A restriction estimate for surfaces with negative Gaussian curvatures
aa r X i v : . [ m a t h . C A ] M a y A RESTRICTION ESTIMATE FOR POLYNOMIALSURFACES WITH NEGATIVE GAUSSIAN CURVATURES
SHAOMING GUO AND CHANGKEUN OH
Abstract.
We prove L p bounds for the Fourier extension operators as-sociated to polynomial surfaces in R with negative Gaussian curvaturesfor every p > . Introduction
For every hypersurface S in R parametrized by(1.1) S = { ( ξ, η, h ( ξ, η ) : ( ξ, η ) ∈ [ − , } , we define an extension operator associated to the surface S by(1.2) E S f ( x , x , x ) = Z [ − , f ( ξ, η ) e (cid:0) ξx + ηx + h ( ξ, η ) x (cid:1) dξdη. Here, h ( ξ, η ) is a C ∞ function and e ( t ) = e πit . One formulation of therestriction problem, introduced by Stein in the 1960s, is to find an optimalrange of ( p, q ) satisfying the following Fourier extension estimate(1.3) k E S f k L p ( R ) ≤ C ( S, p, q ) k f k L q ([ − , ) . It is reasonable to expect that the range of ( p, q ) depends on the choice of thesmooth function h . Stein (see for instance page 345 of [SS11]) conjecturedthat the extension estimate (1.3) holds true if and only if p > q ≥ p ′ with p ′ := p/ ( p − h is non-vanishing, that is,(1.4) det ∂ h∂ξ ∂ h∂ξ∂η∂ h∂ξ∂η ∂ h∂η = 0at every point on [ − , . We refer to [Tao04, Sto19b] for historical back-grounds and applications of the restriction estimates.The condition (1.4) is equivalent to the condition that the Gaussian cur-vature of the surface S is non-vanishing everywhere. As the determinant ofthe Hessian matrix is continuous, once the condition (1.4) is satisfied, thesign of the determinant does not change as the point changes in [ − , .For the case that the sign is positive, there has been a number of signif-icant progress over the past twenty years. Tao, Vargas and Vega [TVV98]proved (1.3) for p ≥ − /
27 and the optimal range of q , and for p ≥ − / and certain range of q . Indeed, they discovered that one can derive thelinear restriction estimate by using certain bilinear restriction estimates.Tao [Tao03] obtained (1.3) for p > / p > /
17. Guth [Gut16] further developed the idea of broad-narrow anal-ysis, successfully applied the polynomial method in the context of Fourierrestriction, and pushed the range to p > .
25 (and q = ∞ ). Later, Shayya[Sha17] and Kim [Kim17] refined the argument of Guth [Gut16] and im-proved his result to the range p > .
25 and q ≥ p ′ . The most recent resultis due to Wang [Wan18], who proved (1.3) for p > /
13. Her method isbased on the polynomial method and the two ends argument that originatedfrom the work of Wolff [Wol01]. For earlier results, we refer to Tao, Vargasand Vega [TVV98], in particular, Table 1 on page 969 of their paper.Regarding the case of negative Gaussian curvatures, the case of the hy-perbolic paraboloid h ( ξ, η ) = ξη has been very well studied. Lee [Lee06]and Vargas [Var05] independently obtained (1.3) for p > / p > .
25 and q = ∞ by using the polynomial method of Guth[Gut16]. Later, Kim [Kim17] refined their argument, and improved theirresult to the range p > .
25 and q > p ′ . More recently, Stovall [Sto19a] ob-tained scale-invariant restriction estimates for p > .
25 (and q = 2 p ′ whichmakes ( p, q ) on the scaling line).However, for general surfaces with negative Gaussian curvatures, theproblem starts to get more complicated. Buschenhenke, M¨uller, and Vargasfirst studied the problem of restriction estimates for perturbed hyperbolicparaboloids in one variable, given by h ( ξ, η ) = ξη + g ( η ) for some smoothfunction g . A typical and model example is g ( η ) = η . They obtained therestriction estimate (1.3) for this typical example in the range p > / p > .
25 in [BMV20c]; for functions g that areof finite types in the range p > / g with g ′′′ monotone in the range p > / p > .
25 forevery polynomial surface with negative Gaussian curvatures. In particular,our result improves over the range p ≥ Theorem 1.1.
Let h ( ξ, η ) be a polynomial such that (1.5) det ∂ h∂ξ ∂ h∂ξ∂η∂ h∂ξ∂η ∂ h∂η < , ESTRICTION ESTIMATES 3 at every point in [ − , . Then for every p > . , q > p ′ , and everyfunction f : [ − , → C , it holds that (1.6) k E S f k L p ( R ) ≤ C h,p,q k f k L q ([ − , ) , where C h,p,q is a constant depending on h, p and q . Let us briefly explain why the case of perturbed hyperbolic paraboloid ismore difficult than that of the hyperbolic paraboloid. In the latter case, wehave two distinguished directions and we have clean scalings that preservethe surface, to be more precise, the surface ( ξ, η, ξη ) is preserved under thescaling ( ξ, η ) ( aξ, bη ) for two arbitrary a, b ∈ R . This scaling structureplays a crucial role in the previously mentioned works [Var05, Lee06, CL17,Sto19a]. However, in the general case, such a scaling structure is no longeravailable. For instance, Buschenhenke, M¨uller and Vargas [BMV20a] ex-plained in their paper that under certain natural scalings, the term η intheir surface ( ξ, η, ξη + η ) could assume a dominant role (see the last equa-tion in page 127 of their paper and the discussion after it) and can no longerbe viewed as a perturbation.Next we briefly mention the novelties of the paper. We follow the frame-work of the polynomial methods of Guth [Gut16], which is partly based onthe method of induction on scales. The major difference comes from the“wall” case (see equation 4.9 and the line below it), in particular, how the L argument of Cordoba, Sj¨olin and Fefferman (see for instance Lemma 7.1of Tao [Tao03]) is applied. When certain “significant” frequency squares (see(4.36)) form “bad lines” or “bad pairs” (see (2.4) and Subsection 2.2), wewill not have good enough L estimate. However, Lemma 2.1 and Lemma2.4, which is one main ingredient of the paper, tell us how bad pairs andbad lines are distributed. Moreover, in contrast to the case of the hyper-bolic paraboloid, bad lines are no longer either horizontal or vertical, but canpoint to a large number of directions. This will make the notion of “broadfunctions” of Guth [Gut16] more complicated (see equation (3.2) and (3.3)).Moreover, as our polynomial h ( ξ, η ) allows perturbations of all different or-ders up to the degree d , when running the method of induction on scales,we will have to pick as many scales K d +1 ≪ K d ≪ · · · ≪ K (see (2.56)) totake care of all the perturbations. Due to the appearance of these differentscales, when defining broad functions, we will have to make sure that ourfunction is broad at every scale. This will make the argument of passingfrom bilinear estimates to linear estimates more involved (see Lemma 4.4).Our main theorem will follow from the following restriction estimate bycombining a standard scaling argument with the argument in [Sha17] (seealso Theorem 5.3 in [Kim17]). SHAOMING GUO AND CHANGKEUN OH
Theorem 1.2.
Let h be given by the polynomial (1.7) h ( ξ, η ) := ξη + a , ξ + a , η + d X i =3 i X j =0 a i,j ξ i − j η j . For sufficiently small ǫ > , the following holds true: Suppose that thepolynomial h satisfies the condition (1.8) | a , | + | a , | + 100 d d X i =3 i X j =0 | a i,j | ≤ ǫ . Then for every / < λ < , ǫ > , R ≥ , ball B R of radius R , andfunction f : [ − , → C , it holds that (1.9) k E S f k L . ( B R ) ≤ C ǫ,d,λ R ǫ k f k − λL ([ − , ) k f k λL ∞ ([ − , ) . Here, the constant C ǫ,d,λ is independent of the choice of a i,j .Proof of Theorem 1.1 by assuming Theorem 1.2. By a standard scaling argu-ment, we may assume that our polynomial h satisfies the assumption (1.8).By Tao’s epsilon removal lemma in [Tao99] (see also [Kim17]), it suffices toprove the local restriction estimate: For every p > . q > p ′ , and ǫ > k E S f k L p ( B R ) ≤ C p,q,ǫ R ǫ k f k L q ([ − , ) . Theorem 1.2 yields the restricted strong type ( p, q ) for the extension opera-tor E S at every p > .
25 and q > p ′ . It suffices to interpolate this with thetrivial L → L ∞ estimate. (cid:3) Notation:
For two non-negative numbers A and A , we write A . A to mean that there exists a constant C , which possibly depends on d , suchthat A ≤ CA . Similarly, we use O ( A ) to denote a number whose absolutevalue is smaller than CA for some constant C .For simplicity, we sometimes use Eg for E S g and use p := 3 .
25. For apolynomial h , we use k h k to denote the supremum of all its coefficients. Forevery set A and a number a >
0, we denote by N a ( A ) the a -neighborhood ofthe set A . We denote by 1 A the characteristic function of A and denote f A by f A . For each square τ with side length K − , let 2 τ denote the squarewith the same center and the side length 2 K − . Let F denote the Fouriertransform. For every K ≥ A ⊂ [ − , we denote by P ( K − , A )the set of all dyadic squares with side length K − in A . We sometimes use P ( K − ) for P ( K − , [ − , ).We will use the dyadic numbers K L , K d , . . . , K , K with(1.11) 1 ≪ K L =: K d +1 ≪ K d ≪ · · · ≪ K ≪ K := K ≪ R. These constants will be determined later.
Acknowledgement.
The authors would like to thank Betsy Stovall for anumber of insightful discussions. S.G. was partially supported by NSF grantDMS-1800274.
ESTRICTION ESTIMATES 5
Note.
Ben Bruce has recently obtained the same extension estimate forthe hyperbolic hyperboloid, which is a non-polynomial surface. Both proofsshare certain features. For this reason, Bruce and the authors have decidedto prepare a joint paper that will include both results. The present manu-script will be uploaded to the arXiv but otherwise remain unpublished.2.
Bad lines and bad pairs
In this section, we introduce the notions of bad lines and bad pairs (offrequency points and frequency squares). These concepts are introducedto study certain degenerate behaviours that originate from higher orderperturbations of the hyperbolic paraboloid (see the higher order terms in(1.7)). Throughout the rest of the paper, we let S be a hypersurface givenby (1.1) with a polynomial h satisfying (1.8).2.1. Bad lines.
Let c ( K − L ) be a small number given by(2.1) c ( K − L ) := 10 − d ǫ K − L . We take a collection D of points on the unit circle S such that D isa maximal c ( K − L )-separated set. For every a ∈ c ( K − L ) Z ∩ [ − , v = ( v , v ) ∈ D with | v | ≤ | v | , let l ,a,v denote the line pass-ing through the point (0 , a ) with direction vector v . Similarly, for every a ∈ c ( K − L ) Z ∩ [ − ,
10] and v = ( v , v ) ∈ D with | v | > | v | , let l ,a,v de-note the line passing through ( a,
0) with direction vector v . We distinguishthese two collections of lines as we will see that the variables ξ and η willbe rescalled differently. This is a phenomenon that was already present inLee [Lee06] and Vargas [Var05].Let us now define bad lines of the hypersurface S . Suppose that a line l = l ,a,v is given. We define an affine transformation M l associated to theline l in the following way. First, let M l, be the action of translation thatsends (0 , a ) to the origin and let M l, be the rotation mapping v to the point(1 , M l := M l, ◦ M l, . We write the polynomial ( h ◦ M − l )( ξ, η ) as(2.3) ( h ◦ M − l )( ξ, η ) = d X i =0 i X j =0 c i,j ξ i − j η j . By the assumption (1.8) it is easy to see that | c , | ≥ − . The line l ,a,v is called a bad line provided that(2.4) max i =2 ,...,d ( | c i, | ) ≤ − d ǫ K − L = 10 d c ( K − L ) . SHAOMING GUO AND CHANGKEUN OH
We define a bad line for l ,a,v in a similar way, with the role of ξ and η in(2.3) and (2.4) exchanged.Take ι ∈ { , } . Let L ι,a,v denote the c ( K − L )-neighborhood of l ι,a,v andcall it a bad strip . We denote the collection of all the bad strips by(2.5) L := { L ι,a,v : l ι,a,v is a bad line; ι = 1 , } . The directions of bad lines change as the polynomial h changes. However,the total number of bad lines can always be controlled. Also, bad linesspread out. Lemma 2.1.
