Two principles of decoupling

Jianhui Li and Tongou Yang Department of Mathematics, Northwestern University
Evanston, IL 60208, United States jianhui.li@northwestern.edu Department of Mathematics, University of California
Los Angeles, CA 90095, United States tongouyang@math.ucla.edu

Abstract.

We put forward a conical principle and a degeneracy locating principle of decoupling. The former generalises the Pramanik-Seeger argument used in the proof of decoupling for the light cone. The latter locates the degenerate part of a manifold and effectively reduces the decoupling problem to two extremes: non-degenerate case and totally degenerate case. Both principles aim to provide a new algebraic approach of reducing decoupling for new manifolds to decoupling for known manifolds.

1. Introduction

Fourier decoupling was first introduced by Wolff in [Wol00] as a tool to prove $L^{p}$ local smoothing estimates for large $p$ . In [Wol00], the decoupling was formulated in the setting of the truncated light cone in $\mathbb{R}^{3}$ . Later, observed by Pramanik and Seeger [PS07], the decoupling inequalities for light cones in $\mathbb{R}^{3}$ can be reduced to that of circles in $\mathbb{R}^{2}$ . The technique is now known as Pramanik-Seeger iteration. In simple terms, a light cone can be approximated by cylinders at intermediate scales. By projection and induction on scales, decoupling for a light cone in $\mathbb{R}^{n}$ is then reduced to decoupling for the unit sphere in $\mathbb{R}^{n-1}$ . Therefore, together with the seminal work of Bourgain and Demeter [BD15] on the sharp decoupling inequalities for the unit sphere (equivalent to that of a compact piece of a paraboloid), satisfactory decoupling results for light cones are obtained.

Based upon decoupling for spheres and light cones, decoupling inequalities for general manifolds have also been largely studied, and we consider two main types of such manifolds. The first type of manifolds carry certain non-degenerate conditions. See Table 1 below for some important examples. See also the following list of decoupling for non-degenerate manifolds in $\mathbb{R}^{n}$ : [BD16][Oh18][DGS19][GOZZK23][GZ19]. Their nondengenacy conditions are slightly harder to state, and we encourage the reader to their papers for details. In general, these results rely on the corresponding multilinear decoupling inequalities derived from the geometry of the manifolds.

Results	$n$	$k$	non-degeneracy	critial exponents
Bourgain-Demeter	$n$	$n-1$	$D^{2}\phi(x)\succ 0$	$\ell^{2}(L^{\frac{2(n+1)}{n-1}})$
[BD15]
Bourgain-Demeter	$n$	$n-1$	$\det D^{2}\phi(x)\neq 0$	$\ell^{\frac{2(n+1)}{n-1}}(L^{\frac{2(n+1)}{n-1}})$
[BD17]
Bourgain-Demeter-	$n$	$1$	$\det\left(\phi^{\prime\prime},\dots,\phi^{(n)}\right)$	$\ell^{2}(L^{n(n+1)})$
Guth [BDG16]			$\neq 0$
Guth-Maldague-	$3$	$2$	$\det D^{2}\phi(x)\neq 0$	$\ell^{2}(L^{4})$
Oh [GMO24]

Table 1. Decoupling inequalities for non-degenerate manifolds in

\mathbb{R}^{n}

as graphs of

\phi:[-1,1]^{k}\to\mathbb{R}^{n-k}

The second type of manifolds shown in Table 2 below are more general as the non-degenerate conditions may not be satisfied. The major approach to these results is to deal with the decoupling for pieces of the manifold on which the non-degenerate factor is small. For instance, for the case of hypersurfaces, the non-degenerate factor is the Gaussian curvature. When compared to Table 1, very limited results are known in higher dimensions $n>3$ or higher co-dimensions $n-k\geq 2$ .

Results	$n$	descriptions
Biswas-Gilula-Li-
Schwend-Xi [BGL⁺20];	2	analytic/smooth planar curves
Demeter [Dem20];
Y. [Yan21]
Bourgain-Demeter	$3$	some surfaces of revolution: $\phi(x)=\gamma(\|x\|)$
-Kemp [BDK20]
Kemp [Kem21]	3	surfaces lacking planar points
Kemp [Kem22]	3	Gaussian curvature vanishing identically
L.-Y. [LY22]	3	mixed-homogeneous polynomials $\phi$
L.-Y. [LY23]; Guth-	3	smooth surfaces
Maldague-Oh [GMO24]
Gao-Li-Zhao-Zheng	$n$	$\phi(x)=\sum_{i}\phi_{i}(x_{i})$ , $\phi_{i}^{\prime\prime}\sim 1$ or $\phi_{i}(x_{i})=x_{i}^{m}$ .
[GLZZ22]

Table 2. Decoupling inequalities for curves/surfaces in

\mathbb{R}^{n}

as graphs of

\phi:[-1,1]^{n-1}\to\mathbb{R}

In this paper, we first generalise the cone-to-sphere reduction to a radial principle that applies to more general manifolds that exhibits some conical structure. Then, we form the degeneracy locating principle by abstracting the idea of reducing the manifolds with possible degeneracy to the non-degenerate counterpart. These results provide an algebraic approach of reducing decoupling for more complicated manifolds to simpler manifolds. In particular, we provide immediate corollaries of these principles, namely, decoupling inequalities for new manifolds.

1.1. Decoupling inequalities

We first formulate decoupling inequalities in a more general setting.

Definition 1.1.

Given a compact subset $S\subseteq\mathbb{R}^{n}$ and a finite collection $\mathcal{R}$ of boundedly overlapping¹¹1To deal with technicalities, we will impose a slightly stronger overlapping condition. See Definition 1.7. parallelograms $R\subseteq\mathbb{R}^{n}$ . For Lebesgue exponents $p,q\in[2,\infty]$ and $\alpha\in\mathbb{R}$ , we define the $(\ell^{q}(L^{p}),\alpha)$ -decoupling constant $\mathrm{Dec}(S,\mathcal{R},p,q,\alpha)$ to be the smallest constant $\mathrm{Dec}$ such that

(1.1)

\left\lVert\sum_{R}f_{R}\right\rVert_{L^{p}(\mathbb{R}^{n})}\leq\mathrm{Dec}\,\,(\#\mathcal{R})^{\frac{1}{2}-\frac{1}{q}+\alpha}\left\lVert\left\lVert f_{R}\right\rVert_{L^{p}(\mathbb{R}^{n})}\right\rVert_{\ell^{q}(R\in\mathcal{R})}

for all smooth test function $f_{R}$ Fourier supported on $R\cap S$ .

Given a subset $S\subseteq\mathbb{R}^{n}$ , we say $S$ can be $(\ell^{q}(L^{p}),\alpha)$ -decoupled into the parallelograms $R\in\mathcal{R}$ at the cost of $K$ , if $S\subseteq\cup\mathcal{R}$ and $\mathrm{Dec}(S,\mathcal{R},p,q,\alpha)\leq K$ .

When $p,q$ are given, the exponent $\alpha$ is most commonly taken to be $0$ . Hence, we also say $S$ can be $\ell^{q}(L^{p})$ -decoupled if $S$ can be $(\ell^{q}(L^{p}),0)$ -decoupled. To simplify the notation further, since $S$ and the exponents $p,q,\alpha$ are usually clear from the context, we will often abbreviate $\mathrm{Dec}(\mathcal{R})=\mathrm{Dec}(S,\mathcal{R},p,q,\alpha)$ , and simply say $S$ can be decoupled into parallelograms $R\in\mathcal{R}$ .

By Hölder’s inequality, $\ell^{q}(L^{p})$ decoupling implies $\ell^{q^{\prime}}(L^{p})$ decoupling if $2\leq q^{\prime}\leq q$ . Thus, together with Plancherel’s identity, we always have $\ell^{q}(L^{2})$ decoupling for $q\geq 2$ . Also, for a fixed $q$ and $p^{\prime}\geq p$ , readers shall keep in mind that $\ell^{q}(L^{p^{\prime}})$ decoupling is generally stronger than $\ell^{q}(L^{p})$ decoupling. This follows from interpolation whenever applicable, or can be usually observed from the proof of decoupling.

1.1.1. Decoupling for manifolds in $\mathbb{R}^{n}$

For $0<\delta\ll 1$ , we often consider the case when $S$ is the $\delta$ -normal neighbourhood of a (compact piece of a) manifold $M\subseteq\mathbb{R}^{n}$ :

N_{\delta}(M):=\{\xi+\eta:\xi\in M,\eta\perp T_{\xi}M,|\eta|<\delta\},\quad T_{\xi}M\text{ is the tangent space of }M\text{ at }\xi.

To formulate decoupling for manifolds, we first introduce the following general definition which is independent of the manifold $M$ .

Definition 1.2 ( $\delta$ -flat subsets).

Let $1\leq m\leq n$ . We say a subset $R\subseteq\mathbb{R}^{n}$ is $\delta$ -flat in dimension $m-1$ , if it is contained inside the $\delta$ -normal neighbourhood of some $(m-1)$ -plane.

With this, we return to the formulation of decoupling for manifolds $M$ . Fix $m\leq n$ . We usually would like to decouple $S=N_{\delta}(M)$ into smaller pieces $S^{\prime}$ , such that the convex hull of $S^{\prime}$ is equivalent to a parallelogram $R\subseteq\mathbb{R}^{n}$ that is $\delta$ -flat in dimension $m-1$ .

We remark that if $M$ is a hypersurface, then the convex hull of $S^{\prime}$ will be essentially the same as $S^{\prime}$ , and we only need to consider the case $m=n$ . On the other hand, if $M$ is a curve, then the convex hull of $S^{\prime}$ will be nearly equivalent to the corresponding piece of an anisotropic neighbourhood²²2See, for instance, [GLYZK21] or Chapter 11 of [Dem20] for definitions of anisotropic neighbourhoods of a curve. of $M$ . We might need to consider all $m\in[1,n]$ .

