Small cap decoupling inequalities: Bilinear methods

We obtain sharp small cap decoupling inequalities associated to the moment curve for certain range of exponents $p$. Our method is based on the bilinearization argument due to Bourgain and Bourgain-Demeter. Our result generalizes theirs to all higher dimensions.

Here x = (x 1 , . . . , x n ) ∈ R n , and e(t) := e it for a real number t. Let r ≥ 1 and δ ∈ (0, 1]. We use B r to denote a ball in R n of radius δ −r . Moreover, for a ball B ⊂ R n of radius r B and center c B , we use w B to stand for a suitable weight essentially support on B: Let D p (n, r, δ) be the smallest real number such that the decoupling inequality Here the sum on the right hand side is over all dyadic intervals ∆ of the form [a, a + δ] with a ∈ δZ. We define the number p n by p n := 2k(k + 1) with k = n 2 when n is even 2(k + 1) 2 with k = n−1 2 when n is odd. Let [x] be the greatest integer less than or equal to x. The main theorem is the following.
Theorem 1.1. Let n ≥ 3. For every 2 ≤ p ≤ p n and ǫ > 0, there exists some positive number C n,p,ǫ < ∞ such that

Bilinearization
In this section, we will first reduce the linear decoupling inequality (1.1) to the bilinear decoupling inequality (2.1) by combining the broad-narrow analysis of Bourgain and Guth [BG11] and the linear decoupling inequalities of Bourgain, Demeter and Guth [BDG16] for larger balls. The argument of Bourgain and Guth [BG11] is carried out via an inductive argument on the radii of balls. However, in our case, a ball shrinks relatively "fast" as we iterate because we start with a smaller ball B r (instead of B n as in [BDG16]). Thus, the inductive argument does not work as efficiently as it does in [BD15] or [BDG16]. Instead of relying only on induction, what we will do is, after applying a "smaller" number of steps of certain inductive hypothesis, to use the decoupling for the moment curve in [BDG16] (see (2.2) below) to decompose the frequency into the desired scale.
Next, we will prove the bilinear decoupling inequality (2.1). Instead of working with a bilinear extension operator for the moment curve, we will apply a change of variables (see (2.7)) and transfer it to a linear extension operator for the two-dimensional manifold (2.8); a very similar argument already appeared in [BD16]. In the end, we will prove the desired decoupling inequality (see (2.10)) for the two dimensional manifold (2.8) in the remaining sections.
To run the method of Bourgain and Guth [BG11], we need to introduce the notion of transversality. Let K ≥ 1 be a large number. Let J 1 , J 2 be dyadic intervals with side length K −1 . These two intervals are called K −1 -transverse if the distance between them is greater than or equal to K −1 . (2.1) The following theorem states the decoupling inequality for the moment curve by Bourgain and Demeter [BD15] and Bourgain, Demeter and Guth [BDG16]. Theorem 1.1 will be deduced by combining Proposition 2.1 and Theorem 2.2.
Theorem 2.2 ( [BD15,BDG16]). Let n ≥ 2. For every 2 ≤ p ≤ n(n + 1) and ǫ > 0, and every integrable function f : [0, 1] → C, we have Proof of Theorem 1.1 assuming Proposition 2.1. As n is always fixed, here and below we will always abbreviate E n,I to E I . We use the broad-narrow analysis from [BG11]. For each x ∈ R n , we consider the collection of significant intervals, defined by By considering two possible cases |C(x)| ≥ 3 and |C(x)| ≤ 2, we obtain the following pointwise estimate: We raise both sides of the last display to the p-th power, replace the max on the right hand side by an l p -norm, integrate over B r , and obtain Next, we apply Proposition 2.1 to the last term and obtain (2.3) In (2.3), the last term is already of the desired form, the form of the right hand side of (1.1). We bound the first term on the right hand side of (2.3) using an iteration argument: We rescale the interval R to the whole interval [0, 1] and apply (2.3) again. To be more precise, let M be the constant such that K −M = δ r/n , and we will prove that for every integer m with 1 ≤ m ≤ M . Here, C n,p,ǫ is the constant in (2.2). Note that if the interval is smaller than δ r/n then by the uncertainty principle the last component of the curve {(t, t 2 , . . . , t n ) : t ∈ [0, 1]} does not play a role on the ball B r . Hence, we cannot apply an induction hypothesis anymore. Thus, we stop iterating if the side length of an interval R reaches δ r/n . We already proved (2.4) when m = 1. Suppose that (2.4) holds true for some m = m 0 < M . We will show that (2.4) holds true for m = m 0 + 1. By the induction hypothesis, we obtain (2.5) We fix an interval R with side length K −m 0 . For the sake of simplicity, we assume that R = [0, K −m 0 ]. We take γ such that δ γ = K −m 0 . By applying a change of variables, we obtain .