Let K L be a large number. The total number of bad lines is atmost d ( c ( K − L )) − . For every square τ L of side length K − L , the numberof bad strips intersecting τ L is at most d ǫ − .Proof of Lemma 2.1. The first statement follows from { l ι,a,v : l ι,a,v is a bad line } ≤ d (2.6)for every ι ∈ { , } and every a ∈ c ( K − L ) Z . Without loss of generality, wetake ι = 1. We write the line l := l ,a,v as(2.7) { ( ξ, η ) : η = (tan θ ) ξ + a } for some θ with | θ | ≤ π/
4. We take the translation M − l, : ( ξ, η ) ( ξ, a + η )and the rotation(2.8) M − l, : ( ξ, η ) ( ξ cos θ − η sin θ, ξ sin θ + η cos θ )so that M − l = M − l, ◦ M − l, .Let us compute the coefficient of ξ of the polynomial ( h ◦ M − l )( ξ, η ).The quadratic part of ( h ◦ M − l, )( ξ, η ) is given by (cid:16) a , + d X i =3 a i,i − a i − (cid:17) ξ + (cid:16) d X i =3 ( i − a i,i − a i − (cid:17) ξη + (cid:16) a , + d X i =3 i ( i − a i,i a i − (cid:17) η . (2.9)The coefficient of ξ of the polynomial ( h ◦ M − l, ◦ M − l, )( ξ, η ) is given by (cid:16) a , + d X i =3 a i,i − a i − (cid:17) (cos θ ) + (cid:16) d X i =3 ( i − a i,i − a i − (cid:17) sin θ cos θ + (cid:16) a , + d X i =3 i ( i − a i,i a i − ) (cid:17) (sin θ ) . (2.10) ESTRICTION ESTIMATES 7
We denote this function by F ( θ ). By focusing only on the quadratic part,(2.6) follows from(2.11) { θ ∈ c ( K − L ) Z ∩ [ − π/ , π/
4] : | F ( θ ) | ≤ − d ǫ K − L } ≤ d . If ǫ / ≤ | θ | ≤ π/
4, by the assumption (1.8), we obtain | F ( θ ) | & l ,a,v is not a bad line. Hence, we may assume that | θ | ≤ ǫ / . Note that F ′ ( θ ) = (cid:16) d X i =3 ( i − a i,i − a i − (cid:17) cos 2 θ + (cid:16) − ( a , + d X i =3 a i,i − a i − ) + ( a , + d X i =3 i ( i − a i,i a i − ) (cid:17) sin 2 θ. (2.12)The assumption (1.8) implies that F ′ ( θ ) ≥ / | θ | ≤ ǫ / . Theinequality (2.11) follows immediately.Let us prove the second statement. Suppose that a square τ L is fixed.Denote by ( c , c ) the left bottom corner of τ L . By (2.6), it suffices to showthat there are at most 10 d ǫ − many a such that l ι,a,v intersects τ L for some v . Without loss of generality, we may assume that ι = 1. We can write l ,a,v intersecting τ L as(2.13) { ( ξ, η ) : η = (tan θ ) ξ + ( c − c tan θ ) + ( e − e tan θ ) } for some e and e with 0 ≤ e , e ≤ K − L . It suffices to note that(2.14) | e − e tan θ | ≤ K − L and the points a are 10 − d ǫ K − L -separated. (cid:3) We fix ǫ , ǫ > / < λ <
1, surface S with a polynomial h satisfying (1.8), R ′ ≤ R/
2, ball B R ′ of radius R ′ , and function f : [ − , → C , we assume(2.15) k E S f k L . ( B R ′ ) ≤ C ǫ,d,λ ( R ′ ) ǫ k f k − λL ([ − , ) k f k λL ∞ ([ − . ) . Under this induction hypothesis, we will prove (1.9).We will use two different types of rescaling argument: Isotropic rescalingand anisotropic rescaling. When a function is supported on a small square,we can make use of the induction hypothesis more effectively via isotropicrescaling. When a function is supported on a bad strip, anisotropic rescalingwill come into play. In particular, such anisotropic rescaling is also compat-ible with our induction hypothesis.For every set A ⊂ [ − , we denote g A by g A . Here, 1 A is the charac-teristic function of A . SHAOMING GUO AND CHANGKEUN OH
Lemma 2.2.
Suppose that d ≤ K ≤ R ′ ≤ R/ . Under the inductionhypothesis (2.15) , for every / < λ < , surface S with a polynomial h satisfying (1.8) , ball B R ′ of radius R ′ , square τ ⊂ [ − , with side length K − , and function f : [ − , → C , we obtain (2.16) k E S f τ k L . ( B R ′ ) ≤ K − − λ + . CC ǫ,d,λ ( R ′ ) ǫ k f τ k − λL k f τ k λL ∞ , where C is some universal constant.Proof of Lemma 2.2. Without loss of generality, we assume that the centerof the ball B R ′ is the origin. Write τ = [ c , c + K − ] × [ c , c + K − ]. Weapply the change of variables:(2.17) ( ξ, η ) (cid:0) K ( ξ − c ) , K ( η − c ) (cid:1) and obtain | E S f τ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K Z [ − , g ( ξ, η ) e (cid:16)(cid:10) M x, ( ξ, η, h ( ξ, η )) (cid:11)(cid:17) dξdη (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.18)where(2.19) g ( ξ, η ) := f τ ( c + ξ/K, c + η/K ) ,M is a linear transformation of the form(2.20) M x = (cid:16) x + O (1) x K , x + O (1) x K , x K (cid:17) and h is the polynomial given by h ( ξ, η ) := (cid:0) A (cid:1) ξη + ( A , ) ξ + ( A , ) η + d X i =3 K i − i X j =0 ( A i,j ) ξ i − j η j , (2.21)with A := d X i =3 i X j =1 ( i − j ) jc i − j − c j − a i,j ,A i,j := d X i ′ = i i ′ X j ′ = j (cid:18) j ′ j (cid:19)(cid:18) i ′ − j ′ i − j (cid:19) c i ′ − j ′ − i + j c j ′ − j a i ′ ,j ′ . (2.22)Note that | A | . ǫ . We define the new surface(2.23) S := { ( ξ, η, (1 + A ) − h ( ξ, η )) : ( ξ, η ) ∈ [ − , } . Via a simple change of variables, we obtain k E S f τ k L . ( B R ′ ) . K − . k E S g k L . ( B R ′ ) . (2.24)The ball B R ′ on the right hand side could have been replaced by a smallerball. However, we will not take advantage of that gain. Let us check that ESTRICTION ESTIMATES 9 the polynomial (1 + A ) − h satisfies the induction hypothesis (1.8). Weneed to check that | A , | + | A , | + 100 d (cid:16) d X i =3 K i − i X j =0 | A i,j | (cid:17) ≤ (cid:0) A (cid:1) ǫ . (2.25)This follows from | A , | + | A , | + ǫ | A | + 100 d (cid:16) d X i =3 K i − i X j =0 | A i,j | (cid:17) ≤ ǫ . (2.26)It is not difficult to show that it holds true by the initial condition on thecoefficients a i,j . Hence, we can apply the induction hypothesis and obtain k E S f τ k L . ( B R ′ ) . K − . k E S g k L . ( B R ′ ) . K − . C ǫ,d,λ ( R ′ ) ǫ k g k − λL k g k λL ∞ . K − − λ + . C ǫ,d,λ ( R ′ ) ǫ k f τ k − λL k f τ k λL ∞ . (2.27)The last inequality follows from k g k L ∞ = k f τ k L ∞ and k g k L = K k f τ k L .This completes the proof. (cid:3) Lemma 2.3.
Under the induction hypothesis (2.15) , for every L ∈ L , largenumbers K L , R ′ with d ≤ K L ≤ R ′ ≤ R/ , / < λ < , surface S with a polynomial h satisfying (1.8) , ball B R ′ of radius R ′ , and function f : [ − , → C , we have (2.28) k E S f L k L . ( B R ′ ) ≤ ( K L ) ( − − λ + . ) C d C ǫ,d,λ ( R ′ ) ǫ k f L k − λL k f L k λL ∞ , for some constant C d depending only on d .Proof of Lemma 2.3. The proof is very similar to that of Lemma 2.2. With-out loss of generality, we assume that the center of the ball B R ′ is the origin.Also, to simplify the presentation of the proof, we assume that our strip L is(2.29) [ − , × [ − c ( K − L ) , c ( K − L )] . The proof of the general case is similar. Since L ∈ L , by the definition of abad line, we obtain(2.30) max k =2 ,...,d {| a k, |} ≤ − d K − L ǫ . We apply the anisometric scaling ( ξ, η ) ( ξ, K L η ). After this scaling, E S f L ( x ) becomes K − L Z [ − , f L ( ξ, K − L η ) e (cid:0)(cid:10) ( x , K − L x , K − L x ) , ( ξ, η, h ( ξ, η )) (cid:11)(cid:1) dξdη. (2.31) Here, the polynomial h ( ξ, η ) is given by ξη + a , K L ξ + a , K − L η + d X i =3 i X j =0 a i,j ( K − L ) j − ξ i − j η j . (2.32)Let us use the notation g ( ξ, η ) = f ( ξ, K − L η ) and let S denote the surfaceassociated with h . We can write(2.33) E S f L ( x ) = K − L E S g (cid:16) x , K − L x , K − L x (cid:17) . We apply the change of variables on the physical variables ( x , x , x ) ( x , K − L x , K − L x ) and obtain(2.34) k E S f L k L . ( B R ′ ) . ( K L ) − . k E S g k L p ( B R ′ ) . The polynomial h ( ξ, η ) may not satisfy the induction hypothesis (1.8). Todeal with this issue, we decompose the square [ − , into smaller squareswith side length 10 − d . By the triangle inequality, we obtain(2.35) k E S g k L p ( B R ′ ) . X q ∈P (10 − d ) k E S g q k L p ( B R ′ ) . sup q ∈P (10 − d ) k E S g q k L p ( B R ′ ) . For convenience, we assume that q = [0 , − d ] . The proof of the gen-eral case is similar. We apply the scaling ( ξ, η ) (10 d ξ, d η ). Then E S g q ( x ) becomes(2.36)10 − d Z [ − , g q (10 − d ξ, − d η ) e (cid:0)(cid:10) ( x ′ , x ′ , x ′ ) , ( ξ, η, h ( ξ, η )) (cid:11)(cid:1) dξdη where ( x ′ , x ′ , x ′ ) = (10 − d x , − d x , − d x ) and h ( ξ, η ) is given by ξη + a , K L ξ + a , K − L η + d X i =3 − i − d i X j =0 a i,j ( K − L ) j − ξ i − j η j . (2.37)This polynomial satisfies the induction assumption (1.8). We apply thechange of variables(2.38) ( x , x , x ) (10 − d x , − d x , − d x ) , and by the induction hypothesis (2.15) we obtain k E S f L k L . ( B R ′ ) . ( K L ) − . k E S g k L p ( B R ′ ) . C d C ǫ,d,λ ( K L ) − . ( R ′ ) ǫ k g k − λL k g k λL ∞ . C d C ǫ,d,λ ( K L ) ( − − λ + . ) ( R ′ ) ǫ k f L k − λ k f L k λ ∞ . (2.39)The last inequality follows from k g k ∞ = k f L k ∞ and k g k = ( K L ) k f L k .This completes the proof of Lemma 2.3. (cid:3) ESTRICTION ESTIMATES 11
Bad pairs.
Let ζ be a point on [ − , . We define an affine transfor-mation T ζ associated to ζ in the following way. First, let T ζ, be the actionof translation that sends ζ to the origin. Consider the resulting polynomial(2.40) ( h ◦ T − ζ, )( ξ, η ) = d X i =0 i X j =0 a ′ i,j ξ i − j η j , for some new coefficient a ′ i,j . We examine its quadratic terms. Via simplecomputation, it is not difficult to see that(2.41) | a ′ , | + | a ′ , | . ǫ , | a ′ , − | . ǫ . Therefore, one can find an affine transformation, called T ζ, , which is a smallperturbation of the identity map, such that(2.42) ( h ◦ T − ζ, ◦ T − ζ, )( ξ, η ) = d X i =0 i X j =0 a ′′ i,j ξ i − j η j , a ′′ , = 1 , a ′′ , = a ′′ , = 0for some coefficients a ′′ i,j . Define(2.43) T ζ := T ζ, ◦ T ζ, . We will use the map T ζ frequently throughout the paper. Define a polyno-mial h ζ , associated to the point ζ , by(2.44) h ζ ( ξ, η ) := ξη + d X i =3 i X j =0 a ′′ i,j ξ i − j η j . Now we consider two points ζ , ζ with | ζ − ζ | > K − L . Denote by ζ = ( ζ , , ζ , ) ∈ R the image of ζ under the map T ζ . Note that | ζ | > (2 K L ) − . Without loss of generality, we assume that | ζ , | ≥ | ζ , | . Underthis assumption, it is easy to see that(2.45) | ∂ h ζ ( ζ ) | ≥ K L and | ∂ h ζ ( ζ ) | ≥ | ∂ h ζ ( ζ ) | . Here we have used the separation assumption on ζ and ζ . We define anauxiliary function associated to the pair ( ζ , ζ ) by(2.46) P ζ ,ζ ( ξ, η ) := (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ζ ( ξ, η ) ∂ h ζ ( ξ, η ) ∂ h ζ ( ζ ) ∂ h ζ ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) . Note that P ζ ,ζ (0) = P ζ ,ζ ( ζ ) = 0. Note also that(2.47) P ζ ,ζ ( ξ, η ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ h ζ ( T ζ ( ζ )) ∂ h ζ ( T ζ ( ζ )) − ∂ h ζ ( T ζ ( ζ )) ∂ h ζ ( T ζ ( ζ )) − ∂ h ζ ( ξ, η ) ∂ h ζ ( ξ, η ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Set the zero set of this polynomial to be(2.48) Z ζ ,ζ := { ( ξ, η ) : P ζ ,ζ ( ξ, η ) = 0 } . Notice that Z ζ ,ζ contains both the origin and ζ . From (2.45), we see that(2.49) | ∂ P ζ ,ζ ( ξ, η ) | ≥ K L and | ∂ P ζ ,ζ ( ξ, η ) | ≥ | ∂ P ζ ,ζ ( ξ, η ) | , for every ( ξ, η ) ∈ [ − , . By the implicit function theorem, we can write Z ζ ,ζ ∩ [ − , as(2.50) { ( ξ, η ζ ,ζ ( ξ )) : ξ ∈ [ − , } , for some smooth function η ζ ,ζ with η ζ ,ζ (0) = η ζ ,ζ ( ζ , ) = 0. Moreover,from (2.49) we can conclude that (2.51) | η ′ ζ ,ζ ( ξ ) | . , for every ξ. Define the function h ζ along the variety by(2.52) H ζ ,ζ ( ξ ) := h ζ ( ξ, η ζ ,ζ ( ξ )) . Define two bad sets B ( ζ , ζ ) := { ( ξ, η ζ ,ζ ( ξ )) ∈ [ − , : | η ′ ζ ,ζ ( ξ ) − η ′ ζ ,ζ (0) | ≤ K − } , B ( ζ , ζ ) := { ( ξ, η ζ ,ζ ( ξ )) ∈ [ − , : | H ′ ζ ,ζ ( ξ ) | ≤ K − } . (2.53)Now we have all the notions we need to define bad pairs. We say that ( ζ , ζ )is a bad pair if(2.54) ζ ∈ B ( ζ , ζ ) , where(2.55) B ( ζ , ζ ) := B ( ζ , ζ ) ∩ B ( ζ , ζ ) . Otherwise we say that ( ζ , ζ ) is a good pair . Note that if ζ ∈ B ( ζ , ζ ) then B ( ζ , ζ ) = B ( ζ , ζ ). Note also that even if ( ζ , ζ ) is a good pair, accordingto our definition, ( ζ , ζ ) may not be a good pair.Suppose that τ , τ are squares with side length K − and dist( τ , τ ) ≥ K − L . We say that ( τ , τ ) is a bad pair if ( ζ , ζ ) is a bad pair for every ζ i ∈ τ i . Otherwise, we say that ( τ , τ ) is a good pair.The total number of bad pairs could be enormous. However, under somecircumstance, we are able to describe their distribution. The followinglemma plays a crucial role in the proof of Theorem 3.1. Lemma 2.4.