By a partition of unity, we may assume that $M$ is given by the graph of an $\mathbb{R}^{n-k}$ -valued function $\phi$ on a compact set $\Omega\subseteq\mathbb{R}^{k}$ with bounded derivatives, the $\delta$ -vertical neighbourhood of $M\subseteq\mathbb{R}^{n}$

(1.2)

N_{\delta}^{\phi}(\Omega):=\{(\xi^{\prime},\phi(\xi^{\prime})+c):\xi^{\prime}\in\Omega,c\in\mathbb{R}^{n-k},|c|<\delta\}

is equivalent to the $\delta$ -normal neighbourhood of $M$ for decoupling considerations (see Section 1.7 for the precise definition of equivalent subsets). In this case, we may consider the projection of the $\delta$ -flat parallelograms $R$ onto the coordinate space.

Definition 1.3.

Let $1\leq m\leq n$ . We say a parallelogram $\omega\subseteq\Omega$ is $(\phi,\delta)$ -flat in dimension $m-1$ , if we have

(1.3)

\sup_{x\in\omega}|\lambda(x,\phi(x))|\leq\delta,

for some affine function $\lambda:\mathbb{R}^{n}\to\mathbb{R}^{n-m+1}$ given by $x\mapsto Ax+b$ where $A\in\mathbb{R}^{(n-m+1)\times n}$ has orthonormal rows, and $b\in\mathbb{R}^{n-m+1}$ .³³3See Appendix 4.3 for more properties of $(\phi,\delta)$ -flatness.

To simplify notation, when $M$ is given by the graph of $\phi$ over $\Omega$ , we say $\Omega$ can be graphically decoupled into $(\phi,\delta)$ -flat parallelograms $\omega\in\mathcal{R}$ , if $N_{\delta}^{\phi}(\Omega)$ can be decoupled into parallelograms $\{R_{\omega}:\omega\in\mathcal{R}\}$ , such that each $R_{\omega}$ is equivalent to the convex hull of $N_{\delta}^{\phi}(\omega)$ .

1.2. A radial principle of decoupling

To push the ideas of Pramanik-Seeger iteration [PS07] further, we formulate and prove a radial principle of decoupling. This is a generalisation of the arguments used in Section 8 of [BD15] (see also Appendix B of [GGG⁺22]). Roughly speaking, under certain non-degenerate assumptions, the decoupling for the manifold parameterised by

(s,r(s)t,r(s)\psi(t)),\quad r(s)\sim 1

can be reduced to the decoupling for

(s,t,r(s)+\psi(t)).

The precise statement of the radial principle of decoupling is in Section 2. We give some examples.

(1)

Consider the surface $M$ given by the truncated cone $\{(t,|t|),1\leq|t|\leq 2\}\subseteq\mathbb{R}^{n}$ . After a finite partition and rotation, we may re-parameterise each part of the cone by

$\{(r,rt^{\prime},r\psi(t^{\prime})):1\leq r\leq 2,|t^{\prime}|\leq 1/10\},$

where $\psi(t^{\prime})=\sqrt{1-|t^{\prime}|^{2}}$ and $|t^{\prime}|\leq 1/10$ . By the radial principle, it suffices to study decoupling for the following cylinder

$(r,t^{\prime},r+\psi(t^{\prime}))$

which is precisely the reduction given in [BD15] using Pramanik-Seeger iteration.
(2)

Consider the radial surface $M$ generated by a smooth curve $\gamma:\mathbb{R}\to\mathbb{R}$ , that is, $M=\{(t,\gamma(|t|)):1\leq|t|\leq 2\}\subset\mathbb{R}^{n}$ under an additional assumption that $\gamma^{\prime}(r)\sim 1$ , where $S$ can be thought of as perturbations of the light cone. The case $n=3$ has been studied in [BDK20]. Similarly, each part of the surface can be reparametrised by $\{(\gamma(r),rt^{\prime},r\psi(t^{\prime})),1\leq r\leq 2,|t^{\prime}|\leq 1/10\}$ , where $\psi(t)=\sqrt{1-|t^{\prime}|^{2}}$ . By the implicit function theorem, we further reparametrise the surface by

$(s,\gamma^{-1}(s)t^{\prime},\gamma^{-1}(s)\psi(t^{\prime})),\quad\gamma^{-1}(s)\sim 1.$

Now, the radial principle of decoupling reduces the original decoupling problem to the decoupling for the following surface:

$(s,t^{\prime},\gamma^{-1}(s)+\psi(t^{\prime})).$

Although for these two examples, we can use elementary arguments to study decoupling for the reduced manifolds, we will instead put them under a more general framework introduced right below.

Note that a common feature of the reduced manifolds is that the degenerate set is separated in the $s$ and $t$ variables. It facilitates the needs of the following degeneracy locating principle of decoupling.

1.3. A degeneracy locating principle of decoupling

We make abstract the idea of reducing the manifolds with possible degeneracy to totally degenerate and non-degenerate cases. To avoid technicalities, we present a simple version here. See Section 3 for the detailed definitions and formulation.

Let $\mathcal{M}$ be a family of manifolds $M_{\phi}$ given by the graph of a vector valued function $\phi:R_{\phi}\to\mathbb{R}^{n-k}$ for some parallelogram $R_{\phi}\subseteq[-1,1]^{k}$ . We say that $H$ is a degeneracy determinant if it quantitatively detects the degeneracy of the manifold. More precisely, $H$ maps $M_{\phi}\in\mathcal{M}$ to a smooth function on $R_{\phi}$ such that the following holds:

•

if $\sup_{R_{\phi}}|HM_{\phi}|\leq\sigma\ll 1$ , then $M_{\phi}$ restricted to $R\times\mathbb{R}^{n-k}$ is within a $C\sigma^{\beta}$ neighbourhood of for some totally degenerate manifold $M_{\psi}\in\mathcal{M}$ , i.e. $HM_{\psi}\equiv 0$ , for some $\beta>0,C\geq 1$ depending on $\mathcal{M}$ .

Let us consider the family $\mathcal{M}$ consisting of graphs of polynomials $\phi$ of degree at most $d$ with bounded coefficients. For $k=1$ and polynomial space curves $(t,\phi_{1}(t),\dots,\phi_{n-1}(t)),t\subseteq I\subseteq[-1,1]$ , it is easy to check that $\sup_{j=1,\dots,n-1}|\phi_{j}^{\prime\prime}(t)|$ and the Wronskian of $\phi^{\prime\prime}$ , namely, $\det\left(\phi^{\prime\prime},\phi^{(3)},\dots,\phi^{(n)}\right)$ , are examples of degeneracy determinants. For $k=2$ and $n-k=1$ , $HM_{\phi}:=\det D^{2}\phi$ is a degeneracy determinant as shown in Proposition 4.8. It generalises Theorem 3.2 of [LY23].

Our degeneracy locating principle says that to decouple $M_{\phi}\in\mathcal{M}$ into parallelograms, it suffices to understand the decoupling of these cases at a cost uniform in $\mathcal{M}$ :

•

decoupling of the sublevel set

$\{x\in R_{\phi}:|HM_{\phi}(x)|\leq\sigma\},$

where $M_{\phi}\in\mathcal{M}$ , $\sigma\leq 1$ , into parallelograms.
•

decoupling of the totally degenerate manifolds $M_{\phi}\in\mathcal{M}$ , i.e. $HM_{\phi}\equiv 0$ , into parallelograms.
•

decoupling of the non-degenerate manifolds $M_{\phi}\in\mathcal{M}$ , i.e. $HM_{\phi}\sim 1$ on $R_{\phi}$ , into parallelograms.

In the examples given in Table 2, the totally degenerate cases are the same problems of at least one dimensional lower and the non-degenerate case follows from the famous theorem of Bourgain and Demeter [BD15, BD17]. When $k=1$ , the sublevel set decoupling for polynomial family $\mathcal{M}$ of degree at most $d$ and bounded coefficients is a direct corollary of fundamental theorem of algebra. For special cases when $k=2$ and $n=3$ in [BDK20] and [Kem21], the sublevel sets are neighbourhoods of circles and parabolas. In [LY23], exactly the same framework is used, where the sublevel set decoupling was called “generalised 2D uniform decoupling”.

1.4. Two different scales of induction

We remark that the proofs of both principles are based on the method of induction on scales. More precisely, for the radial principle, essentially we are iterating a bootstrap inequality of the form

D(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon^{2}}D(\delta^{1-\varepsilon}).

Each time the step $\delta^{-\varepsilon}$ changes as $\delta$ gets larger, and the number of steps is of the order $\varepsilon^{-1}\log\log\delta^{-1}$ .

For the degeneracy locating principle, essentially we are iterating a bootstrap inequality of the form

D(\delta)\leq C_{\varepsilon}K^{\varepsilon}D(K\delta),

where the step $K=\delta^{-\varepsilon}$ is fixed throughout the iteration. The number of steps is of the order $\varepsilon^{-1}$ .

In both cases, we are able to prove that the cost of such reduction $D(\delta)\lesssim_{\varepsilon}\delta^{-\varepsilon}$ for every $\varepsilon\in(0,1)$ .

1.5. Corollaries

As an application of the two principles, we present decoupling inequalities of two new types of manifolds. The proofs below assume the simplified versions above to avoid technicalities. Readers are encouraged to check that a direct modification of the arguments in this subsection applies to the detailed versions of the two principles in Sections 2 and 3 below.

Corollary 1.4 (Decoupling for radial surfaces).

Let $2\leq p\leq\frac{2(n+1)}{n-1}$ . Let $\gamma:[1,2]\to\mathbb{R}$ be a smooth function with $\gamma^{\prime}\sim 1$ . Let $M$ be the hypersurface in $\mathbb{R}^{n}$ given by the graph of the function $\phi(t)=\gamma(|t|)$ over

\Omega=\{1\leq|t|\leq 2,|t_{1}|\geq|t|/2\},\quad\quad t:=(t_{1},t^{\prime}).