By applying the change of variables we obtain .
Here B is a rectangle box of dimension δ −r+γ × · · · × δ −r+nγ . Now we split the rectangular box B into balls B ′ of radius δ −r+nγ , and apply (2.3) to each B ′ . Afterwards, we raise everything to the p-th power, sum over B ′ ⊂ B and take the p-th root. In the end, the first term on the right hand side in (2.5) is bounded by Here f (t) := f (δ γ t). We change all variables back and obtain To see how to further process the second term, we take ∆ = [0, δ 1− nγ r +γ ] as an example. The general case can be handled similarly after making an affine change of variables. In this case, we apply the decoupling inequality in Theorem 2.2 for the moment curve (t, t 2 , . . . , t r ). This can be done by viewing x r+1 , . . . , x n in the phase function tx 1 + · · · + t n x n of the extension operator E ∆ as dummy variables. As a consequence, we obtain Note that for every r ≥ n/2, we have Thus, we obtain By the induction argument, this completes the proof of (2.4).
Recall that K −M = δ r/n . By (2.4) with m = M , we obtain We apply Plancherel's theorem and a trivial bound at L ∞ to control the first term on the right hand side. In the end, we will take K to be large enough and obtain It suffices to note that p for every r ≥ n/2 and p ≥ 2. This finishes the proof of Theorem 1.1 assuming Proposition 2.1.
We prove Proposition 2.1 in the next step. In previous decoupling papers [BD15,BDG16], a large separation of intervals (the transversality constant K −1 ) is essential because of the use of multilinear Kakeya inequalities. However, in this paper, we do not (directly) use any multilinear Kakeya inequality. In fact, we will see that there is certain significant advantage if the separation of intervals is small (see the statement of Proposition 2.3).
This phenomenon is particular to the approach we are using: We will apply a change of variables (see (2.7)) to transfer the problem of bilinear decoupling for the moment curve to the problem of linear decoupling for a two dimensional manifold (given by (2.8)). This change of variables is non-linear. As a consequence, it is hard to find an explicit expression of the manifold, not to mention to prove certain sharp decoupling inequalities. However, we will see that the smaller the transversality constant is, the more the induced manifold will behave like a monimial manifold. Moreover, a sharp decoupling inequality for such a moment manifold has already been established in [GZ18].
The following proposition states a bilinear decoupling inequality with a smaller transversality constant, compared with the one in Proposition 2.1.
Proposition 2.1 follows from Proposition 2.3 via a simple scaling argument. We leave out the details.
It remains to prove Proposition 2.3. Given two intervals J 1 , J 2 ⊂ [0, δ ǫ ] that are δ ǫ K −1transverse, we follow the idea of convolving two measures that are supported on J 1 and J 2 separately, and consider the support of the output measure and the set Under the assumption that J 1 and J 2 are δ ǫ K −1 -transverse, it is not difficult to see that the Jacobian matrix ∂(u,v) ∂(t,s) is non-singular on J 1 × J 2 . This allows us to write t and s as functions of u and v. Furthermore, we can write (2.6) as Given an integer n ′ ≥ 3, smooth functions P 3 , . . . , P n ′ , and a surface we define the associated extension operator for x ′ ∈ R n ′ and a set ⊂ R 2 . Proposition 2.3 follows from the decoupling for the twodimensional surface M given by (2.8).