Fix sufficiently small ǫ > and ǫ > . Let ζ , ζ be pointsin [ − , with | ζ − ζ | > K − L . For sufficiently large dyadic numbers K L , K d , . . . , K , K with (2.56) K L =: K d +1 ≪ K d ≪ · · · ≪ K ≪ K := K We postpone the details to (2.69) in order not to interrupt the discussion.
ESTRICTION ESTIMATES 13 the following holds true: Either there exist squares υ j,i ∈ P ( K − j , R ) suchthat (2.57) B ( ζ , ζ ) ⊂ (cid:16) d [ i =1 T ζ ( υ d +1 ,i ) (cid:17) ∪ (cid:16) d − [ j =2 O ( K j +1 ) [ i =1 T ζ ( υ j,i ) (cid:17) or there exist bad strips L j such that (2.58) N K − L (cid:0) B ( ζ , ζ ) (cid:1) ⊂ [ j =1 T ζ ( L j ) . Note that for each j the number of the squares υ j,i is O ( K j +1 ). Eventhough K j +1 is a large number compared to one, it is still very small com-pared to K j . Hence, the number of squares will not cause any trouble.To prove this lemma, we will consider several cases. In some cases, wewill reduce our problem to a problem involving a one-variable polynomialand apply Lemma 2.5. Recall that for a polynomial p ( ξ ), k p k denote thesupremum of all its coefficients. Lemma 2.5.
Let n ≥ . For every polynomial (2.59) p ( ξ ) = b + b ξ + · · · + b n ξ n with k p k = 1 and for every < γ . , there exist points { ξ i } n ( n +1) / i =1 suchthat (2.60) (cid:8) ξ ∈ [ − ,
4] : | p ( ξ ) | < γ (cid:9) ⊂ n ( n +1)2 [ i =1 (cid:16) [ ξ i − γ n , ξ i + 4 γ n ] (cid:17) . This lemma is a corollary of Lemma 2.14 in [GZ19]. We will give a prooffor the completeness of the paper.
Proof of Lemma 2.5.
Let us use the notation I = [ − ,
4] for simplicity.We claim that for every σ > p with k p ′′ k L ∞ ( I ) ≤ { ξ ∈ I : | p ( ξ ) | ≤ σ } ⊂ { ξ ∈ I : | p ′ ( ξ ) | ≤ σ } ∪ N σ ( Z ( p )) , (2.61)where Z ( p ) denotes the zero set of p . We assume that this claim holds truefor a moment and deduce (2.60). Note that if k p ′ k ≪
1, then by the claimthe left hand side of (2.60) is empty for sufficiently small γ >
0. Hence, wemay assume that k p ′ k ≃
1. By the fundamental theorem of algebra, it is notdifficult to see that the lemma follows by applying the claim repeatedly.It remains to prove this claim. First note that if | p ( ξ ) | ≤ σ and | p ′ ( ξ ) | > σ then(2.62) | p ( ξ ) | − σ | p ′ ( ξ ) | + 12 σ k p ′′ k L ∞ ( I ) < . Our goal to show that in this case dist( ξ, Z ( p )) ≤ σ . Without lose ofgenerality, we may assume that p ( ξ ) >
0. Suppose that dist( ξ, Z ( p )) > σ for a moment and deduce the contradiction. We consider the case that p ′ ( ξ ) >
0. By Taylor’s theorem,(2.63) 0 < p ( ξ − σ ) ≤ p ( ξ ) − σp ′ ( ξ ) + 12 σ k p ′′ k L ∞ ( I ) . This contradicts with (2.62). We consider the other case that p ′ ( ξ ) <
0. ByTaylor’s theorem,(2.64) 0 < p ( ξ + σ ) ≤ p ( ξ ) + σp ′ ( ξ ) + 12 σ k p ′′ k L ∞ ( I ) . This also contradicts (2.62), and therefore completes the proof of Lemma2.5. (cid:3)
Proof of Lemma 2.4.
Recall the map T ζ from (2.43). Denote by ζ =( ζ , , ζ , ) the image of ζ under T ζ . Without loss of generality, we assumethat | ζ , | ≥ | ζ , | . Throughout the proof, for the sake of simplicity, h ζ will be abbreviated to h , H ζ ,ζ simply to H , η ζ ,ζ to η , and B ( ζ , ζ ) to B . Let us first compute the derivative of η ( ξ ). Recall that the function η ( ξ )is implicitly defined by(2.65) P ζ ,ζ ( ξ, η ( ξ )) = ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) − ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) = 0 . We take a derivative with respect to ξ and obtain ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) + ∂ h ( ξ, η ( ξ )) η ′ ( ξ ) ∂ h ( ζ ) − ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) − ∂ h ( ξ, η ( ξ )) η ′ ( ξ ) ∂ h ( ζ ) = 0 . (2.66)By rearranging terms, we obtain(2.67) η ′ ( ξ ) = ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) − ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) − ∂ h ( ξ, η ( ξ )) ∂ h ( ζ ) . Note that(2.68) η ′ (0) = ∂ h ( ζ ) ∂ h ( ζ ) . Moreover, by taking derivatives repeatedly to (2.65), it is not difficult toshow that(2.69) | η ( j )0 ( ξ ) | . ξ ∈ [ − , j = 1 , . . . , d . By the chain rule and (2.65), we obtain | H ′ ( ξ ) | = | ∂ h ( ξ, η ( ξ )) + ∂ h ( ξ, η ( ξ )) η ′ ( ξ ) | = (cid:12)(cid:12)(cid:12) ∂ h ( ξ, η ( ξ )) (cid:0) η ′ ( ξ ) + ∂ h ( ζ ) ∂ h ( ζ ) (cid:1)(cid:12)(cid:12)(cid:12) . (2.70) ESTRICTION ESTIMATES 15
For ( ξ, η ( ξ )) ∈ B , we apply the triangle inequality and obtain | H ′ ( ξ ) | ≥ (cid:12)(cid:12)(cid:12) ∂ h ( ξ, η ( ξ )) (cid:0) η ′ (0) + ∂ h ( ζ ) ∂ h ( ζ ) (cid:1)(cid:12)(cid:12)(cid:12) − K − = 2 (cid:12)(cid:12)(cid:12) ∂ h ( ξ, η ( ξ )) (cid:0) ∂ h ( ζ ) ∂ h ( ζ ) (cid:1)(cid:12)(cid:12)(cid:12) − K − . (2.71)Depending on the size of (cid:12)(cid:12) ∂ h ( ζ ) ∂ h ( ζ ) (cid:12)(cid:12) , we have two cases(2.72) (cid:12)(cid:12)(cid:12) ∂ h ( ζ ) ∂ h ( ζ ) (cid:12)(cid:12)(cid:12) > K − / or (cid:12)(cid:12)(cid:12) ∂ h ( ζ ) ∂ h ( ζ ) (cid:12)(cid:12)(cid:12) ≤ K − / . The quantity that appears in the above display has a geometric meaning,see (2.68).
Case 1. (cid:12)(cid:12)(cid:12) ∂ h ( ζ ) ∂ h ( ζ ) (cid:12)(cid:12)(cid:12) > K − / .In this case, by (2.71), for ( ξ, η ( ξ )) ∈ B we obtain(2.73) 2 K − / | ∂ h ( ξ, η ( ξ )) | − K − < | H ′ ( ξ ) | ≤ K − . Multiplying K / on both sides we obtain(2.74) | ∂ h ( ξ, η ( ξ )) | < K − / , which certainly means that(2.75) B ⊂ { ( ξ, η ( ξ )) ∈ B : | ∂ h ( ξ, η ( ξ )) | < K − / } . By the lower bound (2.45) and the equality (2.65), for every ( ξ, η ( ξ )) ∈ B ,we obtain(2.76) | ∂ h ( ξ, η ( ξ )) | < K − / . By choosing K ≪ K we can take υ ∈ P ( K − , R ) such that T ζ ( υ ) containsthe set on the right hand side of (2.75). This completes the discussion ofCase 1. Case 2. (cid:12)(cid:12)(cid:12) ∂ h ( ζ ) ∂ h ( ζ ) (cid:12)(cid:12)(cid:12) ≤ K − / .In this case, by (2.65), for every ( ξ, η ( ξ )), we obtain(2.77) | ∂ h ( ξ, η ( ξ )) | ≤ K − / . Since ( ξ, η ( ξ )) ∈ B , we obtain from (2.68) and the triangle inequality that(2.78) | η ′ ( ξ ) | ≤ K − / . This, combined with (2.67), implies that(2.79) | ∂ h ( ξ, η ( ξ )) | ≤ K − / ≤ K − . We have shown that B ⊂ (cid:8) ( ξ, η p ( ξ )) ∈ B : | ∂ h ( ξ, η ( ξ )) | ≤ K − / (cid:9) ∩ (cid:8) ( ξ, η p ( ξ )) ∈ B : | ∂ h ( ξ, η ( ξ )) | ≤ K − (cid:9) =: D ∩ D . (2.80)Depending on the size of max i =3 ,...,d (cid:0) | a ′′ i, | (cid:1) , we have two subcases(2.81) max i =3 ,...,d (cid:0) | a ′′ i, | (cid:1) ≤ K − L or max i =3 ,...,d (cid:0) | a ′′ i, | (cid:1) ≥ K − L . Case 2.1. max i =3 ,...,d (cid:0) | a ′′ i, | (cid:1) ≤ K − L .In this case, we have(2.82) sup ξ ∈ [ − , | ∂ h ( ξ, | ≤ d K − L and since ∂ h ( ξ, η ) = η · (1 + O ( ǫ )) + ∂ h ( ξ,
0) we see that(2.83) D ⊂ N d K − L ( { η = 0 } ) . Since we are in the former case of (2.81) and the width of a bad strip is2 · − d ǫ K − L , we can find two bad strips L and L so that the imageof their union under the map T ζ covers the set on the right hand side of(2.83). By (2.80), we obtain(2.84) B ⊂ T ζ ( L ) ∪ T ζ ( L ) . This gives the desired result.