For any $\delta>0$ , let $\mathcal{I}_{\delta}$ be a partition of $[1,2]$ into $(\gamma,\delta)$ -flat intervals $I$ such that $2I$ is not $(\gamma,\delta)$ -flat. Let $\mathcal{R}_{\delta}$ be a covering of $\Omega$ by parallelograms $R$ , each of which is equivalent to the region

\{(r\sqrt{1-|t^{\prime}|^{2}},rt^{\prime}):r\in I,t^{\prime}\in T\}

for $I\in\mathcal{I}_{\delta}$ and $\delta^{1/2}$ -squares $T$ .

Then $\Omega$ can be graphically $\ell^{p}(L^{p})$ decoupled into $(\phi,\delta)$ -flat parallelograms $R\in\mathcal{R}_{\delta}$ at a cost of $O_{\varepsilon,\gamma}(\delta^{-\varepsilon})$ , for any $\varepsilon>0$ . Moreover, if $\phi$ is convex, this $\ell^{p}(L^{p})$ decoupling can be improved to $\ell^{2}(L^{p})$ decoupling.

Proof of Corollary 1.4 using the simplified versions of the two principles.

Following the arguments in Example 2 in Subsection 1.2, the manifold $M$ can be broken into $O(1)$ many pieces, each of which can be rotated and reparameterised by

(1.4)

\{(s,\gamma^{-1}(s)t^{\prime},\gamma^{-1}(s)\psi(t^{\prime})):|t^{\prime}|\leq 1/10,s\in\gamma([1,2])\},

where $\psi(t^{\prime})=\sqrt{1-|t^{\prime}|^{2}}$ . By the radial principle of decoupling, it reduces to the decoupling of the manifold given by

(1.5)

\{(s,t^{\prime},\gamma^{-1}(s)+\psi(t^{\prime})):|t^{\prime}|\leq 1/10,s\in\gamma([1,2])\}.

Since $\gamma$ is smooth with $\gamma^{\prime}\sim 1$ on $[1,2]$ , $\gamma^{-1}$ is smooth on $\gamma([1,2])$ . By the smooth approximation argument in Section 2.3 of [LY23], it suffices to prove the uniform decoupling result for the family $\mathcal{M}=\mathcal{M}(d)$ consisting of manifolds $M_{\phi}$ with $\phi(s,t^{\prime})=P(s)+\psi(t^{\prime})$ where $P$ is a polynomial of degree at most $d$ and with bounded coefficients bounded by $1$ . We pick $HM_{\phi}(s,t^{\prime})=P^{\prime\prime}(s)$ . By the degeneracy locating principle, it suffices to check the following.

•

$H$ is a degeneracy determinant. This follows from the mean value theorem, since $s\in\mathbb{R}$ .
•

The sublevel set

$\{(s,t^{\prime}):|P^{\prime\prime}(s)|\leq\sigma\}$

is already a union of $O_{d}(1)$ many rectangular strips.
•

The totally degenerate manifolds $M_{\phi}\in\mathcal{M}$ are cylinders because $P$ must be an affine function. By cylindrical decoupling, it suffices to decouple

$\{(t^{\prime},\psi(t^{\prime})):|t^{\prime}|\leq 1/10\}$

which has non-vanishing Gaussian curvature everywhere. The decoupling inequalities for these cases are given in [BD15].
•

The non-degenerate manifolds $M_{\phi}\in\mathcal{M}$ has non-vanishing Gaussian curvature everywhere. Their decoupling inequalities are given in [BD15] and [BD17] (for hyperbolic cases).

We have all assumptions for the degeneracy locating principle checked. Therefore, we have obtained the decoupling of the manifold given in (1.5), and hence (1.4), into $\delta$ -flat parallelograms as desired.

It is straightforward to check by the full version of the radial principle, Theorem 2.2, that the family $\mathcal{R}_{\delta}$ is exactly the decoupled $\delta$ -flat parallelograms.

∎

Corollary 1.5 (Decoupling for graphs of additively separable functions).

Let $2\leq p\leq\frac{2(n+1)}{n-1}$ . For $j=1,\dots,J$ , let $n_{j}\in\{1,2\}$ such that $\sum_{j}n_{j}=n-1$ and $\phi_{j}:[-1,1]^{n_{j}}\to\mathbb{R}$ be smooth. Let $\phi:[-1,1]^{n-1}\to\mathbb{R}$ be such that

(1.6)

\phi(x_{1},\dots,x_{J})=\sum_{j=1}^{J}\phi_{j}(x_{j}),\quad x_{j}\in[-1,1]^{n_{j}}.

For any $\varepsilon,\delta>0$ , there exists a $B$ -overlapping⁴⁴4See Definition 1.7; here $B$ depends on $\phi,\varepsilon$ but not $\delta$ . cover $\mathcal{R}$ of $[-1,1]^{n-1}$ by $(\phi,\delta)$ -flat parallelograms $R$ (in dimension $n-1$ ), such that $[-1,1]^{n-1}$ can be graphically $\ell^{p}(L^{p})$ decoupled into $R\in\mathcal{R}$ at a cost of $O_{\varepsilon,\phi}(\delta^{-\varepsilon})$ . Moreover, if $\phi$ is convex, this $\ell^{p}(L^{p})$ decoupling can be improved to $\ell^{2}(L^{p})$ decoupling.

Proof of Corollary 1.5 using the simplified version of degeneracy locating principle.

Fix $d$ . For each $0\leq l\leq J$ , let $\mathcal{M}_{l}^{n}$ be the collection of manifolds $M_{\phi}$ given by graphs of polynomials of the form (1.6), such that each $\phi_{j}$ has degree at most $d$ and coefficients bounded by 1, and $|\det D^{2}\phi_{j}|\sim 1$ on $[-1,1]^{n_{j}}$ for $j=1,\dots,l$ . It suffices to prove the decoupling for $M_{\phi}\in\mathcal{M}_{0}^{n}$ at a cost of $O_{\varepsilon,d}(\delta^{-\varepsilon})$ uniform in the coefficients of the polynomials, by the smooth approximation argument in Section 2.3 of [LY23].

We now induce on $n$ and $l$ . The base case $\mathcal{M}_{J}^{n}$ is exactly given in [BD15, BD17] at a cost of $O_{\varepsilon,d}(\delta^{-\varepsilon})$ . The other base cases $\mathcal{M}_{0}^{1},\mathcal{M}_{0}^{2}$ are exactly in [Yan21] and [LY23] respectively. By induction, we may assume the decoupling of $M_{\phi}\in\mathcal{M}_{1}^{n},\mathcal{M}_{0}^{n-1},\mathcal{M}_{0}^{n-2}$ at a cost of $O_{\varepsilon,d}(\delta^{-\varepsilon})$ .

We apply the degeneracy locating principle on $\mathcal{M}_{0}^{n}$ with $H(M_{\phi})=\det D^{2}\phi_{1}$ .

First, we consider the easy case when $n_{1}=1$ . By the mean value theorem, $H$ is a degeneracy determinant. By the fundamental theorem of algebra, the sublevel set

\{x\in[-1,1]:|\phi_{1}^{\prime\prime}(x)|\leq\sigma\}

is a union of $O_{d}(1)$ many intervals. Totally degenerate $M_{\phi}\in\mathcal{M}^{n}_{0}$ with $\phi_{1}^{\prime\prime}\equiv 0$ are cylinders, which can be solved by cylindrical decoupling and the induction hypothesis on $\mathcal{M}^{n-1}_{0}$ . Non-degenerate $M_{\phi}\in\mathcal{M}^{n}_{0}$ are those with $|\phi_{J}^{\prime\prime}|\sim 1$ , hence also in $\mathcal{M}^{n}_{1}$ . This follows from the induction hypothesis.

Second, we consider the remaining case when $n_{1}=2$ . By Proposition 4.8, $H$ is a degeneracy determinant when $n_{1}=2$ . The sublevel set decoupling is exactly Theorem 3.1 of [LY23]. Totally degenerate $M_{\phi}\in\mathcal{M}^{n}_{0}$ with $\det D^{2}\phi_{1}\equiv 0$ are cylinders, which again can be solved by cylindrical decoupling and the induction hypothesis on $\mathcal{M}^{n-2}_{0}$ . The non-degenerate case is similar to the case when $n_{J}=1$ . ∎

Corollary 1.6 (Decoupling for analytic curves in $\mathbb{R}^{n}$ ).

Let $m\in[1,n]$ be an intermediate dimension and $2\leq p\leq m(m+1)$ . Let $\phi:[-1,1]\to\mathbb{R}^{n-1}$ be real-analytic, and let $M_{\phi}$ be the smooth curve in $\mathbb{R}^{n}$ given by

\{(s,\phi_{1}(s),\dots,\phi_{n-1}(s)):s\in[-1,1]\}.

For each $0<\delta\ll_{\phi}1$ , let $\mathcal{I}_{\delta}$ be a collection of disjoint intervals $I$ that are $(\phi,\delta)$ -flat but not $(\phi,\delta/2)$ -flat in dimension $(m-1)$ . Then, $[-1,1]$ can be graphically $\ell^{2}(L^{p})$ decoupled into $I\in\mathcal{I}_{\delta}$ at a cost $O_{\varepsilon,\phi}(\delta^{-\varepsilon})$ for any $\varepsilon>0$ .

We remark that unless the whole curve lies in an affine space of dimension $m-1$ , such partition $\mathcal{I}_{\delta}$ always exists for $\delta\ll_{\phi}1$ . The interested reader may refer to Proposition 3.1 of [Yan21] for a proof in the case $n=2$ ; the higher dimensional cases are similar.

To prove the new version, we observe that if $W_{n}\phi$ has rank at most $m-2$ , $\phi$ stays in some $(m-1)$ dimensional affine subspace. In this case, $[-1,1]$ is $(\phi,\delta)$ -flat for all $\delta>0$ . It is impossible to have such a collection $\mathcal{I}_{\delta}$ .