Proposition 2.4. Let n ≥ 3 and ǫ > 0. Let K ≥ 1 be sufficiently large. Let 0 < δ < K −1 . Let J 1 , J 2 ⊂ [0, δ ǫ ] be δ ǫ K −1 -transverse. For every 2 ≤ p ≤ p n /2 and every integrable function Here, the sum runs over squares It is not difficult to find a collection of 10 4 square grids {G i } 1≤i≤10 4 satisfying the followings: (1) Each square in each grid G i has a dyadic side length 16δ, Also, a simple computation shows that there exists a small positive constant c K independent of the choice of ∆ 1 , ∆ 2 and the parameter δ such that for some a, b and Here, B (X, Y ), r denotes the ball of radius r centered at the point (X, Y ). We denote by Q ∆ 1 ,∆ 2 the square from some grid G i that contains L(∆ 1 , ∆ 2 ). By the property (2.11), for each square from a grid G i , the number of sets of the form We use the change of variables: u = t+s and v = t 2 +s 2 . Let

Proposition 2.3 follows from
(2.12) By using the grids constructed at the beginning of the proof, we obtain Therefore, . (2.13) We apply Proposition 2.4 and bound (2.13) by By the property that |{(∆ 1 , ∆ 2 ) : Q ∆ 1 ,∆ 2 = }| = O(δ −ǫ ) for each ∈ G i , the last expression can be further bounded by , we obtain By applying the Cauchy-Schwarz inequality twice, the right hand side can be bounded by Therefore, we obtain the inequality (2.12) and this completes the proof of Proposition 2.3 assuming Proposition 2.4.
For the rest of the paper, we give a proof of Proposition 2.4.

Some linear algebra
In this section, we will make some preparation for the proof of Proposition 2.4. To be more precise, we will show that, after certain affine transformations, the manifold M (defined in (2.8)) is very close to some moment manifold (see Theorem 3.1 and (4.7)).
For each (a, b) ∈ [0, 1] 2 , we define the manifold M (a,b) to be Here, the polynomials p i are defined in (2.9). Next we define a relation between two manifolds. For i = 1, 2, let M i be a manifold given by where P j,i is a real-valued function for each i and j. We say that M 1 ∼ = M M 2 if there exist an invertible linear transformation M : R n → R n and some vector b ∈ R n such that (u, v, P 3,2 (u, v), . . . , P n,2 (u, v)) ⊺ = M u, v, P 3,1 (u, v), . . . , P n,1 (u, v) Here, the superscript ⊺ refers to a transpose.
The main result in this section is the following. Recall that the functions u, v are defined in (2.7).
Theorem 3.1. Let n ≥ 3, K ≥ 100 and 0 ≤ ζ ≤ 1. Suppose that (a, b) = (u(α, β), v(α, β)) for some 0 ≤ α, β ≤ ζ with |α − β| ≥ ζK −1 . Then there exists an invertible linear transformation M such that M (a,b) ∼ = M (u, v, q 3 (u, v), . . . , q n (u, v)) , where (3.1) To obtain good approximation formula for p k (u + a, v + b), a natural idea is to apply Taylor's expansion. Taking partial derivatives of p k in terms of u and v can get very complicated. We will instead compute partial derivatives of p k in terms of t and s, and then apply formulas for derivatives of implicit functions. In this approach, the following lemma will be particularly useful.
Here, every * denotes a number that we will not keep track of. The matrix B α,β has the following form Here, A i×i is an i × i matrix and 0 i×j is an i × j matrix whose components are all zero. Thus, to prove Lemma 3.2, it suffices to show that det (A i×i ) = 0 for all i.
We define the polynomials r j (t, s) := (t + 2αs) i−1−j (t + 2βs) j for j = 0, . . . , i − 1. Then the matrix (A i×i ) ⊺ can be expressed as Without loss of generality, we may assume that β = 0. We apply a change of variables: By direct computations, we obtain We compute the derivatives of r j and obtain Thus, we obtain det (A i×i ) = (2β − 2α) , and this is non-vanishing whenever α = β. This completes the proof of the lemma.
Let us continue with the proof of Theorem 3.1. We first consider the case ζ = 0. Note that in this case (a, b) = (0, 0), and what we need to prove becomes Here, for k ≥ 1, e 2k+1 ,ē 2k+1 are some polynomials whose lowest degree is greater than or equal to k + 2. The functions e 1 ,ē 1 are defined to be identically zero. Here,ē 2k+1 does not indicate the complex conjugation of e 2k+1 . We prove (3.2) by an inductive argument. The base cases i = 1, 2 of the induction are trivial. Note that in this case k = 0. Next, by Newton's identity, for every i ≥ 3, we have Suppose that k 0 ≥ 0 and (3.2) holds true for all k with 0 ≤ k ≤ 2k 0 . We apply the above identity and the induction hypothesis, and obtain and Note that e 2k 0 +1 ,ē 2k 0 +1 are polynomials whose lowest degrees are at least k 0 + 2. This closes the induction, and therefore finishes the proof of (3.2).