Case 2.2. max i =3 ,...,d (cid:0) | a ′′ i, | (cid:1) ≥ K − L .Define the sets(2.85) D j := { ( ξ, η ( ξ )) ∈ B : (cid:12)(cid:12)(cid:12) ∂ j h ∂ξ j ( ξ, η ( ξ )) (cid:12)(cid:12)(cid:12) ≤ K − j } for every j = 3 , . . . , d . We claim that(2.86) B ⊂ (cid:16) d \ j =1 D j (cid:17) ∪ (cid:16) d − [ j =2 O ( K j +1 ) [ i =1 T ζ ( υ j,i ) (cid:17) for some υ j,i ∈ P ( K − j , R ).Let us prove the claim inductively. Suppose that we have obtained that(2.87) B ⊂ (cid:16) k \ j =1 D j (cid:17) ∪ (cid:16) k − [ j =2 O ( K j +1 ) [ i =1 T ζ ( υ j,i ) (cid:17) for some k ∈ { , . . . , d − } . Note that the case k = 2 follows from (2.80).Note first that(2.88) k \ j =1 D j ⊂ (cid:0) k +1 \ j =1 D j (cid:1) ∪ (cid:0) k \ j =1 D j ∩ ( D k +1 ) c (cid:1) . ESTRICTION ESTIMATES 17
We will show that(2.89) k \ j =1 D j ∩ ( D k +1 ) c ⊂ O ( K k +1 ) [ i =1 T ζ ( υ k,i ) . This gives the proof of (2.87) for k + 1 and by an inductive argument itfinishes the proof of the claim (2.86).Let us prove (2.89). Let ( ξ , η ( ξ )) be a point belonging to the set onthe left hand side of (2.89). It is not difficult to see that (2.89) follows fromthe following geometric observation: All points ( ξ, η ( ξ )) ∈ B with(2.90) CK k +1 K − k ≤ | ξ − ξ | ≤ C ′ K − k +1 do not belong to T kj =1 D j ∩ ( D k +1 ) c . Here C, C ′ are two constant that areto be chosen. Divide (2.65) by ∂ h ( ζ ). By taking derivatives with respectto ξ , for every j ≥
2, we obtain(2.91) ∂ j − ∂ξ j − (cid:16) ∂ h ( ξ, η ( ξ )) (cid:17) = ∂ j − ∂ξ j − (cid:16) ∂ h ( ξ, η ( ξ )) (cid:17) ∂ h ( ζ ) ∂ h ( ζ ) . Since we are considering the latter case of (2.72), the right hand side isbounded by O ( K − / ). By the chain rule, we obtain ∂ j h ∂ξ j ( ξ, η ( ξ )) + ∂ h ∂ξ∂η ( ξ, η ( ξ )) η ( j − ( ξ )+ Error( ξ, η (0)0 ( ξ ) , . . . , η ( j − ( ξ )) = O ( K − / ) . (2.92)where(2.93) | Error( ξ, η (0)0 ( ξ ) , . . . , η ( j − ( ξ )) | . j − X l =1 | η ( l )0 ( ξ ) | . Hence, by an inductive argument, we see that(2.94) | η ( j − ( ξ ) | ≃ (cid:12)(cid:12)(cid:12) ∂ j h ∂ξ j ( ξ , η ( ξ )) (cid:12)(cid:12)(cid:12) ≤ K − j and(2.95) | η ( k ) ( ξ ) | ≃ (cid:12)(cid:12)(cid:12) ∂ k +1 h ∂ξ k +1 ( ξ , η ( ξ )) (cid:12)(cid:12)(cid:12) > K − k +1 . for every j = 2 , . . . , k . By (2.94) and the previous discussion, it suffices toshow that for every point ( ξ, η ( ξ )) ∈ B with CK k +1 K − k ≤ | ξ − ξ | ≤ C ′ K − k +1 it holds that(2.96) | η ( k − ( ξ ) | > CK − k . Let us prove (2.96). By the fundamental theorem of calculus and (2.94) with j = k , we obtain η ( k − ( ξ ) = Z ξξ η ( k )0 ( x ) dx + η ( k − ( ξ )= Z ξξ (cid:0) Z xξ η ( k +1)0 ( y ) dy + η ( k )0 ( ξ ) (cid:1) dx + η ( k − ( ξ )= Z ξξ Z xξ η ( k +1)0 ( y ) dydx + ( ξ − ξ ) η ( k )0 ( ξ ) + O ( K − k )= Z ξξ ( ξ − y ) η ( k +1)0 ( y ) dy + ( ξ − ξ ) η ( k )0 ( ξ ) + O ( K − k ) . (2.97)We use (2.95), (2.97), and a trivial bound (2.69), and obtain(2.98) | η ( k − ( ξ ) | & K − k +1 | ξ − ξ | − c | ξ − ξ | − cK − k , for some small constant c . It is not difficult to see that if 4 cK k +1 K − k ≤| ξ − ξ | ≤ C ′ K − k +1 then the above lower bound is greater than 2 cK − k andthis finishes the proof of (2.96), and thus (2.87) and (2.86).It remains to prove that(2.99) d \ j =1 D j ⊂ d [ i =1 T ζ ( υ d +1 ,i )for some υ d +1 ,i ∈ P ( K − d +1 , R ). For simplicity, we first write(2.100) d \ j =1 D j ⊂ n ( ξ, η ) ∈ [ − , : max j =1 ,...,d (cid:12)(cid:12)(cid:12) ∂ j h ∂ξ j ( ξ, η ) (cid:12)(cid:12)(cid:12) ≤ K − d o . By Taylor’s theorem with respect to ξ -variable, we obtain(2.101) ∂h ∂ξ (0 , η ) = d X j =1 ( − j − ξ j − ( j − ∂ j h ∂ξ j ( ξ, η ) . Thus, the set on the right hand side of (2.100) is contained in(2.102) n ( ξ, η ) ∈ [ − , : max (cid:16)(cid:12)(cid:12)(cid:12) ∂h ∂ξ ( ξ, η ) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ∂h ∂ξ (0 , η ) (cid:12)(cid:12)(cid:12)(cid:17) . K − d o . Since ∂h ∂ξ (0 , η ) = η · (1 + O ( ǫ )), if a point ( ξ, η ) belongs to the above set,we obtain | η | . K − d . Hence, by treating η as an error, the above set iscontained in(2.103) n ( ξ, η ) ∈ [ − , : | η | . K − d , (cid:12)(cid:12)(cid:12) ∂h ∂ξ ( ξ, (cid:12)(cid:12)(cid:12) . K − d o . Since we are in the latter case of (2.81), the norm of the polynomial ∂ h ( ξ, & K − L and we apply Lemma 2.5 and obtain (2.99). This completesthe proof. (cid:3) ESTRICTION ESTIMATES 19 Basic setup: Broad function and wave packet decomposition
In this section, we follow the broad-narrow analysis of Bourgain and Guth[BG11] and Guth [Gut16], and introduce the notation of broad points. Oneadvantage of introducing this notion is that with it we can start making useof multilinear restriction estimates (bilinear in our case).Our notion of broad points is slightly more complicated than those in[Gut16, CL17], see (3.2) and (3.3). First of all, it involves many differentscale, including K d +1 , · · · , K . In other words, we need to make sure thatour function | Ef ( x ) | is “broad” at every scale K j . This is necessary as inLemma 2.4 bad pairs appear at every scale. Next, the notion of broad pointsinvolves bad strips. A similar notion was already used in [CL17]. We needto get rid of bad strips as it will be difficult to obtain a bilinear restrictionestimate as good as the one in [Gut16], if certain main contributions comefrom a bad strip. However, a technical issue arises here. In [CL17], badstrips are either horizontal or vertical. Two bad strips are either disjointor intersect at a square. In our case, orientations of bad strips are morecomplicated and we need to take into account all the possible intersectionsof bad strips.We also review wave packet decomposition in this section.3.1. Broad function.
Take dyadic numbers(3.1) K L = K d +1 ≪ K d ≪ · · · ≪ K ≪ K = K. Let M := 10 d ǫ − . Recall that P ( K − , A ) is a collection of all dyadicsquares with side length K − in a set A and we sometimes use P ( K − )for P ( K − , [ − , ). Recall also that Lemma 2.1 states that for everysquare with side length K − d +1 there are at most M bad strips with width2 · − d ǫ K − L intersecting the square.For every ( d + 2)-tuple α = ( α , . . . , α d +1 ) ∈ (0 , d +2 , we say that x ∈ R is an α -broad point of Ef if(3.2) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ α d +1 | Ef ( x ) | and max τ j ∈P ( K − j ) | Ef τ j ( x ) | + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ α j | Ef ( x ) | (3.3)for every j = 0 , . . . , d + 1. Here, the bad strips L i do not need to be distinct.Also, the second term on the left hand side of (3.3) is introduced for sometechnical reason.For α ∈ (0 , d +2 and r ∈ R , we use the notation rα = ( rα , . . . , rα d +1 ).We let Br α Ef ( x ) denote the function which is | Ef ( x ) | if x is an α -broad point of Ef and 0 otherwise. Theorem 1.2 follows from the following broadfunction estimate. Theorem 3.1.
For every ǫ > there exist dyadic numbers K, K , . . . , K d , K L with (3.4) K L = K d +1 ( ǫ ) ≪ K d ( ǫ ) ≪ · · · ≪ K ( ǫ ) ≪ K ( ǫ ) = K and a small number δ trans ∈ (0 , ǫ ) so that the following holds true: For every α = ( α , . . . , α d +1 ) with ≥ α j ≥ K − ǫj and j = 0 , . . . , d + 1 , R ≥ , ball B R ⊂ R of radius R , and function f : [ − , → C , it holds that k Br α Ef k L . ( B R ) ≤ C ǫ,d R δ trans ( P d +1 j =0 log ( K ǫj α j )) R ǫ k f k + ǫL max θ ∈P ( R − / ) k f k − ǫL ( θ ) . (3.5) Here, the averaged L norm is defined by (3.6) k f k L ( θ ) := (cid:16) | θ | Z θ | f ( x ) | dx (cid:17) / . Moreover, lim ǫ → K d +1 ( ǫ ) → + ∞ , and δ trans can be taken to be ǫ . Let us assume Theorem 3.1 and finish the proof of Theorem 1.2.
Proof of Theorem 1.2.
For simplicity, we use the notation p = 3 .
25. Fix3 / < λ < /
13. The other case where 5 / ≤ λ ≤ ǫ > < d ǫ < λ − . Our proof relies on an induction argument on the radius R . We assume that(1.9) holds true for all the radii smaller than R/ R . Take α j = K − ǫj . By the definition of the broad function, weobtain | Ef ( x ) | ≤ | Br α Ef ( x ) | + d +1 X j =0 α − j max τ j ∈P ( K − j ) | Ef τ j ( x ) | + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12)! + ( α d +1 ) − max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (3.8) ESTRICTION ESTIMATES 21
We raise both sides to the p -th power, integrate over B R , replace the maxby l p -norms, and obtain Z B R | Ef | p ≤ C Z B R | Br α Ef | p + C d +1 X j =0 α − p j X τ j ∈P ( K − j ) Z B R | Ef τ j | p + X υ j ∈P ( K − j ) X L ,...,L M ∈ L Z B R (cid:12)(cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ (cid:12)(cid:12)(cid:12)(cid:12) p ! + C ( α d +1 ) − p X L ,...,L M ∈ L Z B R (cid:12)(cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ (cid:12)(cid:12)(cid:12)(cid:12) p , (3.9)for some universal constant C . We apply Theorem 3.1 to the first term andobtain Z B R | Br α Ef | p ≤ CC p ǫ,d R p ǫ k f k p + ǫp L max θ ∈P ( R − / ) k f k p − ǫp L ( θ ) ≤ − dp C p ǫ,d,λ R p ǫ k f k p (1 − λ ) L k f k p λ ∞ , (3.10)provided that C ǫ,d,λ in Theorem 1.2 is chosen such that(3.11) 10 dp CC ǫ,d ≤ C ǫ,d,λ . The second inequality of (3.10) follows from H¨older’s inequality and fromthe fact that 12 / > − λ . This takes care of the contribution from thefirst term.Let us bound the second term. Note that p (1 − λ ) >
2. We apply Lemma2.2 and the embedding l p (1 − λ ) ֒ → l to obtain α − p j X τ j ∈P ( K − j ) Z B R | Ef τ j | p ≤ ( K j ) − ǫ C p ǫ,d,λ R p ǫ X τ j ∈P ( K − j ) k f τ j k p (1 − λ ) L k f τ j k p λL ∞ ≤ ( K j ) − ǫ C p ǫ,d,λ R p ǫ k f k p (1 − λ ) L k f k p λL ∞ . (3.12)Hence, the second term is also harmless as K j is sufficiently large. The thirdterm can be dealt with by following the same argument. We leave out thedetails. Let us bound the fourth term. By Lemma 2.3 and the embedding l p (1 − λ ) ֒ → l , we obtain α − p d +1 X L ,...,L M ∈ L Z B R (cid:12)(cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ (cid:12)(cid:12)(cid:12)(cid:12) p ≤ ( K d +1 ) − ǫ C p ǫ,d,λ R p ǫ X L ,...,L M ∈ L k f ∩ Mi =1 L i k p (1 − λ ) L k f k p λL ∞ ≤ ( K d +1 ) − ǫ C p ǫ,d,λ R p ǫ (cid:16) X L ,...,L M ∈ L k f ∩ Mi =1 L i k L (cid:17) p − λ )2 k f k p λL ∞ . (3.13)By Lemma 2.1, it is further bounded by(3.14) ǫ − p dp ( K d +1 ) − ǫ C p ǫ,d,λ k f k p (1 − λ ) L k f k p λL ∞ . By taking K d +1 to be sufficiently large, we finish the analysis of the lastterm. This completes the proof of Theorem 1.2. (cid:3) Wave packet decomposition.