Proof of Corollary 1.6 using the simplified version of degeneracy locating principle..

The case $m=1$ is trivial. For each $2\leq m\leq n$ , we define the generalised $m\times n$ Wronskian matrix

(1.7)

W_{m}\phi(s):=\left[\begin{array}[]{ccc}\rule[2.15277pt]{10.76385pt}{0.5pt}&\phi^{\prime\prime}(s)&\rule[2.15277pt]{10.76385pt}{0.5pt}\\ \rule[2.15277pt]{10.76385pt}{0.5pt}&\phi^{(3)}(s)&\rule[2.15277pt]{10.76385pt}{0.5pt}\\ &\vdots&\\ \rule[2.15277pt]{10.76385pt}{0.5pt}&\phi^{(m)}(s)&\rule[2.15277pt]{10.76385pt}{0.5pt}\end{array}\right].

Since $\phi$ is real analytic, it follows from elementary linear algebra (see for instance [B.00]) that $\det(W_{m}\phi(W_{m}\phi)^{T})(s)\equiv 0$ if and only if the curve $\phi$ lies in an $(m-1)$ -plane.

Let $\mathcal{M}$ be the collection of manifolds $M_{\phi}$ given by graphs of polynomials $\phi:[-1,1]\to\mathbb{R}^{n-1}$ of bounded coefficients of degree at most $d$ satisfying either of the followings:

•

$M_{\phi}$ is a straight line;
•

the determinant of the $(m-1)\times(m-1)$ matrix $W_{m}\phi(s)(W_{m}\phi(s))^{T}$ , as a polynomial of $s$ , has some coefficient $\sim$ 1. Here and throughout this proof, all implicit constants depend on $n,d$ only.

Define $HM_{\phi}:=\det(W_{m}\phi(W_{m}\phi)^{T})$ . To apply the degeneracy locating principle, we need to check the followings:

•

$H$ is a degeneracy determinant. Suppose that $0<\sup_{I}|HM_{\phi}|\leq\sigma$ for some interval $I$ . Since $HM_{\phi}$ is a univariate polynomial of degree at most $2dn$ having some coefficients $\sim 1$ , by Corollary 4.3,

$1\sim\sup_{[-1,1]}|HM_{\phi}|\lesssim|I|^{-2dn}\sup_{I}|HM_{\phi}|\leq|I|^{-2dn}\sigma.$

Hence, $|I|\lesssim\sigma^{1/(2dn)}$ . On the other hand, $\sup_{s\in I}\|\phi(s)-\phi(s_{0})\|_{\infty}\lesssim_{d}|I|$ , for some $s_{0}\in I$ . Thus, $M_{\phi}$ restricted to $I\times\mathbb{R}^{n-k}$ is inside $O(\sigma^{1/(2dn)})$ neighbourhood of $M_{\psi}$ , where $\psi(s)\equiv\phi(s_{0})$ and $HM_{\psi}\equiv 0$ .
•

Decoupling of sublevel sets. The sublevel set $\{s:|HM_{\phi}(s)|\leq\sigma\}$ is an union of $O_{d}(1)$ many intervals. No decoupling is needed.
•

Decoupling of totally degenerate manifolds $M_{\phi}\in\mathcal{M}$ . These manifolds are straight lines. No decoupling is needed.
•

Decoupling of non-degenerate manifolds $M_{\phi}\in\mathcal{M}$ . By partition of unity into $O_{n,d}(1)$ many pieces, we may assume that $W(\phi^{\prime\prime}_{1},\dots,\phi^{\prime\prime}_{m-1})$ has determinant $\sim 1$ . The decoupling of these manifolds follows from direct applications of cylindrical decoupling and decoupling for moment curves in $\mathbb{R}^{m}$ by Bourgain-Demeter-Guth [BDG16].

With all conditions checked, we conclude that $N_{\delta}(M_{\phi})$ , $M_{\phi}\in\mathcal{M}$ , can be uniformly $\ell^{2}(L^{p})$ decoupled into $\delta$ -flat parallelograms $N^{\phi}_{\delta}(I)$ (see (1.2)), $I\in\mathcal{I}_{\delta}$ , at a cost $O_{\varepsilon,\mathcal{M}}(\delta^{-\varepsilon})$ in dimension $m-1$ . By the smooth approximation argument in Section 2.3 of [LY23], we upgrade the uniform decoupling results for families of polynomial curves to the desired decoupling results for smooth curves. ∎

1.6. Outline of the paper

In Section 2 we state and prove the radial principle. In Section 3 we state and prove the degeneracy locating principle. Section 4 is the appendix, which discusses some fundamental concepts about parallelograms, polynomials and flatness in decoupling.

1.7. Notation

We first introduce some standard notation that will be used in this paper.

(1)

We use the standard notation $a=O_{C}(b)$ , or $|a|\lesssim_{C}b$ to mean there is a constant $C$ such that $|a|\leq Cb$ . In many cases, such as results related to polynomials, the constant $C$ will be taken to be absolute or depend on the polynomial degree only, and so we may simply write $a=O(b)$ or $|a|\lesssim b$ . Similarly, we define $\gtrsim$ and $\sim$ .
(2)

Given a manifold $M\subseteq\mathbb{R}^{n}$ , we define the $\delta$ -normal neighbourhood $N_{\delta}(M)$ to be

$N_{\delta}(M):=\{\xi+\eta:\xi\in M,\eta\perp T_{\xi}M,|\eta|<\delta\},$

where $T_{\xi}M$ is the tangent space of $M$ at $\xi$ .
(3)

Given a continuous function $\phi:\Omega\subseteq\mathbb{R}^{k}\to R^{n-k}$ , we define the $\delta$ -vertical neighbourhood $N^{\phi}_{\delta}(\Omega)$ to be

$N_{\delta}^{\phi}(\Omega):=\{(\xi^{\prime},\phi(\xi^{\prime})+c):\xi^{\prime}\in\Omega,c\in\mathbb{R}^{n-k},|c|<\delta\}.$

1.7.1. Equivalence of geometric objects

Decoupling inequalities are formulated using parallelograms in $\mathbb{R}^{n}$ . To deal with technicalities, it is crucial that we make clear some fundamental geometric concepts.

(1)

A parallelogram $R\subseteq\mathbb{R}^{n}$ is defined to be a set of the form

$\{x+b\in\mathbb{R}^{n}:|x\cdot u_{i}|\leq l_{i}\},$

where $\{u_{i}:1\leq i\leq n\}$ is a basis of unit vectors in $\mathbb{R}^{n}$ , $b\in\mathbb{R}^{n}$ and $l_{i}\geq 0$ . If $\{u_{i}:1\leq i\leq n\}$ is in addition orthogonal, we say $R$ is a rectangle. We say two parallelograms are disjoint if their interiors are disjoint.
(2)

Unless otherwise specified, for a parallelogram $S\subseteq\mathbb{R}^{n}$ and $C>0$ , we denote by $CS$ the concentric dilation of $S$ by a factor of $C$ .
(3)

Given a subset $S\subseteq\mathbb{R}^{n}$ , a parallelogram $R$ and $C\geq 1$ , we say $S,R$ are $C$ -equivalent if $C^{-1}R\subseteq S\subseteq CR$ . If the constant $C$ is unimportant, we simply say $S,R$ are equivalent.
(4)

By the John ellipsoid theorem [Joh14], every parallelogram in $\mathbb{R}^{n}$ is $C_{n}$ -equivalent to a rectangle in $\mathbb{R}^{n}$ , for some dimensional constant $C_{n}$ . For this reason and in view of the enlarged overlap introduced right below, we do not distinguish rectangles and parallelograms.

1.7.2. Enlarged overlap

Let $\mathcal{R}$ be a family of parallelograms. In all decoupling inequalities in literature, we hope the parallelograms in $\mathcal{R}$ do not overlap too much. Since we often deal with constant enlargement of parallelograms, we will need a slightly stronger version of overlap condition.

Definition 1.7.

Given a family $\mathcal{R}$ of parallelograms in $\mathbb{R}^{n}$ . Given a function $B:[1,\infty)\to[1,\infty)$ , we say the family is $B$ -overlapping, if for every constant $C\geq 1$ we have

\sum_{R\in\mathcal{R}}1_{CR}\leq B(C).

For all decoupling at some scale $\delta\in(0,1)$ in this paper, the function $B$ depends on many parameters such as $\varepsilon$ and the family of the manifolds we are decoupling, but most importantly, it is independent of $\delta$ . We simply say $\mathcal{R}$ is boundedly overlapping if the function $B$ depends on unimportant parameters that are clear from the context.

1.8. Acknowledgement

The first author would like to thank Betsy Stovall for preliminary discussions on an early stage of the work. The first author is supported by AMS-Simons Travel Grants. The second author is supported by the Croucher Fellowships for Postdoctoral Research. He would like to thank Terry Tao for numerous suggestions on both the content and the presentation of this article.

2. The radial principle

In this section, we are going to introduce and prove the radial principle of decoupling.

2.1. Statement of theorem

We will need some notation to make the statement precise. First we impose a standard regularity assumption on the partitions.

Definition 2.1 (Dyadic mesh).

For each $\delta\in 2^{-\mathbb{N}}$ , let $\mathcal{R}_{\delta}$ be a cover of $[-1,1]^{n}$ by parallelograms of variable directions and dimensions. Given a function $B:[1,\infty)\to[1,\infty)$ , we say the collection $\{\mathcal{R}_{\delta}:\delta\in 2^{-\mathbb{N}}\}$ is a dyadic mesh with $B$ -overlap, if the followings hold:

(1)

Each $\mathcal{R}_{\delta}$ is $B$ -overlapping as in Definition 1.7.
(2)

For dyadic $0<\delta<\sigma<1$ , $C\geq 1$ and each subset $R_{\delta}\in\mathcal{R}_{\delta}$ , there exists $C^{\prime}=C^{\prime}(C,n)$ such that $CR_{\delta}$ is contained in the union of at most $O_{B,C,n}(1)$ subsets $C^{\prime}R_{\sigma}$ where $R_{\sigma}\in\mathcal{R}_{\sigma}$ .
(3)

$\#\mathcal{R}_{\delta}\leq\delta^{-C_{n}}$ for some dimensional constant $C_{n}$ .

