ ( z ) p φ ( z ) p dz (cid:1) p dz dz (cid:21) p . sup z ,z Z B R | Ef ( z − z ) | p | Ef ( z − z ) | p dz . sup z ,z Z B R | E e f ( z ) | p | E e f ( z ) | p dz where e f i is a modulation of f i that depends on z i . Note that k e f i k L = k f τ i k L . Since e f i is still supported in τ i and the pair ( τ , τ ) is strongly separated, wemay apply Theorem 3 to conclude that X Q ∈B k Ef ∗ φ k p L ∞ ( C Q ) k Ef ∗ φ k p L ∞ ( C Q ) ≤ K O (1) k f k pL , which completes the proof. (cid:3) Let Q be a broad cube and first suppose that there is no strongly separatedpair of caps in S p ( Q ). Then by Lemma 2.3 there exists an m -dimensionalaffine space V such that τ ⊂ N cK − ( V ) for all τ ∈ S p ( Q ) . But this forcesthe directions G ( τ ) to be in an O ( K − ) neighborhood of the ( m + 1)-plane W in R d given by W = G ( V ) , G ( ω ) = | ( ω, − | G ( ω )(note that the angle between G ( ω ) and G ( ω ) is proportional to the distance | ω − ω | if the centers of the caps are O ( K − )-separated). Therefore Q ∈ N ,assuming we have chosen A appropriately depending only on the constantfrom Lemma 2.3. Since we are assuming Q ∈ B this cannot happen and sothere must be two strongly separated caps τ , τ ∈ S p ( Q ). By the definitionof S p ( Q ) we then have k Ef k L p ( Q ) ≤ K O (1) k Ef τ k L p ( Q ) k Ef τ k L p ( Q ) . The pair ( τ , τ ) depends on Q but we may make this estimate uniform bysumming over all possible strongly-separated pairs (note the number of suchpairs is O ( K d − )). We then apply Lemma 3.1 to conclude that X Q ∈B k Ef k pL p ( Q ) ≤ K O (1) X ( τ ,τ )strongly sep. X Q ∈B k Ef τ k p L p ( Q ) k Ef τ k p L p ( Q ) ≤ CR ǫp k f k pL p (provided δ = δ ( ǫ ) is chosen small enough, e.g. δ = ǫ ).3.2. The narrow case.
We now estimate the contribution of the narrowcubes. Suppose Q ∈ N and let W be an ( m + 1)-plane in R d such thatAngle( G ( τ ) , W ) ≤ AK − for each τ ∈ S p ( Q ). Then there is an m -dimensional affine space V in R d − such that τ ⊂ N cK − ( V ) for each τ ∈ S p ( Q ). In particular we can take V = { ω ∈ R d − : G ( ω ) ∈ W } . We choose a minimal collection Θ V of θ covering N cK − ( V ). Note that Θ V contains cK m caps θ . Applying flat decoupling and then H¨older’s inequalitywe obtain k Ef k L p ( Q ) ≤ CK m ( − p )1 (cid:0) X θ ∈ Θ V k Ef θ k L p ( w Q ) (cid:1) ≤ CK m (1 − p )1 (cid:0) X θ ∈ Θ V k Ef θ k pL p ( w Q ) (cid:1) p ≤ CK m (1 − p )1 (cid:0) X θ k Ef θ k pL p ( w Q ) (cid:1) p . Since X Q w Q . w B R we can sum over Q to conclude that(14) (cid:0) X Q ∈N k Ef k pL p ( Q ) (cid:1) p ≤ CK m (1 − p )1 (cid:0) X θ k Ef θ k pL p ( w BR ) (cid:1) p . We will now use induction on scales. By Proposition 1.1, for each θ we canfind a function g θ supported in B d − (0 ,
2) such that k f θ k L p = K − ( d − p k g θ k L p and such that k Ef θ k L p ( w BR ) ≤ K − ( d − d +1 p k Eg θ k L p ( w BR/K ) . By induction on scales we then obtain k Ef θ k L p ( w BR ) ≤ C ǫ R ǫ K − ǫ K − ( d − d +1 p K d − p k f θ k L p . YPERBOLOID RESTRICTION 13
After applying this argument for each θ we see from (14) that (cid:0) X Q ∈N k Ef k pL p ( Q ) (cid:1) p ≤ C ǫ R ǫ K − ǫ K m (1 − p )1 K − ( d − d +1 p K d − p k f k L p . The induction closes provided(15) m (1 − p ) − ( d −
1) + 2 dp ≤ , since we may assume K is large enough that C ǫ K − ǫ ≤ . Note that (15) isequivalent to p ≥ d − m ) d − m − . Some algebra shows that 2( d − m ) d − m − ≤ d + 2) d if and only if m ≤ d − . We have assumed this is true for m , and so the narrow case of Theorem 1follows. Remark . In the narrow case above we have used flat decoupling indimension m . This has nothing to do with the curvature of H and is truefor any extension operator E ′ f when f is supported in a thin neighborhoodof an m -plane. If one instead uses the stronger ℓ decoupling result provenby Bourgain and Demeter in [2] there is no gain in our argument, since thisstill leads to a loss of K m ( − p )1 in the first step. This is related to the factthat the surface H contains subsets which are affine spaces of dimension m , even though the curvature of H is nonzero. The ℓ decoupling does notdistinguish the difference, since we can imagine that Ef is supported in asmall neighborhood of one of these affine spaces; in this case the K m ( − p )1 loss is sharp.We further elaborate on the last claim by considering the special case d = 5 , m = 2. Note in this case m = d − and so our argument in the narrowcase does not apply. Fix a K -cube Q and suppose there is no pair of caps( τ , τ ) which are strongly separated and in S p ( Q ). Then by Lemma 2.3 thesupport of f must be contained in an O ( K − )-neighborhood of an m -plane V . If we assume there is at least one significant τ ∈ S p ( Q ) that contains theorigin then from the proof of Lemma 2.3 we see that V ∩ B (0 ,