We briefly review the standard wavepacket decomposition. Details can be found in Tao [Tao03] and Guth[Gut16]. We decompose the square [ − , into smaller dyadic squares θ ofside length R − / . Let w θ denote the left bottom corner of θ . Let v θ denotethe normal vector to our surface S at the point ( w θ , h ( w θ )). Let T ( θ ) denotea set of tubes covering B R , that are parallel to v θ with radius R / δ andlength CR . Denote T := ∪ θ ∈P ( R − / ) T ( θ ) and T ( τ ) := ∪ θ ∈P ( R − / ,τ ) T ( θ ).For each T ∈ T ( θ ), let v ( T ) denote the direction v θ of the tube. Proposition 3.2 (Wave packet decomposition) . If f ∈ L ([ − , ) thenfor each T ∈ T we can choose a function f T so that the following holds true: (1) If T ∈ T ( θ ) then supp( f T ) ⊂ θ ; (2) If x ∈ B R \ T , then | Ef T ( x ) | ≤ R − k f k ; (3) For any x ∈ B R , | Ef ( x ) − P T ∈ T Ef T ( x ) | ≤ R − k f k L ; (4) If T , T ∈ T ( θ ) and T , T are disjoint, then | R f T ¯ f T | ≤ R − R θ | f | ; (5) P T ∈ T ( θ ) R [ − , | f T | . R θ | f | . For the proof of the wave packet decomposition, we refer to that of Propo-sition 2.6 in [Gut16].We decompose [ − , into smaller squares τ with side length K − . Write f = P τ ∈P ( K − ) f τ and apply the above wave packet decomposition to thefunction f τ . To simplify notation, ( f τ ) T will be abbreviated to f τ,T . Lemma 3.3.
Suppose that T k ⊂ T indexed by k ∈ A for some index set A .If each tube T belongs to at most µ of the subsets { T k } k ∈ A then for every θ , (3.15) X k ∈ A Z θ (cid:12)(cid:12)(cid:12) X T ∈ T k f τ,T (cid:12)(cid:12)(cid:12) . µ Z θ | f τ | . ESTRICTION ESTIMATES 23
The proof of Lemma 3.3 is identical to that of Lemma 2.7 in [Gut16].Hence, we leave out the details. We record a special case of Lemma 3.3.
Lemma 3.4.
For every cap θ , τ , and subcollection T k ⊂ T , (3.16) Z θ (cid:12)(cid:12)(cid:12)(cid:12) X T ∈ T k f τ,T (cid:12)(cid:12)(cid:12)(cid:12) . Z θ | f τ | . Proof of Theorem 3.1
In this section we will prove Theorem 3.1. We may assume that (3.5) holdstrue for α d +1 ≥ − d by taking K d +1 large enough and for α j ≥ K − j +1 for j = 0 , . . . , d by taking K j large enough. The constants K d +1 , · · · , K will satisfy(4.1) 1 ≪ K d +1 ≪ K d ≪ · · · ≪ K ≪ K . We will introduce other parameters, called δ , δ trans , δ deg , such that(4.2) δ trans ≪ δ deg ≪ δ ≪ ǫ. More explicitly, we take(4.3) δ trans = ǫ , δ deg = ǫ , and δ = ǫ . Moreover, recall from the last section that M = 10 d ǫ − .The proof is via an induction argument on the radius R . We assume that(3.5) holds true for all radii ≤ R/ ≤ R .We will utilize the polynomial partitioning lemma in [Gut16]. Theorem 4.1 (Corollary 1.7 in [Gut16]) . Let W be a non-negative L func-tion on R n . Then for any D ∈ Z + , there is a non-zero polynomial P of degreeat most D so that R n \ Z ( P ) is a disjoint union of ≃ D n open sets O i , andthe integrals R O i W agree up to a factor of 2. Moreover, the polynomial P is a product of non-singular polynomials. We apply Theorem 4.1 to the function 1 B R (Br α ( Ef )) p with D = R δ deg .Then there exists a polynomial P such that(4.4) R \ Z ( P ) = ≃ D G i =1 O i and(4.5) Z O i ′ ∩ B R (Br α Ef ) p ≃ Z O i ∩ B R (Br α Ef ) p for every i and i ′ . Here, P is a product of non-singular polynomials.We define a wall W to be the R / δ -neighborhood of the variety Z ( P ).Define cells O ′ i := ( O i ∩ B R ) \ W . Denote the collection of all the tubesintersecting O ′ i by(4.6) T i := { T ∈ T : T ∩ O ′ i = ∅} . For every τ ∈ P ( K − ) and τ j ∈ P ( K − j ) with j = 1 , . . . , d + 1, define(4.7) f τ,i := X T ∈ T i f τ,T , f τ j ,i := X τ ∈P ( K − ,τ j ) f τ,i , f i := X τ ∈P ( K − ) f τ,i . To prove (3.5), we decompose our ball B R into the cells and the wall andobtain(4.8) Z B R (Br α Ef ) p = ≃ D X i =1 Z B R ∩ O ′ i (Br α Ef ) p + Z B R ∩ W (Br α Ef ) p . There are two scenarios. We say that we are in the cellular case if(4.9) ≃ D X i =1 Z B R ∩ O ′ i (Br α Ef ) p ≥ Z B R ∩ W (Br α Ef ) p . Otherwise, we say that we are in the wall case .4.1.
Cellular case.
In this case, there exists a subcollection T of the indexset of the cells with cardinality ≃ D such that for all i ∈ T (4.10) Z B R (Br α Ef ) p ≃ D Z B R ∩ O ′ i (Br α Ef ) p . We refer to Lemma 4.1 in [CL17] for more details.
Lemma 4.2.
For every i ∈ T and every x ∈ O ′ i , it holds that (4.11) Br α Ef ( x ) ≤ Br α Ef i ( x ) + O ( R − k f k L ) . Proof of Lemma 4.2.
Suppose that x ∈ O ′ i is an α -broad point of Ef.
Byitem (3) of Proposition 3.2, we obtain(4.12) Ef τ ( x ) = X T ∈ T Ef τ,T ( x ) + O ( R − k f τ k ) . If x ∈ T , then T intersects O ′ i and T ∈ T i . If x / ∈ T , then by item (2) ofProposition 3.2, we have | Ef τ,T ( x ) | ≤ R − k f τ k . Hence, we have(4.13) Ef τ ( x ) = Ef τ,i ( x ) + O ( R − k f τ k ) . By summing over τ , we obtain(4.14) Ef ( x ) = Ef i ( x ) + O ( R − k f k ) . It remains to show that x is a 2 α -broad point of Ef i . We may assumethat | Ef ( x ) | ≥ R − k f k and by the above inequality, we have | Ef i ( x ) | ≥ R − k f k . We need to show that(4.15) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ,i ( x ) (cid:12)(cid:12)(cid:12) ≤ α d +1 | Ef i ( x ) | ESTRICTION ESTIMATES 25 and max τ j ∈P ( K − j ) | Ef τ j ,i ( x ) | + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ,i ( x ) (cid:12)(cid:12)(cid:12) ≤ α j | Ef i ( x ) | (4.16)for every j = 0 , . . . , d + 1.Let us first prove (4.15). By (4.13), the left hand side of (4.15) is boundedby(4.17) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + O ( R − k f k )which is further bounded by(4.18) 1110 α d +1 | Ef ( x ) | as x is an α -broad point of Ef . By (4.14), the above display is furtherbounded by(4.19) 1110 α d +1 | Ef i ( x ) | + O ( R − k f k ) . Since the error term is bounded by α d +1 | Ef i ( x ) | , this gives (4.15). Thesecond inequality (4.16) can be proved in the exactly same way. We leaveout the details. (cid:3) We apply Lemma 4.2 to each cell O ′ i and obtain(4.20) Z B R ∩ O ′ i (cid:16) Br α Ef ( x ) (cid:17) p . Z B R (cid:16) Br α Ef i ( x ) (cid:17) p + O ( R − p k f k p L ) . If the error dominates the first term on the right hand side, we automaticallyobtain Theorem 3.1. Hence, we may assume that the first term dominatesthe error. We decompose the ball B R into smaller balls of radius R/ R and obtain Z B R ∩ O ′ i (cid:16) Br α Ef ( x ) (cid:17) p . R δ trans ( P j log ( K ǫj α j )) p R ǫp k f i k ǫp L max θ ∈P ( R − / ) k f i k − ǫp L ( θ ) . (4.21)We apply Lemma 3.4 to k f i k L ( θ ) and see that the above term is boundedby R δ trans ( P j log ( K ǫj α j )) p R ǫp k f i k ǫp L max θ ∈P ( R − / ) k f k − ǫp L ( θ ) . R δ trans ( P j log ( K ǫj α j )) p R ǫp (cid:0) X τ ∈P ( K − ) k f τ,i k L (cid:1) ǫp max θ ∈P ( R − / ) k f k − ǫp L ( θ ) . (4.22) As a line can intersect Z ( P ) at most ( D + 1)-times (recall that the degreeof the polynomial P is D ), we see that each tube T ∈ T intersects at most( D + 1) many cells O ′ i . By this observation and Lemma 3.3, we have(4.23) X i ∈T k f τ,i k L . D k f τ k L . Since the cardinality of T is ≃ D , we can find some i ∈ T such that(4.24) X τ ∈P ( K − ) k f τ,i k L . D − X τ ∈P ( K − ) k f τ k L . D − k f k L . By combining (4.10), (4.21), (4.22) and (4.24) with i = i , we obtain Z B R (Br α Ef ) p . D − ǫp R Cδ trans R δ trans ( P j log ( K ǫj α j )) p R ǫp k f k ǫp L max θ ∈P ( R − / ) k f k − ǫp L ( θ ) . (4.25)It suffices to note that(4.26) D − ǫ R Cδ trans = R − ǫδ deg + Cδ trans ≪ R . This completes the proof of the cellular case.4.2. Wall case.
In the wall case, we need to show that k Br α Ef k L . ( B R ∩ W ) ≤ − C ǫ,d R δ trans ( P j log ( K ǫj α j )) R ǫ k f k + ǫL max θ ∈P ( R − / ) k f k − ǫL ( θ ) . (4.27)We will cover B R with ≃ R δ balls B k of radius R − δ . Let T k, tang denotethe collection of all tubes T ∈ T such that B k ∩ W ∩ T = ∅ and(4.28) Angle( v ( T ) , T z ( Z ( P ))) ≤ R − / δ , for every z that is a non-singular point of Z ( P ) lying in 2 B k ∩ T . Here T z ( Z ( P )) is the tangent plane of Z ( P ) at the point z . Let T k, trans denotethe collection of all tubes T ∈ T such that B k ∩ W ∩ T = ∅ and(4.29) Angle( v ( T ) , T z ( Z ( P ))) > R − / δ , for some non-singular point z of Z ( P ) lying in 2 B k ∩ T . We use thenotations T k, − and T k, + for T k, tang and T k, trans , respectively. It is easy tosee that each tube T with B k ∩ W ∩ T = ∅ belongs to either T k, + or T k, − .Denote T k, + ( τ ) := T ( τ ) ∩ T k, + and T k, − ( τ ) := T ( τ ) ∩ T k, − . We will use twogeometric estimates. Lemma 4.3 (Lemma 3.5 and 3.6 in [Gut16]) . (1) Each tube T ∈ T belongs to at most R O ( δ deg ) different sets T k, + . (2) For each k , the number of different θ with T k, − ∩ T ( θ ) = ∅ is at most R / O ( δ ) . ESTRICTION ESTIMATES 27
For a subcollection I of the squares τ , we define f I by(4.30) f I = X τ ∈ I f τ , and define(4.31) f τ,k, + := X T ∈ T k, + ( τ ) f T , f k, + := X τ ∈P ( K − ) f τ,k, + , f I,k, + := X τ ∈ I f τ,k, + . Moreover, define f τ,k, − , f k, − , f I,k, − similarly. We also define the bilinearfunction(4.32) Bil( Ef ) := X ( τ,τ ′ ) ∈P ( K − ) ×P ( K − ):( τ,τ ′ ) is a good pair | Ef τ | / | Ef τ ′ | / . Now we are ready to run the broad-narrow analysis of Bourgain and Guth[BG11] and Guth [Gut16]. We will use geometric lemma 2.4 here and seethe motivation of our definition of broad points.
Lemma 4.4.
Suppose that α ∈ (0 , d +2 satisfies (4.33) K − ǫj ≤ α j ≤ K − dj +1 , K − ǫd +1 ≤ α d +1 ≤ − d for every j = 0 , . . . , d . Then for every x ∈ B k ∩ W Br α Ef ( x ) ≤ X I Br d α Ef I,k, + ( x )+ K Bil( Ef k, − )( x ) + R − k f k . (4.34) The summation P I runs over all possible collections of squares with side-length K − .Proof of Lemma 4.4. Suppose that x is an α -broad point of Ef . We mayassume that(4.35) | Ef ( x ) | ≥ R − k f k . Otherwise, the inequality is trivial. We define the non-significant tangentialpart(4.36) C = (cid:8) τ ∈ P ( K − ) : | Ef τ,k, − ( x ) | < K − | Ef ( x ) | (cid:9) . We consider several cases.