The third condition is very mild. For instance, it is trivially true when each $R_{\delta}$ has all dimensions $\geq\delta$ .

Let $k,l\geq 1$ . Given a $C^{1}$ function $r(s)$ defined in $[-1,1]^{k}$ and map into $[1,2]$ . Given a $C^{2}$ function $\psi(t)$ defined in $[-1,1]^{l}$ and map into $\mathbb{R}$ .

The canonical example of $\psi(t)$ is given by $\psi(t)=\sqrt{1-|t|^{2}}$ . A slightly different interesting example is $\psi(t)=1-|t|^{2}$ . Both motivate the terminology “radial principle”. However, for future use, we will actually study much more general functions $\psi$ that may not have radial structure; see the beginning of Theorem 2.2 right below.

Given $S\subseteq[-1,1]^{k}$ and $T\subseteq[-1,1]^{l}$ in the parameter space, we define

(2.1)		$\displaystyle\Sigma_{0}(S,T)$	$\displaystyle:=N_{\delta}(\{(s,t,r(s)+\psi(t)):s\in S,t\in T\})$
(2.2)		$\displaystyle\Pi_{0}(S,T)$	$\displaystyle:=N_{\delta}(\{(s,r(s)t,r(s)\psi(t)):s\in S,t\in T\}).$

By Proposition 4.7, if $S$ is $(r,\delta)$ -flat and $T$ is $(\psi,\delta)$ -flat, all in the first sense (4.1), then there exist parallelograms $\Sigma(S,T)$ , $\Pi(S,T)$ which contain and are equivalent to $\Sigma_{0}(S,T)$ , $\Pi_{0}(S,T)$ , respectively.

Theorem 2.2 (Radial principle of decoupling).

Suppose that either

(1)

$r(s)$ is affine, or
(2)

$L\psi(t):=\psi(t)-t\cdot\nabla\psi(t)$ is away from $0$ .

Assume there are $B$ -overlapping dyadic meshes $\{\mathcal{S}_{\delta}:\delta\in 2^{-\mathbb{N}}\}$ , $\{\mathcal{T}_{\delta}:\delta\in 2^{-\mathbb{N}}\}$ , covering $[-1,1]^{k}$ , $[-1,1]^{l}$ respectively, and obeying the following conditions:

(1)

Each $S\in\mathcal{S}_{\delta}$ is $(r,\delta)$ -flat in the sense of (4.1), namely, there exist affine functions $\lambda(s)$ , $\Lambda(s)$ such that

(2.3) $|r(s)-\lambda(s)|\leq\delta,\quad\forall s\in S.$

(2)

For each $T\in\mathcal{T}_{\delta}$ , $2T$ is $(\psi,\delta)$ -flat in the sense of (4.3), namely,

(2.4)

\sup_{\begin{subarray}{c}t\in 2T,v\in\mathbb{S}^{l-1}\\ a\in\mathbb{R}:t+av\in 2T\end{subarray}}\left|v^{T}D^{2}\psi(t)v\right||a|^{2}\leq\delta.

(3)

Let $\Sigma$ be the subset of $\mathbb{R}^{k+l+1}$ given by

\Sigma=\{(s,t,r(s)+\psi(t)):(s,t)\in[-1,1]^{k}\times[-1,1]^{l}\}.

Thus the collection $\{\Sigma(S,T):S\in\mathcal{S}_{\delta},T\in\mathcal{T}_{\delta}\}$ is a $\tilde{B}$ -overlapping cover of $\Sigma$ by parallelograms, where $\tilde{B}$ is independent of $\delta$ .

Fix $p,q,\alpha$ . Assume for each $\delta>0$ , the following $\ell^{q}(L^{p})$ decoupling inequality holds: for test functions $f_{S,T}$ each Fourier supported in $\Sigma(S,T)$ , we have

(2.5)

\left\lVert\sum_{S\in\mathcal{S}_{\delta}}\sum_{T\in\mathcal{T}_{\delta}}f_{S,T}\right\rVert_{p}\leq C_{\varepsilon}\delta^{-\varepsilon}(\#\mathcal{S}_{\delta}\#\mathcal{T}_{\delta})^{\alpha}\left\lVert\left\lVert f_{S,T}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S}_{\delta},T\in\mathcal{T}_{\delta})},\quad\forall\varepsilon>0,

for some constant $C_{\varepsilon}$ depending possibly on other parameters but independent of the test functions $f_{S,T}$ . Here and throughout this section, we abbreviate $\left\lVert\cdot\right\rVert_{p}=\left\lVert\cdot\right\rVert_{L^{p}(\mathbb{R}^{k+l+1})}$ .

Consider the subset $\Pi\subseteq\mathbb{R}^{k+l+1}$ given by

(2.6)

\Pi=\{(s,r(s)t,r(s)\psi(t)),\,(s,t)\in[-1,1]^{k}\times[-1,1]^{l}\}.

Thus the collection $\{\Pi(S,T):S\in\mathcal{S}_{\delta},T\in\mathcal{T}_{\delta}\}$ is a $\bar{B}$ -overlapping cover of $\Pi$ by parallelograms, where $\bar{B}$ is independent of $\delta$ . Moreover, for each $\delta>0$ , the following $\ell^{q}(L^{p})$ decoupling inequality holds: for test functions $g_{S,T}$ Fourier supported in $\Pi(S,T)$ , we have

(2.7)

\left\lVert\sum_{S\in\mathcal{S}_{\delta}}\sum_{T\in\mathcal{T}_{\delta}}g_{S,T}\right\rVert_{p}\leq C^{\prime}_{\varepsilon}\delta^{-\varepsilon}(\#\mathcal{S}_{\delta}\#\mathcal{T}_{\delta})^{\alpha}\left\lVert\left\lVert g_{S,T}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S}_{\delta},T\in\mathcal{T}_{\delta})},\quad\forall\varepsilon>0,

where $C^{\prime}_{\varepsilon}$ depends at most on

\varepsilon,C_{\varepsilon},\left\lVert r\right\rVert_{C^{1}},\left\lVert\psi\right\rVert_{C^{2}},\inf|L\psi(t)|,k,l,B,\tilde{B},\bar{B},p,q,\alpha.

Moreover, all implicit constants in the big-O notation above depend at most on $\left\lVert r\right\rVert_{C^{1}},\left\lVert\psi\right\rVert_{C^{2}},\inf|L\psi(t)|,k,l,B,\tilde{B},\bar{B}$ .

For example, in the case of the standard truncated light cone in $\mathbb{R}^{3}$ studied in Section 8 of [BD15], we have the following choices:

	$\displaystyle\cdot k=l=1,\quad r(s)=s+2,\quad\psi(t)=\sqrt{1-t^{2}},$
	$\displaystyle\cdot\mathcal{S}_{\delta}\text{ is the trivial partition $\{[-1,1]\}$},$
	$\displaystyle\cdot\mathcal{T}_{\delta}\text{ is the partition of $[-1,1]$ into intervals of length $\delta^{1/2}$},$
	$\displaystyle\cdot B(C)=\tilde{B}(C)=\bar{B}(C)=(100C)^{100},$
	$\displaystyle\cdot\alpha=0,\quad p=6,\quad q=2.$

Thus we have the following chain of reduction of decoupling:

		$\displaystyle(s,st,s\sqrt{1-t^{2}})$
	$\displaystyle\implies$	$\displaystyle(s,t,s+\sqrt{1-t^{2}})\quad\text{by radial principle},$
	$\displaystyle\implies$	$\displaystyle(s,t,\sqrt{1-t^{2}})\quad\text{by affine equivalence},$
	$\displaystyle\implies$	$\displaystyle(t,\sqrt{1-t^{2}})\quad\text{by projection principle},$

which can be dealt with using Bourgain-Demeter decoupling Theorem (Theorem 1 of [BD15]).

The rest of this section is devoted to the proof of Theorem 2.2.

2.2. Preliminary reductions

Denote by $D(\delta)$ the smallest constant such that for test functions $g_{S,T}$ Fourier supported in $\Pi(S,T)$ , we have

(2.8)

\left\lVert\sum_{S\in\mathcal{S}_{\delta}}\sum_{T\in\mathcal{T}_{\delta}}g_{S,T}\right\rVert_{p}\leq D(\delta)(\#\mathcal{S}_{\delta}\#\mathcal{T}_{\delta})^{\alpha}\left\lVert\left\lVert g_{S,T}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S}_{\delta},T\in\mathcal{T}_{\delta})}.

Our goal is to show that

(2.9)

D(\delta)\lesssim_{\varepsilon}\delta^{-\varepsilon},\quad\forall\varepsilon>0.

Let $\varepsilon>0$ . We first partition $[-1,1]^{k}$ into cubes $R_{0}$ of length $\delta^{\varepsilon}$ . Since we allow $\varepsilon$ losses, we may just decouple each $R_{0}$ . By translation in $s$ , we may assume without loss of generality that the centre of $R_{0}$ is the origin. By dividing the last two coordinates by $r(0)$ , we may assume without loss of generality that $r(0)=1$ .

We record that

(2.10)

|r(s)-1|\lesssim\delta^{\varepsilon},\quad\forall s\in R_{0}.

By abuse of notation we may also regard $D(\delta)$ as the best decoupling constant adapted to the smaller cubes.