2) can betaken to be a subset of the surface C defined in Section 2. Moreover V canbe assumed to be a vector space.Let { v, u } be an orthonormal basis for V . Since v − u ∈ V ∩ B (0 , ⊂ C the argument in Lemma 2.1 implies that M v · u = 0 and hence M u · v = 0.We also know by hypothesis that M v · v = 0 and M u · u = 0. Therefore { v, u, M v, M u } is an orthonormal basis for R with V ⊥ = span { M v, M u } . Now let A be the orthonormal matrix with inverse A − = (cid:2) v u M v M u (cid:3) , so that A maps V to the 2-plane determined by η = 0 and η = 0. Applyingthe change of coordinates determined by A shows that k Ef k L p ( Q ) = k e Ef A k L p ( Q A ) where f A is the natural transform of f and e E is the extension operator withphase x · η + t ( M A η · η ) M A = ( A − ) T M A − = In particular e E is the extension operator associated to the hyperbolic surface H = { η ∈ R : η = η η + η η } . Since f is supported in a K − -neighborhood of V it follows that f A is sup-ported in a K − -neighborhood of the 2-plane where η = 0 , η = 0. As aconsequence de Ef A is supported in a K − neighborhood of the 2-plane V A = { η ∈ R : η = ( η , η , , , , } . Note that V A ⊂ H and therefore we can choose f so that the loss of K − p )1 in our first decoupling step is sharp for general f . This can be seen forexample by taking f so that de Ef A is essentially the indicator function of V A ∩ B (0 , η , η , η , η , η ) → ( η , η , K η , K η , K η )associated to H and then argue by induction on scales (since such a transfor-mation will map the support of f A to a cube of side-length O (1) but shrinkthe size of Q A ). This gives a favorable result for each individual Q , butremember that V can vary depending on Q and may not even be a vectorspace. We have not found a way to effectively deal with the contribution ofdifferent V , mainly because K − -neighborhoods of different V can intersectin complicated ways and naive estimates give a loss in K that is much toolarge to close the induction. A similar issue arises in higher dimensions when d is odd and m = d − . Remark . The idea of using a broad-narrow analysis to deduce linear re-striction theorems from multilinear restriction theorems dates back to Bour-gain and Guth in [3]. They prove restriction estimates for the paraboloid byusing k -linear restriction ([1]) in the broad case and an induction procedurein the narrow case. Their argument works in a range of p that is larger thanwhat Tao proved in [13] using bilinear restriction theorems. When d = 3 YPERBOLOID RESTRICTION 15 their methods also adapt to the hyperbolic surface ξ = ξ ξ and prove The-orem 1 in this case. If d ≥ p ≥ d +2) d by directly using the k -linear Bennet-Carbery-Tao es-timate with k = d + 1, along with a flat decoupling and induction-on-scalesargument. In the narrow case in odd dimensions this procedure is not aseffective since one needs to use a smaller k .Recall that the intersection of H with a hyperplane can have zero Gaussiancurvature. This complicates any induction-on-dimension procedure whencompared to the elliptic case, where the intersection of a paraboloid witha hyperplane is a paraboloid of lower dimension. The case d = 3 for H is special since you can only lose curvature if the hyperplane is (almost)parallel to the ξ or ξ axis. In this case case one can instead exploit non-isotropic scaling symmetries ( ξ , ξ , ξ ξ ) → ( aξ , bξ , abξ ξ ) to close theinduction. We have not found a way to carry this argument out in higherdimensions, except in the localized setting summarized at the end of theprevious remark. References [1] J. Bennet, A. Carbery, and T. Tao,
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Department of Mathematics, University of Illinois Urbana-Champaign, Ur-bana, IL 61801, USA
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