Case 1. C c is empty.In this case, by the triangle inequality, we obtain | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) + X τ ∈P ( K − ) | Ef τ,k, − ( x ) | + R − k f k ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) + K − | Ef ( x ) | + R − k f k . (4.37) By rearranging the terms, we obtain(4.38) | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) . It remains to show that x is a 200 d α -broad point of Ef k, + . By definition,it follows from(4.39)max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) ≤ d α d +1 (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) and max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) + max τ j ∈P ( K − j ) | X τ ∈P ( K − ,τ j ) Ef τ,k, + ( x ) | ≤ d α j (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) (4.40)for every j = 0 , . . . , d + 1. Let us prove (4.39). By (4.38) it suffices to showthat(4.41) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) ≤ d α d +1 | Ef ( x ) | . It follows by the fact that τ ∈ C and x is an α -broad point of Ef ( x ). Thisgives the proof of (4.39). The inequality (4.40) can be proved in exactly thesame way. We leave out the details. Case 2. C c is not empty.We pick τ ∗ ∈ C c and take the dyadic square containing τ ∗ with sidelength K − d +1 . We take a square υ with sidelength 8 K − d +1 so that it contains thedyadic square and the distance between the boundary of the dyadic squareand the boundary of υ becomes greater than 2 K − d +1 . We may assume thatthe square υ is a union of 64 dyadic squares with side length K − d +1 . We havetwo subcases. Case 2.1. S τ ∈C c τ ⊂ υ. Note that all the elements τ ⊂ υ c belong to C . Since x is an α -broadpoint of Ef and x ∈ B k , we obtain | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ c ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ α d +1 | Ef ( x ) | + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ c ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ (cid:0) α d +1 + K − (cid:1) | Ef ( x ) | + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ c ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) + R − k f k . (4.42) ESTRICTION ESTIMATES 29
Recall that α d +1 ≤ − d . This implies(4.43) | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ c ) Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) . It remains to show that x is a 200 d α -broad point of the function P τ ⊂ υ c Ef τ,k, + .By definition and by applying (4.43) and the fact that τ ∈ C , it suffices toshow(4.44) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ( ∩ Mi =1 L i ) ∩ υ c ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d α d +1 | Ef ( x ) | and max τ j ∈P ( K − j ) | Ef τ j ( x ) | + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d α j | Ef ( x ) | , (4.45)for every j = 0 , . . . , d + 1. Note that (4.45) immediately follows from theassumption that x is an α -broad point of Ef ( x ). Let us prove (4.44). Bythe triangle inequality, the left hand side of (4.44) is bounded bymax L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (4.46)Since υ is a union of 64 dyadic squares with side length K − d +1 and x is an α -broad point of Ef , by the triangle inequality, it is further bounded by65 α d +1 | Ef ( x ) | and this gives the proof of (4.44). Case 2.2.
There exists τ ∗∗ ∈ C c such that dist( τ ∗ , τ ∗∗ ) ≥ K − d +1 .If C c contains two squares τ and τ such that the pair ( τ , τ ) is a goodpair, then we obtain(4.47) | Ef ( x ) | ≤ K | Ef τ ,k, − ( x ) | / | Ef τ ,k, − ( x ) | / ≤ K Bil( Ef k, − )( x ) . This gives the desired result. Hence, we may assume that there does notexist two squares τ , τ ∈ C c such that ( τ , τ ) is a good pair.Since x ∈ W and non-singular points are dense on Z ( P ), by the definitionof tangent wave packets, we can find a non-singular point z of Z ( P ) suchthat(4.48) | x − z | ≤ R / δ and Angle( v ( T ) , T z ( Z ( P ))) ≤ R − / δ for every T ∈ T k, − with z ∈ T . Since τ ∗ , τ ∗∗ ∈ C c , they contain points suchthat the angles between the normal vectors of the surface associated with h at the points and T z ( Z ( P )) are smaller than 2 R − / δ . We denote thepoints in τ ∗ and τ ∗∗ by ζ ∗ and ζ ∗∗ , respectively. Write ζ ∗∗ := ( ζ ∗∗ , ζ ∗∗ ) := T ζ ∗ ( ζ ∗∗ ). Here, T ζ ∗ is the map defined in (2.43). Without loss of generality,we may assume that | ζ ∗∗ | ≥ | ζ ∗∗ | . Note that { θ ∈ P ( R − / ) : Angle( T z ( Z ( P )) , v θ ) ≤ R − / δ }⊂ { T − ζ ∗ ( ξ, η ) : | P ζ ∗ ,ζ ∗∗ ( ξ, η ) | < R − / O ( δ ) } (4.49)because (2.47) says that | P ζ ∗ ,ζ ∗∗ ( ξ, η ) | is comparable to the volume of theparallelogram generated by the normal vectors of the surface associated with h at ζ ∗ , ζ ∗∗ , and T − ζ ∗ ( ξ, η ). Here, v θ is the normal vector to the surface S associated with h at the point ( w θ , h ( w θ )) and w θ is the bottom corner of θ . Recall that(4.50) P ζ ∗ ,ζ ∗∗ ( ξ, η ) = ∂ h ζ ∗ ( ξ, η ) ∂ h ζ ∗ ( ζ ∗∗ ) − ∂ h ζ ∗ ( ξ, η ) ∂ h ζ ∗ ( ζ ∗∗ ) . Since ∂ h ζ ∗ ( ξ, η ) = η · (1 + O ( ǫ )) + ∂ h ζ ∗ ( ξ,
0) and ∂ h ζ ∗ ( ξ, η ) = ξ · (1 + O ( ǫ )) + ∂ h ζ ∗ (0 , η ), by (2.45), the set on the second line of (4.49) is con-tained in(4.51) T − ζ ∗ ( N R − / O ( δ ) ( Z ( P ζ ∗ ,ζ ∗∗ ))) . Hence, for every τ ∈ C c with dist( τ ∗ , τ ) ≥ K − d +1 , there exists(4.52) ζ = ζ ( τ ) ∈ τ ∩ T − ζ ∗ ( Z ( P ζ ∗ ,ζ ∗∗ )) . Since T ζ ∗ ( ζ ∗ ) , T ζ ∗ ( ζ ∗∗ ) , T ζ ∗ ( ζ ) ∈ Z ( P ζ ∗ ,ζ ∗∗ ), we obtain(4.53) det ∂ h ζ ∗ ( T ζ ∗ ( ζ ∗ )) ∂ h ζ ∗ ( T ζ ∗ ( ζ ∗ )) − ∂ h ζ ∗ ( T ζ ∗ ( ζ ∗∗ )) ∂ h ζ ∗ ( T ζ ∗ ( ζ ∗∗ )) − ∂ h ζ ∗ ( T ζ ∗ ( ζ )) ∂ h ζ ∗ ( T ζ ∗ ( ζ )) − = 0 . Since ( τ ∗ , τ ) is a bad pair, ( ζ ∗ , ζ ) is a bad pair. By (2.47) and (4.53), weobtain Z ( P ζ ∗ ,ζ ∗∗ ) = Z ( P ζ ∗ ,ζ ) and B ( ζ ∗ , ζ ) = B ( ζ ∗ , ζ ∗∗ ). Since ( ζ ∗ , ζ ) is abad pair, T ζ ∗ ( ζ ) ∈ B ( ζ ∗ , ζ ) = B ( ζ ∗ , ζ ∗∗ ). We apply Lemma 2.4. We considerthe first case in Lemma 2.4: There exist some squares υ ′ j,i ∈ P ( K − j , R )such that { T ζ ∗ ( ζ ) = T ζ ∗ ( ζ ( τ )) : τ ∈ C c , dist( τ ∗ , τ ) ≥ K − d +1 }⊂ B ( ζ ∗ , ζ ∗∗ ) ⊂ (cid:16) d [ i =1 T ζ ∗ ( υ ′ d +1 ,i ) (cid:17) ∪ (cid:16) d − [ j =2 O ( K j +1 ) [ i =1 T ζ ∗ ( υ ′ j,i ) (cid:17) . (4.54)We take T − ζ ∗ on the both sides, enlarge squares υ ′ j,i , include the square υ defined at the discussion at the beginning of Case 2, and we obtain(4.55) [ τ ∈C c τ ⊂ (cid:16) d [ i =1 υ ′ d +1 ,i (cid:17) ∪ (cid:16) d − [ j =2 K j +1 [ i =1 υ ′ j,i (cid:17) =: Badfor some υ ′ j,i ∈ P ( K − j , [ − , ). ESTRICTION ESTIMATES 31
By the triangle inequality,(4.56) | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , Bad) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) . By the definition of an α -broad point, the upper bound of α j , and thetriangle inequality, we obtain (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , Bad) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d X i =1 (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ′ d+1 , i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + d − X j =2 K j +1 X i =1 (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ′ j,i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ (cid:0) d α d +1 + d − X j =2 K j +1 α j (cid:1) | Ef ( x ) | ≤ − | Ef ( x ) | . (4.57)Hence, by combining (4.56) and (4.57), and by x ∈ B k , we obtain(4.58) | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ): τ ∩ Bad c = ∅ Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) . It remains to show that x is a 200 d α -broad point of (cid:12)(cid:12)(cid:12) P τ ∩ Bad c = ∅ Ef τ,k, + ( x ) (cid:12)(cid:12)(cid:12) .By the above inequality (4.58) and by the fact that all the squares τ with τ ∩ Bad c = ∅ belong to C , it suffices to show that(4.59) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d α d +1 | Ef ( x ) | and max τ j ∈P ( K − j ) | Ef τ j ( x ) | + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d α j | Ef ( x ) | (4.60)for every j = 0 , . . . , d + 1.Let us prove (4.59) first. By the triangle inequality, the left hand side of(4.59) is bounded bymax L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ( ∩ Mi =1 L i ) ∩ Bad) Ef τ (cid:12)(cid:12)(cid:12) . (4.61) Since x is an α -broad point of Ef , as in (4.57), it is further bounded by(4.62) (cid:16) d α d +1 + d − X j =2 K j +1 α j (cid:17) | Ef ( x ) | . Since this is bounded by 100 d α d +1 | Ef ( x ) | by the lower bound and upperbound of α j , this gives the proof of (4.59).Let us prove (4.60). The first term on the left hand side of (4.60) can bebounded by α j | Ef ( x ) | by the definition of an α -broad point. By the triangleinequality, the second term is bounded bymax υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i ) ∩ Bad) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (4.63)The first term of (4.63) can be bounded by α j | Ef ( x ) | by the definition ofan α -broad point. For j = 0 , ,
2, Bad does not play a role and the secondterm of (4.63) can be dealt with by the definition of an α -broad point andit is bounded by α j | Ef ( x ) | . For j = 3 , . . . , d , by the triangle inequality, thesecond term of (4.63) is bounded bymax υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + j − X l =2 K l +1 X i =1 max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ,υ l,i ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (4.64)We can apply the definition of an α -broad point and it is bounded by(4.65) (cid:16) α j + j − X l =2 K l +1 α l (cid:17) | Ef ( x ) | . This is futher bounded by 200 d α j | Ef ( x ) | . For j = d + 1, by the triangleinequality, the second term of (4.63) is bounded by(4.66) (cid:16) d α d +1 + d − X l =2 K l +1 α l (cid:17) | Ef ( x ) | . This is futher bounded by 200 d α d +1 | Ef ( x ) | . This completes the proof of(4.60) and finishes the discussion for the first case of Lemma 2.4.Let us consider the second case in Lemma 2.4: There exist bad strips L j such that(4.67) [ τ ∈C c τ ⊂ υ ∪ L ′ ∪ L ′ =: Bad , ESTRICTION ESTIMATES 33 where υ is the square with side length 8 K − d +1 defined at the beginning ofthe discussion of Case 2. By the triangle inequality,(4.68) | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , Bad ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) . By inclusion-exclusion principle, (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , Bad ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , L ′ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , L ′ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , L ′ ∩ L ′ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , L ′ ∩ υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , L ′ ∩ υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , L ′ ∩ L ′ ∩ υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (4.69)By the definition of an α -broad point, it is futher bounded by 300 α d +1 | Ef ( x ) | .Hence, we obtain(4.70) | Ef ( x ) | ≤ (cid:12)(cid:12)(cid:12) X τ ∈P ( K − ): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) . It remains to show that x is a 200 d α -broad point of (cid:12)(cid:12)(cid:12) P τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) .It can be proved as in (4.59) and (4.60). By definition and (4.70), it sufficesto prove(4.71) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d α d +1 | Ef ( x ) | and max τ j ∈P ( K − j ) | Ef τ j ( x ) | + max υ j ∈P ( K − j ) max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ j ∩ ( ∩ Mi =1 L i )): τ ∩ Bad c = ∅ Ef τ ( x ) (cid:12)(cid:12)(cid:12) ≤ d α j | Ef ( x ) | (4.72)for every j = 0 , . . . , d + 1. Let us first prove (4.71). By inclusion-exclusion principle, the left handside of (4.71) is bounded bymax L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + X j =1 max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ( ∩ Mi =1 L i ) ∩ L ′ j ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ( ∩ Mi =1 L i ) ∩ L ′ ∩ L ′ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + X j =1 max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ( ∩ Mi =1 L i ) ∩ L ′ j ∩ υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ( ∩ Mi =1 L i ) ∩ L ′ ∩ L ′ ∩ υ ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (4.73)Since L ′ and L ′ are bad strips and at each point x there are at most M badstrips intersecting x , the above term is further bounded by4 max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , ∩ Mi =1 L i ) Ef τ ( x ) (cid:12)(cid:12)(cid:12) + 4 max L ,...,L M ∈ L (cid:12)(cid:12)(cid:12) X τ ∈P ( K − , υ ∩ ( ∩ Mi =1 L i )) Ef τ ( x ) (cid:12)(cid:12)(cid:12) . (4.74)Since x is an α -broad point of Ef , it is bounded by 300 α d +1 | Ef ( x ) | andfinishes the proof of (4.71). The proof of (4.72) is identical to that for (4.71).We leave out the details. (cid:3) By raising to the p -th power on both sides of Lemma 4.4, integratingover B k ∩ W and summing over all the balls B k , we obtain Z B R ∩ W | Br α Ef ( x ) | p . K X k,I Z B k ∩ W (cid:16) Br d α Ef I,k, + (cid:17) p + K X k Z B k ∩ W Bil( Ef k, − ) p + R − k f k p . (4.75)Hence, by the inequality(4.76) max θ ∈P ( R − / ) k f k L ( θ ) . R / k f k , ESTRICTION ESTIMATES 35 the wall estimate (4.27) follows from the estimate of the transverse wavepackets X k,I Z B k ∩ W (cid:16) Br d α Ef I,k, + (cid:17) p ≤ C ( K ) R − δǫ R δ trans ( P j log ( K ǫj α j )) p R ǫp k f k ( + ǫ ) p L max θ ∈P ( R − / ) k f k ( − ǫ ) p L ( θ ) , (4.77)and the estimate of the bilinear operator(4.78) X k Z B k ∩ W Bil( Ef k, − ) p ≤ C ( K ) R Cδ k f k ( ) p L max θ ∈P ( R − / ) k f k ( ) p L ( θ ) . The estimates for the transverse wave packets.