2.3. Intermediate scale decoupling

Let $\sigma=\delta^{1-\varepsilon}$ . Since $N_{\delta}(\Pi)\subseteq N_{\sigma}(\Pi)$ , applying the definition of $D(\sigma)$ gives

(2.11)

\left\lVert\sum_{S\in\mathcal{S}_{\delta}}\sum_{T\in\mathcal{T}_{\delta}}g_{S,T}\right\rVert_{p}\leq(\#\mathcal{S}_{\sigma}\#\mathcal{T}_{\sigma})^{\alpha}D(\sigma)\left\lVert\left\lVert g_{S^{\prime},T^{\prime}}\right\rVert_{p}\right\rVert_{\ell^{q}(S^{\prime}\in\mathcal{S}_{\sigma},T^{\prime}\in\mathcal{T}_{\sigma})},

where $g_{S^{\prime},T^{\prime}}:=\sum_{S\in\mathcal{S}_{\delta}(S^{\prime})}\sum_{T\in\mathcal{T}_{\delta}(T^{\prime})}g_{S,T}$ , and where $\mathcal{S}_{\delta}(S^{\prime})$ the collection of subsets $S\in\mathcal{S}_{\delta}$ that lie within $S^{\prime}$ , and similarly for $T^{\prime}$ . By Part (2) in the definition of a dyadic mesh, it then suffices to decouple $g_{S^{\prime},T^{\prime}}$ for each fixed pair of intermediate scale subsets $S^{\prime},T^{\prime}$ .

2.4. Approximation

This subsection will be the key to this proof. The main idea as follows. First, we argue by affine invariance that we can let $T^{\prime}$ be centred at $0$ , and that $\nabla\psi(0)=0$ . Next, also by affine invariance, we can let $\nabla r(0)=0$ . Then by Taylor expansion, we can write

r(s)\psi(t)=(1+E(s))(\psi(0)+Q(t)),

where $E(s)$ and $Q(t)$ are $\sigma$ -flat over $S^{\prime}$ and $T^{\prime}$ , respectively, and their gradients are $0$ at $0$ . Thus the last coordinate can be further simplified to $E(s)\psi(0)+Q(t)$ , since $E(s)Q(t)=O(\sigma^{2})\leq\delta$ . But then reversing the affine invariance, the decoupling of $E(s)\psi(0)+Q(t)$ is equivalent to $r(s)L\psi(0)+\psi(t)$ . More precisely, we now need to decouple the manifold parametrised by

(s,t,r(s)L\psi(0)+\psi(t)).

If $r(s)$ is affine, then the decoupling problem reduces to that for

(s,t,\psi(t)).

If $L\psi(0)\neq 0$ , the decoupling problem reduces that for

(s,t,r(s)+\psi(t)).

We now provide the details. Let $T^{\prime}$ be centred at $c$ . Denote by $\phi(t)=\psi(t+c)$ . By a translation, the manifold $(s,r(s)t,r(s)\psi(t))$ can be written as

(2.12)

(s,r(s)(t+c),r(s)\phi(t)),

where $t\in T^{\prime}$ and $T^{\prime}$ is centred at $0$ .

2.4.1. Affine invariance in $\phi$

Write $\phi(t)=\phi(0)+\nabla\phi(0)\cdot t+Q(t)$ . By a linear transformation, we may transform the decoupling problem to the manifold

(2.13)

(s,r(s)(t+c),r(s)(\phi(0)-\nabla\phi(0)\cdot c+Q(t))).

Note that $\phi(0)-\nabla\phi(0)\cdot c=\psi(c)-c\cdot\nabla\psi(t)=L\psi(c)$ . Also, note that $\nabla Q(0)=0$ by this reduction.

2.4.2. Affine invariance in $r$

Write $r(s)=1+\nabla r(0)s+E(s)$ , where $|E(s)|\leq\sigma$ . Using an affine transformation, we may transform the decoupling problem to the manifold

(2.14)

(s,r(s)t+E(s)c,r(s)(L\psi(c)+Q(t))).

2.4.3. Approximation

We now denote $\tau=r(s)t+E(s)c$ . We first claim that $\tau\in 2T^{\prime}$ . Indeed, for any $v\in\mathbb{S}^{l-1}$ , we compute

	$\displaystyle\|(\tau-t)\cdot v\|$	$\displaystyle=\|(r(s)-1)t\cdot v+E(s)c\cdot v\|$
		$\displaystyle\lesssim\delta^{\varepsilon}\|\mathrm{proj}_{v}(T^{\prime})\|+\sigma$
		$\displaystyle\sim\delta^{\varepsilon}\|\mathrm{proj}_{v}(T^{\prime})\|,$

since $T^{\prime}$ has all dimensions at least $\sigma^{1/2}=\delta^{\frac{1-\varepsilon}{2}}$ . This means that $\tau-t\in\delta^{\varepsilon}T^{\prime}$ , and in particular, $\tau\in 2T^{\prime}$ .

We then approximate the manifold in (2.14) by

(2.15)

(s,\tau,r(s)L\psi(c)+Q(\tau)).

Indeed, one need to compute the difference

	$\displaystyle r(s)(L\psi(c)+Q(t))-(r(s)L\psi(c)+Q(\tau))$
	$\displaystyle=(r(s)-1)Q(t)+Q(t)-Q(\tau).$

By (2.4), we have in particular $Q(t)=O(\sigma)$ . Thus by (2.10), we have $(r(s)-1)Q(t)=O(\sigma\delta^{\varepsilon})=O(\delta)$ .

It remains to estimate $Q(t)-Q(\tau)$ . We first apply an affine rescaling $l^{\prime}:[-1,1]^{l}\to 2T^{\prime}$ , and denote $Q^{\prime}=Q\circ l^{\prime}$ . Also, denote $t^{\prime}=l^{\prime-1}t$ , $\tau^{\prime}=l^{\prime-1}\tau$ . Since $\tau-t\in\delta^{\varepsilon}T^{\prime}$ , we have $|\tau^{\prime}-t^{\prime}|\lesssim\delta^{\varepsilon}$ . Also, by (2.4) and affine invariance (see Proposition 4.4), we have that

(2.16)

\sup_{v\in\mathbb{S}^{l-1}}\sup_{x\in[-1,1]^{l}}|v^{T}D^{2}Q^{\prime}(x)v|\lesssim\sigma.

Then by the elementary Lemma 2.3 below, we have

	$\displaystyle\|Q(t)-Q(\tau)\|$	$\displaystyle=\|Q^{\prime}(t^{\prime})-Q^{\prime}(\tau^{\prime})\|$
		$\displaystyle\lesssim\sup_{v\in\mathbb{S}^{l-1}}\sup_{x\in[-1,1]^{l}}\|v^{T}D^{2}Q^{\prime}(x)v\|\|t^{\prime}-\tau^{\prime}\|$
		$\displaystyle\lesssim\sigma\delta^{\varepsilon}=\delta.$

This finishes the approximation assuming the lemma.

Lemma 2.3.

Let $T\subseteq[-1,1]^{l}$ be a parallelogram centred at $0$ . Let $Q:T\to\mathbb{R}$ be a $C^{2}$ function with $\nabla Q(0)=0$ . For all $t\neq\tau\in T$ , denote by $v$ the unit vector in the same direction as $t-\tau$ . Then we have

|Q(t)-Q(\tau)|\lesssim\sup_{t^{\prime}\in T}|v^{T}D^{2}Q(t^{\prime})v||t-\tau|(|t|+|\tau|).

Proof of lemma.

Given $t\neq\tau\in T$ . Then

	$\displaystyle\|Q(t)-Q(\tau)\|$	$\displaystyle=\left\|\int_{0}^{1}\nabla Q((1-u)\tau+ut)\cdot(t-\tau)du\right\|$
		$\displaystyle=\left\|\int_{0}^{1}\nabla Q((1-u)\tau+ut)\cdot(t-\tau)du-\int_{0}^{1}\nabla Q(0)\cdot(t-\tau)du\right\|$
		$\displaystyle\leq\|t-\tau\|\int_{0}^{1}\|\nabla Q((1-u)\tau+ut)-\nabla Q(0)\|du$
		$\displaystyle\lesssim\sup_{t^{\prime}\in T}\|v^{T}D^{2}Q(t^{\prime})v\|\|t-\tau\|(\|t\|+\|\tau\|).$

∎

Therefore, we have reduced the decoupling of $g_{S^{\prime},T^{\prime}}$ to that for the manifold

(2.17)

\{(s,\tau,r(s)L\psi(c)+Q(\tau)):s\in S^{\prime},\tau\in 2T^{\prime}\}.

If either $|L\psi(t)|\sim 1$ or $r(s)$ affine, then we can reverse the affine invariance in the $\tau$ variable, and so we have reduced to the decoupling for $r(s)+\psi(t)$ .

2.5. Applying decoupling assumption

Now we can apply (2.5) with $\varepsilon^{2}$ in place of $\varepsilon$ to get that

(2.18)

\left\lVert g_{S^{\prime},T^{\prime}}\right\rVert_{p}\leq C_{\varepsilon^{2}}\delta^{-\varepsilon^{2}}(\#\mathcal{S}_{\delta}(S^{\prime})\#\mathcal{T}_{\delta}(T^{\prime}))^{\alpha}\left\lVert\left\lVert g_{S,T}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S}_{\delta}(S^{\prime}),T\in\mathcal{T}_{\delta}(T^{\prime}))},\quad\forall\varepsilon>0.

By Part (3) in the definition of a dyadic mesh, plus dyadic decomposition and the triangle inequality (see Proposition 4.9), we may assume that $\#\mathcal{S}_{\delta}(S^{\prime})$ is essentially a constant for each $S^{\prime}$ at a cost of $\log\delta^{-1}\lesssim_{\varepsilon}\delta^{-\varepsilon^{2}}$ . Similarly, we may assume $\#\mathcal{T}_{\delta}(T^{\prime})$ is essentially a constant for each $T^{\prime}$ .