Next, we consider the case that (a, b) = (u(α, β), v(α, β)) for some 0 ≤ α, β ≤ ζ with |α − β| ≥ ζK −1 , where ζ > 0. Let h be an arbitrary polynomial of two variables u, v: We define a truncation of the polynomial h at the degree l by For every function g : R 2 → C and a, b ∈ R, we define the function g a,b (u, v) to be g a,b (u, v) := g(a + u, b + v).
We will show that for every k ≥ 1 and j = 0, 1 there exist w j,k = (w 1,j,k , · · · , w 2k,j,k ) ∈ R 2k with |w i,j,k | K ζ and some constant C a,b,2k+1+j such that holds for every u, v. Let us first accept this claim. By an affine transformation, we can replace the surface M (a,b) by for j = 0, 1 and k ≥ 1. Here, w j,k = (w 1,j,k , . . . , w 2k,j,k ) is the vector satisfying the claim (3.3). By the claim (3.3), we obtain Note that the error is harmless. We first consider the case when j = 1. By (3.2), we obtain Sinceē a,b 2k+1 is a polynomial of degree greater than or equal to k + 2 and |a|, |b| ζ, we obtain ). Hence, we finally obtain We next consider the case when j = 0. By (3.2), we obtain Since e a,b 2k+1 is a polynomial of degree greater or equal to k + 2, we get . Hence, we finally obtain This finishes the proof of Theorem 3.1, modulo the proof of the claimed representation (3.3), which we carry out now.
We reformulate this problem by using partial derivatives. Recall that the gradient ∇ u,v (f ) defined in (3.1) is a column vector. We will show that for every k ≥ 1 and j = 0, 1 there exists (3.4) We rewrite the matrix M ′ by By applying Lemma 3.2 and multiplying the matrix A −1 a,b on both sides of (3.4), it suffices to show that there exists Since p i (t, s) = t i + s i , by direct computations, we obtain that ∇ t,s ∇ 2 t,s · · · ∇ k t,s ⊺ p 1 p 2 · · · p 2k p 2k+1+j is equal to To simplify the notation, we reorder the rows by applying a linear transformation, and we may assume that P is the matrix defined by  We rewrite the matrix P by where γ(t) = (t, t 2 , . . . , t 2k , t 2k+1+j ) ⊺ , and γ (i) indicates the ith derivative of γ(t).
To proceed, we need to compute the determinant of a submatrix of the matrix P . This will rely on a formula of the determinant of the generalized Vandermonde matrix due to Kalman [K84].
This completes the proof of Theorem 3.1.

Proof of Proposition 2.4
In this section we will finish the proof of Proposition 2.4. Here we will see the motivation of restricting both intervals J 1 and J 2 to the small interval [0, δ ǫ ]. Roughly speaking, when both J 1 and J 2 are close to the origin, we are able to approximate the relevant manifold M (defined in (2.8)) by the moment manifold M 0 (see (4.7)). One advantage of working with the manifold M 0 is that it is translation-invariant. Moreover, certain sharp decoupling inequalities for such a manifold have already been established in [GZ18]. The proof there relies crucially on the fact that the manifold is a moment manifold and a translation-invariant manifold. In sharp contrast, the manifold M is neither a moment manifold nor a translation-invariant manifold.
Having a small parameter δ ǫ as above will create enough room for a bootstrapping argument (see (4.2)). Such kind of a bootstrapping argument can be dated back to the work of Pramanik and Seeger [PS07].
Recall that we need to show that (4.1) Here it is important to keep in mind that the boxes that appear in the right hand side of (4.1) all have non-empty intersections with L(J 1 , J 2 ). To prove (4.1), we will apply an inductive argument and prove for every integer m with r ≤ m ≤ rǫ −1 . The desired inequality (4.1) follows from (4.2) with m = rǫ −1 .
Let us start with proving the base case of (4.2), that is, the case m = r. This follows from L 2 orthogonality and interpolation with a trivial L ∞ bound: E M L(J 1 ,J 2 ) g L p (w B r ) ≤ C p,K,ǫ δ −Cǫ R∩L(J 1 ,J 2 ) =∅:l(R)=δ ǫ Suppose that we have proven (4.2) for some m = m 0 < rǫ −1 . We will show that (4.2) holds true for m = m 0 + 1. By the induction hypothesis, we have Fix a square R intersecting L(J 1 , J 2 ) with side length δ m 0 ǫ/r . For simplicity, we put γ = m 0 ǫ r . We claim that E M R g L p (w B r ) ≤ (C p,ǫ C p,K,ǫ ) 4 (δ This claim, combined with (4.3) will finish the proof of (4.2) with m = m 0 + 1, thus close the induction.