Let us prove (4.77).Recall that B k is a ball of radius R − δ . By the induction on the radius R and Lemma 3.4, we obtain Z B k ∩ W | Br d α Ef I,k, + ( x ) | p ≤ CC p ǫ R Cδ trans R δ trans ( P j log ( K ǫj α j )) p R (1 − δ ) ǫp × k f I,k, + k ǫp L max θ ∈P ( R − / ) k f k . − ǫp L ( θ ) . (4.79)By summing over B k ⊂ B R and by embedding l ǫp ֒ → l , we obtain X k Z B k ∩ W | Br d α Ef I,k, + ( x ) | p ≤ CC p ǫ R Cδ trans R p δ trans ( P j log ( K ǫj α j )) R (1 − δ ) ǫp (cid:16) X k,τ k f τ,k, + k L (cid:17) ǫp max θ ∈P ( R − / ) k f k − ǫp L ( θ ) . (4.80)We apply Lemma 4.3 with Lemma 3.3 and obtain(4.81) X k Z | f τ,k, + | . R Cδ deg Z | f τ | + R − k f τ k . By combining (4.80) and (4.81), we obtain X k Z B k ∩ W | Br d α Ef I,k, + ( x ) | p . C p ǫ R Cδ deg R Cδ trans R p δ trans ( P j log ( K ǫj α j )) R (1 − δ ) ǫp k f k ǫp L max θ ∈P ( R − / ) k f k − ǫp L ( θ ) . (4.82)Note that(4.83) R Cδ deg R Cδ trans R − δǫp ≪ R − δǫ . This completes the proof of (4.77).
The estimates for the bilinear operator.
Let us prove (4.78). By re-placing the summation over k by the supremum, it suffices to show that(4.84) Z B k ∩ W Bil( Ef k, − ) p ≤ C ( K ) R Cδ k f k ( ) p L max θ ∈P ( R − / ) k f k ( ) p L ( θ ) for every k . We cover B k ∩ W with balls Q of radius R / . For each Q , let(4.85) T k, − ,Q := { T ∈ T k, − : T ∩ Q = ∅} . On a ball Q , we obtain(4.86) Ef τ,k, − ( x ) = X T ∈ T k, − ,Q Ef τ,T ( x ) + O ( R − k f k ) . Since a wall could be curved, the tangent wave packets are not necessarilycoplanar. However, the tangent wave packets intersecting a ball Q are alwayscoplanar in the sense that they are contained in some R / δ -neighborhoodof a hyperplane by the following reason. Since Q ∩ W is non-empty, wecan choose a point z ∈ Z ( P ) in the R / δ -neighborhood of Q . Since thenon-singular points are dense in Z ( P ) as P is a product of non-singularpolynomials, we may assume that z is a non-singular point. By definition,for every T ∈ T k, − ,Q and z ∈ T ∩ B k ∩ Z ( P ), the angle between v ( T ) and T z ( Z ( P )) is O ( R − / O ( δ ) ). Hence, all the elements of T k, − ,Q are containedin the R / O ( δ ) -neighborhood of the hyperplane T z ( Z ( P )).Since the wave packets T ∈ T k, − ,Q are coplanar, the support of f τ,T and f τ ′ ,T must be clustered near some variety. We have defined a good pair ( τ, τ ′ )so that it enjoys the following property: Either the hypersurface along thevariety or the variety itself is curved in such a way that we can perform L -argument either way. This idea will be clear in the proof of Proposition4.5.Let ψ Q be a smooth function such that | ψ Q ( x ) | ≃ x ∈ Q and F ( ψ Q )is supported on B cR − / (0) for some small number c >
0. Here, F is theFourier transform. Proposition 4.5.
Suppose that ( τ, τ ′ ) is a good pair. For any ball Q ofradius R / intersecting B k ∩ W , Z Q (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q Ef τ,T (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q Ef τ ′ ,T (cid:12)(cid:12)(cid:12) . C ( K ) R Cδ X T ∈ T k, − ,Q X T ∈ T k, − ,Q Z R | ψ Q | | Ef τ,T | | Ef τ ′ ,T | . (4.87) Proof of Proposition 4.5.
To apply the Fourier transform, we first replacethe strict cut-off by a smooth cut-off and bound the left hand side of (4.87)
ESTRICTION ESTIMATES 37 by Z R | ψ Q | (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q Ef τ,T (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q Ef τ ′ ,T (cid:12)(cid:12)(cid:12) . (4.88)By the discussion above Proposition 4.5 and the discussion at the beginningof Case 2.2 of the proof of Lemma 4.4, there exist points ζ ∈ τ and ζ ′ ∈ τ ′ such that supports of f τ,T and f τ ′ ,T are contained in the set(4.89) T − ζ ( N R − / O ( δ ) ( Z ( P ζ,ζ ′ ))) . Let us put ζ ′ := ( ζ ′ , ζ ′ ) := T ζ ( ζ ′ ). Without loss of generality, we mayassume that | ζ ′ | ≥ | ζ ′ | . By a change of variables we obtain | Ef τ,T ( x ) | = (cid:12)(cid:12)(cid:12) Z [ − CK − ,CK − ] f τ,T ( T − ζ ( ξ, η )) e (cid:0) ( Ax ) · ( ξ, η, h ζ ( ξ, η ) (cid:1) dξdη (cid:12)(cid:12)(cid:12) = | E S g T ζ ( τ ) ,T ( Ax ) | (4.90)for some linear transformation A with determinant ≃
1. Here S is thesurface corresponding to the polynomial h ζ ( ξ, η ) and(4.91) g T ζ ( τ ) ,T := f τ,T ◦ T − ζ . We apply the same change of variables given by the matrix T − ζ to Ef τ ′ ,T and we define(4.92) g T ζ ( τ ′ ) ,T := f τ ′ ,T ◦ T − ζ . Put φ Q ( x ) := ψ Q ( A − x ). After this change of variables, we obtain(4.88) . Z R | φ Q | (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q E S g T ζ ( τ ) ,T (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q E S g T ζ ( τ ′ ) ,T (cid:12)(cid:12)(cid:12) . (4.93)We expand the L -norm and obtain X T ,T ′ ∈ T k, − ,Q X T ,T ′ ∈ T k, − ,Q Z R (cid:0) ( φ Q E S g T ζ ( τ ) ,T )( φ Q E S g T ζ ( τ ′ ) ,T ) (cid:1) × (cid:0) ( φ Q E S g T ζ ( τ ) ,T ′ )( φ Q E S g T ζ ( τ ′ ) ,T ′ ) (cid:1) . (4.94)We apply Plancherel’s theorem and the integration becomes Z R (cid:0) F ( φ Q E S g T ζ ( τ ) ,T ) ∗ F ( φ Q E S g T ζ ( τ ′ ) ,T ) (cid:1) × (cid:0) F ( φ Q E S g T ζ ( τ ) ,T ′ ) ∗ F ( φ Q E S g T ζ ( τ ′ ) ,T ′ ) (cid:1) . (4.95)We will perform an L argument. Denote by w ( T i ) and w ( T ′ i ) the projec-tions to the ξ -axis of the centers of the supports of g T ζ ( τ ) ,T i and g T ζ ( τ ′ ) ,T ′ i . The integration vanishes unless w ( T ) + w ( T ) = w ( T ′ ) + w ( T ′ ) + O ( R − / O ( δ ) ) , (4.96) η ζ,ζ ′ ( w ( T )) + η ζ,ζ ′ ( w ( T )) = η ζ,ζ ′ ( w ( T ′ )) + η ζ,ζ ′ ( w ( T ′ )) + O ( R − / O ( δ ) ) , (4.97) H ζ,ζ ′ ( w ( T )) + H ζ,ζ ′ ( w ( T )) = H ζ,ζ ′ ( w ( T ′ )) + H ζ,ζ ′ ( w ( T ′ )) + O ( R − / O ( δ ) ) . (4.98)We claim that either(4.99) | η ′ ζ,ζ ′ ( ζ ′ ) − η ′ ζ,ζ ′ (0) | > (2 K ) − or(4.100) | H ′ ζ,ζ ′ ( ζ ′ ) | > (2 K ) − . Recall that ( τ, τ ′ ) is a good pair. Thus, there exist µ ∈ τ and µ ′ ∈ τ ′ such that ( µ, µ ′ ) is a good pair. Let us take µ ′ := ( µ ′ , µ ′ ) := T µ ( µ ′ ). Bythe implicit function theorem, the set Z ( P µ,µ ′ ) can be written as a graph ofsome function η µ,µ ′ . Since ( µ, µ ′ ) is a good pair, the claim follows from | η ′ ζ,ζ ′ ( ζ ′ ) − η ′ µ,µ ′ ( µ ′ ) | . K d +1 K − , (4.101) | η ′ ζ,ζ ′ (0) − η ′ µ,µ ′ (0) | . K d +1 K − , (4.102) | H ′ ζ,ζ ′ ( ζ ′ ) − H ′ µ,µ ′ ( µ ′ ) | . K d +1 K − , (4.103)provided that K is sufficiently large, compared to K . Before we enter theproof, let us mention that by continuity and by Taylor’s expansion, we have(4.104) k T ζ − T µ k . K − and k h ζ − h µ k . K − . Let us first prove (4.101). By (2.67), | η ′ ζ,ζ ′ ( ζ ′ ) − η ′ µ,µ ′ ( µ ′ ) | is equal to(4.105) (cid:12)(cid:12)(cid:12) ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) − ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) − ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) − ∂ h µ ( µ ′ ) ∂ h µ ( µ ′ ) − ∂ h µ ( p ′ ) ∂ h µ ( µ ′ ) ∂ h µ ( µ ′ ) ∂ h µ ( µ ′ ) − ∂ h µ ( µ ′ ) ∂ h µ ( µ ′ ) (cid:12)(cid:12)(cid:12) . Let us write the above term as AB − CD = A ( D − B )+( A − C ) BBD . Since | ∂ h ζ ( ζ ′ ) | ≃| ∂ h µ ( µ ′ ) | ≃
1, by (2.45), we obtain | BD | − . K d +1 . It suffices to provethat | B − D | . K − and | A − C | . K − . This easily follows by the in-equalities k h ζ − h µ k . K − and | ζ ′ − µ ′ | ≤ K − . This finishes the proof of(4.101).Let us next prove (4.102). By the definition of η ζ,ζ ′ and η µ,µ ′ , | η ′ ζ,ζ ′ (0) − η ′ µ,µ ′ (0) | = (cid:12)(cid:12)(cid:12) ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) − ∂ h µ ( µ ′ ) ∂ h µ ( µ ′ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂ h ζ ( ζ ′ ) ∂ h µ ( µ ′ ) − ∂ h µ ( µ ′ ) ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) ∂ h µ ( µ ′ ) (cid:12)(cid:12)(cid:12) . (4.106) ESTRICTION ESTIMATES 39
This is equal to(4.107) (cid:12)(cid:12)(cid:12) ∂ h ζ ( ζ ′ ) (cid:0) ∂ h µ ( µ ′ ) − ∂ h ζ ( ζ ′ ) (cid:1) + (cid:0) ∂ h ζ ( ζ ′ ) − ∂ h µ ( µ ′ ) (cid:1) ∂ h ζ ( ζ ′ ) ∂ h ζ ( ζ ′ ) ∂ h µ ( µ ′ ) (cid:12)(cid:12)(cid:12) . By (2.45), it is bounded by(4.108) . K d +1 (cid:16) | ∂ h µ ( µ ′ ) − ∂ h ζ ( ζ ′ ) | + | ∂ h ζ ( ζ ′ ) − ∂ h µ ( µ ′ ) | (cid:17) . Since k h ζ − h µ k . K − and | ζ ′ − µ ′ | ≤ K − , it is further bounded by K d +1 K − and finishes the proof of (4.102).Lastly, let us prove (4.103). By the chain rule, | H ′ ζ,ζ ′ ( ζ ′ ) − H ′ µ,µ ′ ( µ ′ ) | = | ∂ h ζ ( ζ ′ ) + ∂ h ζ ( ζ ′ ) η ′ ζ,ζ ′ ( ζ ′ ) − ∂ h µ ( µ ′ ) − ∂ h µ ( µ ′ ) η ′ µ,µ ′ ( µ ′ ) |≤ | ∂ h ζ ( ζ ′ ) − ∂ h µ ( µ ′ ) | + | ∂ h ζ ( ζ ′ ) η ′ ζ,ζ ′ ( ζ ′ ) − ∂ h µ ( µ ′ ) η ′ µ,µ ′ ( µ ′ ) | . (4.109)It is further bounded by K d +1 K − by the inequalities k h ζ − h µ k . K − ,and | ζ ′ − µ ′ | ≤ K − , and (4.101). This finishes the proof of the claim.By the claim, either (4.99) or (4.100) holds true. Since the side length of τ and τ ′ is K − , we obtain that η ′ ζ,ζ ′ ( ξ ) = η ′ ζ,ζ ′ (0) + O ( K − ) for every ( ξ, η ζ,ζ ′ ( ξ )) ∈ T ζ ( τ ) , (4.110) η ′ ζ,ζ ′ ( ξ ′ ) = η ′ ζ,ζ ′ ( ζ ′ ) + O ( K − ) for every ( ξ ′ , η ζ,ζ ′ ( ξ ′ )) ∈ T ζ ( τ ′ ) . (4.111)Hence, we obtain either(4.112) | η ′ ζ,ζ ′ ( w ( T )) − η ′ ζ,ζ ′ ( w ( T )) | > K for every T i ∈ T k, − ,Q , or(4.113) | H ′ ζ,ζ ′ ( w ( T )) − H ′ ζ,ζ ′ ( w ( T )) | > K for every T i ∈ T k, − ,Q . We consider only the case that (4.112) holds true. The other case (4.113)can be dealt with in exactly the same way. By the relation (4.96), we mayassume that w ( T ) < w ( T ′ ) < w ( T ′ ) < w ( T ). We take two intervals I = [ w ( T ) , w ( T ′ )] and I ′ = [ w ( T ′ ) , w ( T )]. Note that | I | = | I ′ | + O ( R − / O ( δ ) ). By (4.112), we obtain(4.114) | I | + | I ′ | . C ( K ) (cid:12)(cid:12)(cid:12) Z I η ′ ζ,ζ ′ − Z I ′ η ′ ζ,ζ ′ (cid:12)(cid:12)(cid:12) + R − / O ( δ ) , and by the fundamental theorem of calculus, it is further bounded by C ( K ) (cid:12)(cid:12)(cid:12) η ζ,ζ ′ ( w ( T ′ )) − η ζ,ζ ′ ( w ( T )) − η ζ,ζ ′ ( w ( T ′ )) + η ζ,ζ ′ ( w ( T )) (cid:12)(cid:12)(cid:12) + R − / O ( δ ) . (4.115) By the relation (4.97), this is bounded by C ( K ) R − / O ( δ ) and we obtain | w ( T i ) − w ( T ′ i ) | . R − / O ( δ ) . Therefore,(4.88) . R Cδ X T ∈ T k, − ,Q X T ∈ T k, − ,Q Z R | φ Q | (cid:12)(cid:12) E S g T ζ ( τ ) ,T (cid:12)(cid:12) (cid:12)(cid:12) E S g T ζ ( τ ′ ) ,T (cid:12)(cid:12) . (4.116)We change back to the original variables and it gives the desired result. (cid:3) Lemma 4.6.