Combined with (2.11), we thus have for some $\tilde{C}_{\varepsilon}$ that

(2.19)

\left\lVert\sum_{S\in\mathcal{S}_{\delta}}\sum_{T\in\mathcal{T}_{\delta}}f_{S,T}\right\rVert_{p}\leq\tilde{C}_{\varepsilon}\delta^{-\varepsilon^{2}}D(\sigma)(\#\mathcal{S}_{\delta}\#\mathcal{T}_{\delta})^{\alpha}\left\lVert\left\lVert f_{S,T}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S}_{\delta},T\in\mathcal{T}_{\delta})}.

Recall $\sigma=\delta^{1-\varepsilon}$ . This implies the following bootstrap inequality:

(2.20)

D(\delta)\leq D(\delta^{1-\varepsilon})\tilde{C}_{\varepsilon}\delta^{-\varepsilon^{2}}.

Iterating this bootstrap inequality gives us $D(\delta)\lesssim_{\varepsilon}\delta^{-2\varepsilon}$ .

3. The degeneracy locating principle

In this section, we introduce and prove the degeneracy locating principle of decoupling. We need a little notation to make this principle more general.

3.1. The setup

Let $1\leq k\leq n-1$ . Let $\mathcal{M}_{0}=\mathcal{M}_{0}(n,k)$ be the collection of manifolds in $\mathbb{R}^{n}$ consisting of $M_{\phi}$ given by the graph of a vector valued $C^{1}$ ⁵⁵5Typically, we consider smooth manifolds. The minimal requirement here is that the $C^{1}$ norms of the functions $\phi$ are bounded uniformly. function $\phi:R_{\phi}\to\mathbb{R}^{n-k}$ for some parallelogram $R_{\phi}\subseteq[-1,1]^{k}$ .

Throughout the rest of of this section, we fix some integer $m\in[1,n]$ , and all flatness conditions are stated in dimension $m-1$ (see Definitions 1.2 and 1.3).

Definition 3.1 (Affine equivalence).

Define an equivalence relation $\sim$ on $\mathcal{M}_{0}$ as follows. $M_{\phi}\sim M_{\psi}$ if there exist affine transformations $L_{1},L_{2}$ with $|\det L_{1}|=1$ such that $L_{1}\circ M_{\phi}\circ L_{2}=M_{\psi}$ . In other words, $M_{\phi}\sim M_{\psi}$ if there exist $A\in\mathbb{R}^{(n-k)\times k}$ with $|\det A|=1$ and $b\in\mathbb{R}^{k}$ such that $\phi(L_{2}(x))-(Ax+b)=\psi(x)$ for all $x\in R_{\psi}$ .

Note that by the formulation of decoupling in Definition 1.1, it is invariant under affine transformations. We summarise this into the following lemma.

Lemma 3.2.

Suppose that $M_{\phi},M_{\psi}\in\mathcal{M}_{0}$ are equivalent. For $\vec{c}\in\mathbb{R}^{n-k}$ with $\sup_{i}|c_{i}|\leq 1$ and $c:=\inf_{i}|c_{i}|>0$ , denote by $\vec{c}\psi$ the $\mathbb{R}^{n-k}$ -valued function

(c_{1}\psi_{1},\dots,c_{n-k}\psi_{n-k}).

For $\delta>0$ and fixed $p,q,\alpha$ , the following two statements are equivalent:

(1)

$N_{\delta}^{\phi}(R_{\phi})$ can be $(\ell^{q}(L^{p}),\alpha)$ decoupled into parallelograms at cost $C$ .
(2)

$N_{\delta}^{\psi}(R_{\psi})$ can be $(\ell^{q}(L^{p}),\alpha)$ decoupled into parallelograms at cost $C$ .

Either statement above implies:

$\quad\,\,\,\,\mathrm{(3)}$ $N^{\vec{c}\psi}_{c\delta}(R_{\psi})$ can be $(\ell^{q}(L^{p}),\alpha)$ decoupled into parallelograms at cost $C$ .

Furthermore, if $c_{i}=c$ for all $1\leq i\leq n-k$ , then all the above are equivalent to

$\quad\,\,\,\,\mathrm{(4)}$ $N_{c\delta}(M_{\vec{c}\psi})$ can be $(\ell^{q}(L^{p}),\alpha)$ decoupled into parallelograms at cost $C$ .

Let $\mathcal{R}_{0}$ be the collection of all parallelograms contained in $[-1,1]^{k}$ .

Definition 3.3 (Rescaling invariance).

Let $\mathcal{M}$ be a subfamily of $\mathcal{M}_{0}$ . A collection $\mathcal{R}\subseteq\mathcal{R}_{0}$ of parallelograms in $[-1,1]^{k}$ is said to be rescaling invariant with respect to $\mathcal{M}$ if the following holds. For every $\sigma\in(0,1]$ , $M_{\phi}\in\mathcal{M}$ and every $(\phi,\sigma)$ -flat parallelogram $R\in\mathcal{R}$ with $R\subseteq R_{\phi}$ , the manifold $M_{\phi}\cap(R\times\mathbb{R}^{n-k})$ , as an element of $\mathcal{M}_{0}$ , is equivalent to $M_{\vec{\sigma}\psi}$ for some $M_{\psi}\in\mathcal{M}$ and $\vec{\sigma}\in\mathbb{R}^{n-k}$ with $\inf_{i}\sigma_{i}=\sigma$ .

Moreover, a collection $\mathcal{R}\subseteq\mathcal{R}_{0}$ of parallelograms in $[-1,1]^{k}$ is said to be non-rotational rescaling invariant with respect to $\mathcal{M}$ if we further assume that the transformation $L_{2}$ in the equivalence definition $M_{\phi}\cap(R\times\mathbb{R}^{n-k})\sim M_{\vec{\sigma}\psi}$ does not involve rotation, i.e. $DL_{2}$ is a diagonal matrix.

In simple words, every flat piece of a manifold in $\mathcal{M}$ over $R\in\mathcal{R}$ can be rescaled to become another manifold in $\mathcal{M}$ . We illustrate Lemma 3.2 and Definition 3.3 using the following example. Consider the moment curve $M_{\phi}$ given by $(s,\phi(s))=(s,s^{2},s^{3})$ , $s\in[-1,1]$ in $\mathbb{R}^{3}$ . Let $\sigma\in(0,1)$ .

•

The interval $R=[0,\sigma^{1/3}]$ is $(\phi,\sigma)$ -flat in dimension $2$ . The manifold $M_{\phi}\cap(R\times\mathbb{R}^{2})$ can be rescaled to become

$\{(s,\sigma^{2/3}s^{2},\sigma s^{3}):s\in[-1,1]\},$

which is $M_{\vec{\sigma}\phi}$ where $\vec{\sigma}:=(\sigma^{2/3},\sigma)$ so that $\inf_{i}\sigma_{i}=\sigma$ . A similar argument shows that other $(\phi,\sigma)$ -flat intervals behave similarly. Therefore, the collection of all closed intervals is rescaling invariance with respect to the singleton family $\mathcal{M}=\{\mathcal{M}_{\phi}\}$ . In this case, Lemma 3.2 is exactly the key rescaling property of the moment curve used in [BDG16].
•

The interval $R=[0,\sigma^{1/2}]$ is $(\phi,\sigma)$ -flat in dimension $1$ . The manifold $M_{\phi}\cap(R\times\mathbb{R}^{2})$ can be rescaled to become

$\{(s,\sigma s^{2},\sigma^{3/2}s^{3}):s\in[-1,1]\},$

which is $M_{\vec{\sigma}\phi}$ where $\vec{\sigma}:=(\sigma,\sigma)$ so that $\inf_{i}\sigma_{i}=\sigma$ , and $\psi(s)=(s^{2},\sigma^{1/2}s^{3})$ . The idea here is that we are ignoring the torsion given by the third coordinate $s^{3}$ . We can still get $\ell^{2}(L^{6})$ decoupling in this case. This can be seen by projection principle and decoupling for the parabola. In this setting, the collection of all closed intervals is rescaling invariance with respect to the family $\mathcal{M}$ containing all manifolds $M_{\phi}$ such that $R_{\phi}=[-1,1]$ , and $\phi$ is of the form $(s^{2},\phi_{2}(s))$ , where $\phi_{2}(s)$ is an arbitrary continuous function.

Definition 3.4 (Degeneracy determinant).

Let $\mathcal{M}$ be a subfamily of $\mathcal{M}_{0}$ and $\mathcal{R}$ be rescaling invariant with respect to $\mathcal{M}$ . We say an operator $H$ on $\mathcal{M}$ that sends $M_{\phi}\in\mathcal{M}$ to a smooth function on $R_{\phi}$ is a degeneracy determinant if the following holds.

(1)

For every $M_{\phi}\in\mathcal{M}$ , $HM_{\phi}$ is Lipschitz, namely,

$|HM_{\phi}(x)-HM_{\phi}(y)|\leq C_{0}|x-y|,\quad\forall x,y\in R_{\phi},$

where $C_{0}\geq 1$ depends on $\mathcal{M},\mathcal{R}$ only.
(2)

There are constants $C\geq 1$ and $\beta\in(0,1]$ , both depending on $\mathcal{M},\mathcal{R}$ only, such that for any $M_{\phi}\in\mathcal{M}$ and $R\in\mathcal{R}$ , if we have $\left\lVert HM_{\phi}\right\rVert_{L^{\infty}(R)}\leq\sigma$ , then there exists $M_{\psi}\in\mathcal{M}$ such that $R\subseteq R_{\psi}$ , $H\psi\equiv 0$ , and

(3.1) $\|\phi_{i}-\psi_{i}\|_{L^{\infty}(R)}\leq C\,\sigma^{\beta},\quad\forall 1\leq i\leq n-k.$

3.2. Statement of theorem

Theorem 3.5.