Suppose that ( τ, τ ′ ) is a good pair. For any ball Q of radius R / intersecting B k ∩ W , Z Q | Ef τ,k, − | | Ef τ ′ ,k, − | . C ( K ) R O ( δ ) R − / (cid:16) X T ∈ T k, − ,Q k f τ,T k (cid:17)(cid:16) X T ∈ T k, − ,Q k f τ ′ ,T k (cid:17) . (4.117) Proof.
By Lemma 4.5, we obtain Z Q (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q Ef τ,T (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) X T ∈ T k, − ,Q Ef τ ′ ,T (cid:12)(cid:12)(cid:12) . R Cδ X T ∈ T k, − ,Q X T ∈ T k, − ,Q Z R | ψ Q | | Ef τ,T | | Ef τ ′ ,T | . (4.118)We take a smooth function ψ B R such that | ψ B R ( x ) | ≃ x ∈ B R and F ( ψ B R ) is supported on B cR − (0) for some small number c >
0. It sufficesto show that(4.119) Z R | ψ B R Ef τ,T | | ψ B R Ef τ ′ ,T | . C ( K ) R − / k f τ,T k k f τ ′ ,T k . By Plancherel’s theorem, it amounts to showing that(4.120) Z R |F ( ψ B R Ef τ,T ) ∗ F ( ψ B R Ef τ ′ ,T ) | . C ( K ) R − / k f τ,T k k f τ ′ ,T k . Since the normal vectors of T and T are separated by & /K L by theseparation condition on τ and τ , the measure of the intersection of trans-lations of the support of F ( ψ B R Ef τ,T ) and F ( ψ B R Ef τ ′ ,T ) is . K O (1) L R − / .Hence, we obtain two estimates kF ( ψ B R Ef τ,T ) ∗ F ( ψ B R Ef τ ′ ,T ) k ∞ . K O (1) L R − / k f τ,T k ∞ k f τ ′ ,T k ∞ , kF ( ψ B R Ef τ,T ) ∗ F ( ψ B R Ef τ ′ ,T ) k . k f τ,T k k f τ ′ ,T k , (4.121)and interpolating these two inequalities gives the desired inequality. (cid:3) ESTRICTION ESTIMATES 41
We are ready to prove (4.84). Let ( τ , τ ) be a good pair. We decompose B k ∩ W into smaller balls Q of raidus R / and obtain(4.122) Z B k ∩ W | Ef τ ,k, − | | Ef τ ,k, − | = X Q ⊂ B k ∩ W Z Q | Ef τ ,k, − | | Ef τ ,k, − | . By applying Lemma 4.6 to each ball Q , we obtain Z B k ∩ W | Ef τ ,k, − | | Ef τ ,k, − | . C ( K ) R Cδ R − / X Q ⊂ B k ∩ W Y l =1 (cid:16) X T l ∈ T k, − ,Q χ T l ( Q ) k f τ l ,T l k (cid:17)! . (4.123)Here, χ T l ( Q ) = 1 if T l intersects Q and χ T l ( Q ) = 0 otherwise. By inter-changing the sums, it is further bounded by C ( K ) R Cδ R − / X T ∈ T k, − X T ∈ T k, − k f τ ,T k k f τ ,T k X Q ⊂ B k ∩ W χ T ( Q ) χ T ( Q ) . C ( K ) R Cδ R − / X T ∈ T k, − X T ∈ T k, − k f τ ,T k k f τ ,T k . (4.124)By summing over all the good pairs, we obtain(4.125) k Bil( Ef k, − ) k L ( B k ∩ W ) . C ( K ) R Cδ R − / (cid:0) X τ ∈P ( K − ) k f τ,k, − k (cid:1) / . By a trivial estimate, we obtain k Bil( Ef k, − ) k L ( B k ∩ W ) . C ( K ) (cid:16) X τ ∈P ( K − ) k Ef τ,k, − k L ( B k ) (cid:17) / . C ( K ) R / (cid:0) X τ ∈P ( K − ) k f τ,k, − k (cid:1) / . (4.126)By interpolating these inequalities via H¨older’s inequality, we obtain k Bil( Ef k, − ) k L p ( B k ∩ W ) . C ( K ) k Bil( Ef k, − ) k L ( B k ∩ W ) k Bil( Ef k, − ) k L ( B k ∩ W ) . R Cδ R (cid:0) X τ ∈P ( K − ) k f τ,k, − k (cid:1) / . (4.127)By Lemma 4.3, we obtain(4.128) k f τ,k, − k . R − / O ( δ ) max θ ∈P ( R − / ) k f τ k L avg ( θ ) . Hence, by combining this with Lemma 3.4, we obtain k Bil( Ef k, − ) k L p ( B k ∩ W ) . C ( K ) R Cδ (cid:0) X τ ∈P ( K − ) k f τ,k, − k (cid:1) max θ ∈P ( R − / ) k f τ k L ( θ ) . C ( K ) R Cδ k f k max θ ∈P ( R − / ) k f τ k L ( θ ) . (4.129)This completes the proof of (4.84). References [BCT06] Jonathan Bennett, Anthony Carbery, and Terence Tao. On the multilinearrestriction and kakeya conjectures.
Acta Math. , 196(2):261–302, 2006.[BG11] Jean Bourgain and Larry Guth. Bounds on oscillatory integral operators basedon multilinear estimates.
Geom. Funct. Anal. , 21(6):1239–1295, 2011.[BMV19] Stefan Buschenhenke, Detlef M¨uller, and Ana Vargas. On fourier restrictionfor finite-type perturbations of the hyperboloid. arXiv:1902.05442 , 2019.[BMV20a] Stefan Buschenhenke, Detlef M¨uller, and Ana Vargas. A Fourier restrictiontheorem for a perturbed hyperbolic paraboloid.
Proc. Lond. Math. Soc. (3) ,120(1):124–154, 2020.[BMV20b] Stefan Buschenhenke, Detlef M¨uller, and Ana Vargas. Partitions of flat one-variate functions and a fourier restriction theorem for related perturbations ofthe hyperbolic paraboloid.
To appear in J. Geom. Anal.; arXiv:2002.08726 ,2020.[BMV20c] Stefan Buschenhenke, Detlef Mller, and Ana Vargas. A fourier restrictiontheorem for a perturbed hyperbolic paraboloid: polynomial partitioning. arXiv:2003.01619 , 2020.[CL17] Chu-Hee Cho and Jungjin Lee. Improved restriction estimate for hyperbolicsurfaces in R . J. Funct. Anal. , 273(3):917–945, 2017.[Gut16] Larry Guth. A restriction estimate using polynomial partitioning.
J. Amer.Math. Soc. , 29(2):371–413, 2016.[GZ19] Shaoming Guo and Pavel Zorin-Kranich. Decoupling for certain qua-dratic surfaces of low co-dimensions.
To appear in J. Lond. Math. Soc.(2).;arXiv:1902.03450 , 2019.[Kim17] Jongchon Kim. Some remarks on fourier restriction estimates. arXiv:1702.01231 , 2017.[Lee06] Sanghyuk Lee. Bilinear restriction estimates for surfaces with curvatures ofdifferent signs.
Trans. Amer. Math. Soc. , 358(8):3511–3533, 2006.[Sha17] Bassam Shayya. Weighted restriction estimates using polynomial partitioning.
Proc. Lond. Math. Soc. (3) , 115(3):545–598, 2017.[SS11] Elias M. Stein and Rami Shakarchi.
Functional analysis , volume 4 of
PrincetonLectures in Analysis . Princeton University Press, Princeton, NJ, 2011. Intro-duction to further topics in analysis.[Ste86] E. M. Stein. Oscillatory integrals in Fourier analysis. In
Beijing lectures inharmonic analysis (Beijing, 1984) , volume 112 of
Ann. of Math. Stud. , pages307–355. Princeton Univ. Press, Princeton, NJ, 1986.[Sto19a] Betsy Stovall. Scale-invariant Fourier restriction to a hyperbolic surface.
Anal.PDE , 12(5):1215–1224, 2019.[Sto19b] Betsy Stovall. Waves, spheres, and tubes: a selection of Fourier restrictionproblems, methods, and applications.
Notices Amer. Math. Soc. , 66(7):1013–1022, 2019.
ESTRICTION ESTIMATES 43 [Tao99] Terence Tao. The Bochner-Riesz conjecture implies the restriction conjecture.
Duke Math. J. , 96(2):363–375, 1999.[Tao03] Terence Tao. A sharp bilinear restrictions estimate for paraboloids.
Geom.Funct. Anal. , 13(6):1359–1384, 2003.[Tao04] Terence Tao. Some recent progress on the restriction conjecture. In
Fourieranalysis and convexity , Appl. Numer. Harmon. Anal., pages 217–243.Birkh¨auser Boston, Boston, MA, 2004.[Tom75] Peter A. Tomas. A restriction theorem for the Fourier transform.
Bull. Amer.Math. Soc. , 81:477–478, 1975.[TVV98] Terence Tao, Ana Vargas, and Luis Vega. A bilinear approach to the restrictionand Kakeya conjectures.
J. Amer. Math. Soc. , 11(4):967–1000, 1998.[Var05] Ana Vargas. Restriction theorems for a surface with negative curvature.
Math.Z. , 249(1):97–111, 2005.[Wan18] Hong Wang. A restriction estimate in R using brooms. arXiv:1802.04312 ,2018.[Wol01] Thomas Wolff. A sharp bilinear cone restriction estimate. Ann. of Math. (2) ,153(3):661–698, 2001.
Department of Mathematics, University of Wisconsin-Madison and the In-stitute for Advanced Study
Email address : [email protected] of Mathematics, University of Wisconsin-Madison