Let $H$ be a degeneracy determinant defined with respect to a pair $\mathcal{M},\mathcal{R}$ . Let $p,q\in[2,\infty]$ , $\alpha\in[0,\infty)$ , $\varepsilon>0$ , and suppose we have the following assumptions, where all implicit constants and overlap bounds $B$ below depend only on $\mathcal{M},\mathcal{R},\varepsilon,p,q,\alpha$ .

(1)

(Sublevel set decoupling) For each $M_{\phi}\in\mathcal{M}$ and each $\sigma\in(0,1]$ , the set

$\{x\in R_{\phi}:|HM_{\phi}(x)|\leq\sigma\}$

can be $(\ell^{q}(L^{p}),\alpha)$ decoupled into parallelograms uniformly at a cost of $O_{\varepsilon}(\sigma^{-\varepsilon})$ , on each of which $|HM_{\phi}|\lesssim\sigma$ .
(2)

(Degenerate decoupling) If $M_{\phi}\in\mathcal{M}$ is such that $HM_{\phi}\equiv 0$ , then for each $\sigma\in(0,1]$ , $R_{\phi}$ can be graphically $(\ell^{q}(L^{p}),\alpha)$ -decoupled into $(\phi,\sigma)$ -flat parallelograms $R\in\mathcal{R}_{deg}(\phi)\subseteq\mathcal{R}$ at a cost of $O(\sigma^{-\varepsilon}$ ), and each $\mathcal{R}_{deg}(\phi)$ is $B$ -overlapping.
(3)

(Nondegnerate decoupling) If $M_{\phi}\in\mathcal{M}$ such that $\inf_{R_{\phi}}|HM_{\phi}|\geq K^{-1}$ , then for each $\delta>0$ , $R_{\phi}$ can be graphically decoupled into $(\phi,\delta)$ -flat parallelograms $R\in\mathcal{R}_{non}(\phi)$ at a cost of $K^{O(1)}\delta^{-\varepsilon}$ , and each $\mathcal{R}_{non}(\phi)$ is $B$ -overlapping.

Then for each $M_{\phi}\in\mathcal{M}$ , $[-1,1]^{k}$ can be graphically $(\ell^{q}(L^{p}),\alpha)$ decoupled into $(\phi,\delta)$ -flat parallelograms $\mathcal{R}(\phi)$ , with the implicit constants and overlap bounds depending only on $\mathcal{M},\mathcal{R},\varepsilon,p,q,\alpha$ .

Moreover, if the collection $\mathcal{R}$ is non-rotational rescaling invariant with respect to $\mathcal{M}$ , and each collection $\mathcal{R}_{deg}(\phi),\mathcal{R}_{non}(\phi)$ in Assumptions (2) and (3) above has non-overlapping interiors, the final collection $\mathcal{R}(\phi)$ has non-overlapping interiors.

Note that only the second assumption requires decoupling into the rescaling invariant subcollection $\mathcal{R}$ .

It is straight forward to check that the following applications of the degeneracy locating principle in Corollaries 1.4 to 1.6 satisfy the full assumptions in Theorem 3.5.

Corollary	Manifolds $M_{\phi}\in\mathcal{M}$	Parallelograms in $\mathcal{R}$	$HM_{\phi}$
1.4	$\phi(s,t^{\prime})=P(s)+\psi(t^{\prime})$ ,	$(s,t^{\prime})\in I\times Q$ ,	$P^{\prime\prime}(s)$
	$P$ polynomial, $\psi(t^{\prime})=\sqrt{1-\|t^{\prime}\|^{2}}$	$Q$ axis-parallel square
1.5	$\phi(x_{1},x_{2})=\phi_{1}(x_{1})+\phi_{2}(x_{2})$ ⁶⁶6For simplicity, we only write down the case for the family $\mathcal{M}_{1}^{n}$ , $J=2,n=3,4,5$ . The other families are straightforward generalisations of this case.	$(x_{1},x_{2})\in R\times Q$ ,	$\det D^{2}\phi_{1}$
	polynomial, $\|\det D^{2}\phi_{2}\|\sim 1$	$Q$ axis-parallel square
1.6	$\phi:\mathbb{R}\to\mathbb{R}^{n-1}$ polynomials,	all intervals	$\det((W_{m}\phi)$
	$\|HM_{\phi}\|\sim 1$ or $M_{\phi}$ is straight line	i.e. $\mathcal{R}_{0}$	$(W_{m}\phi)^{T})$

Table 3. Applications of the degeneracy locating principles in the corollaries. Here polynomial means a family of polynomial in appropriate number of variables of degree at most

d

and bounded coefficients.

Regarding overlapping conditions, it is clear that the families $\mathcal{R}$ in Corollaries 1.4 and 1.6 above are non-rotational invariant with respect to the corresponding families $\mathcal{M}$ . Therefore, these manifolds can be decoupled into a collection of rectangles having non-overlapping interiors. By a similar construction to in Proposition 3.1 of [Yan21], $(\phi,\delta)$ -flat rectangles are essentially those in the statements of the corollaries.

The rest of this section is devoted to the proof of Theorem 3.5. It features an induction on scales argument similar to that of Section 5 of [LY23]. The only essential difference here is that we choose a different scale $K$ (the scale $M$ in [LY23]), which allows us to improve the overlapping bound of the parallelograms from $C_{\varepsilon}\delta^{-\varepsilon}$ to $C_{\varepsilon}$ . This improvement is nontrivial, in view of Appendix A of [GMO24].

For the rest of this section, we suppress the subscripts $j,k,p,q,\alpha$ unless otherwise specified.

3.3. The base case

We may assume $\delta\ll_{\varepsilon}1$ , otherwise the decoupling inequality is trivial.

Let $K=\delta^{-\varepsilon}$ . Our induction hypothesis is that the conclusion holds at all coarser scales $\delta^{\prime}\gtrsim K^{\beta}\delta$ , where $\beta$ is as in Definition 3.4.

3.4. Degenerate and nondegenerate parts

Partition $R_{\phi}$ into two parts $R_{\text{non}}$ and $R_{\text{degen}}$ as follows:

	$\displaystyle R_{\text{non}}:=\{x\in R_{\phi}:\|HM_{\phi}(x)\|>K^{-1}\}$
	$\displaystyle R_{\text{degen}}:=\{x\in R_{\phi}:\|HM_{\phi}(x)\|\leq K^{-1}\}.$

We first deal with the subset $R_{\text{non}}$ . We can cover it by squares $Q$ of side length $\delta^{1/2}$ , and if $\delta\ll 1$ , then by the Lipschitz assumption in Definition 3.4, we have $|HM_{\phi}(x)|\geq K^{-1}/2$ on each such square. Then we can decouple $R_{\text{non}}$ by these squares $Q$ , by the nondegenerate decoupling assumption (3).

3.5. The degenerate subset

Now we deal with the degenerate subset, with the hope of reducing to the induction hypothesis.

3.5.1. Sublevel set decoupling

We apply the sublevel set decoupling assumption (1) at scale $\sigma:=K^{-1}$ to decouple $S_{\text{degen}}$ into parallelograms $R$ on each of which $\|HM_{\phi}\|_{L^{\infty}(R)}\lesssim K^{-1}$ . It remains to further decouple each $R$ into smaller $(\phi,\delta)$ -flat parallelograms.

3.5.2. Approximation and Degenerate decoupling

By the definition of degeneracy determinant $H$ , since $\|HM_{\phi}\|_{L^{\infty}(R)}\lesssim K^{-1}$ , there exists $M_{\psi}\in\mathcal{M}$ such that $H\psi\equiv 0$ and $\left\lVert\phi-\psi\right\rVert_{\infty}\lesssim K^{-\beta}$ . Apply the degenerate decoupling assumption (2) at scale $\sigma\sim K^{-\beta}$ to the function $\psi$ to graphically decouple $R$ into $(\psi,O(K^{-\beta}))$ -flat parallelograms $R^{\prime}\in\mathcal{R}$ . Thus $R^{\prime}$ is also $(\phi,O(K^{-\beta}))$ -flat.

3.5.3. Rescaling

By the definition of rescaling invariance of $\mathcal{R}$ , we see that $M_{\phi}\cap(R^{\prime}\times\mathbb{R}^{n-k})$ is equivalent to $M_{CK^{-\beta}\psi}$ for some $M_{\psi}\in\mathcal{M}$ . Hence, by Lemma 3.2, the graphical decoupling of $R_{\psi}$ into $(\psi,CK^{\beta}\delta)$ -flat parallelograms is implied by the graphical decoupling of $R^{\prime}$ into $(\phi,\delta)$ -flat parallelograms. Thus we have reduced to decoupling at a larger scale, for which the induction hypothesis can be applied.

3.6. Decoupling inequality

We briefly describe the proof of the required decoupling inequality, and point out the differences from Section 5.3 of [LY23].

Denote by $D(\delta)$ the cost of graphical decoupling of $[-1,1]^{k}$ into $(\phi,\delta)$ -flat parallelograms. Our goal is to show that $D(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon}$ .

The first difference is that we now need to combine $\ell^{q}(L^{p})$ decoupling inequalities obtained at each step, which is done by a simple dyadic decomposition argument with a logarithmic loss $O(\log K)$ at each step (see Proposition 4.9). This is acceptable since we already lose $K^{\varepsilon}$ at each step of the induction. The second difference is the choice of $K=\delta^{-\varepsilon}$ instead of $K\sim_{\varepsilon}1$ in [LY23] (where we used $M$ in place of $K$ ). Carrying out the same proof and combining the decoupling inequalities obtained from the nondegenerate and degenerate parts in Sections 3.4 and 3.5, respectively, we obtain the following bootstrap inequality:

(3.2)

D(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon}+C_{\varepsilon}D(K^{\beta}\delta)K^{\varepsilon}\leq C_{\varepsilon}\delta^{-\varepsilon}+C_{\varepsilon}D(K^{\beta}\delta)\delta^{-\varepsilon^{2}}.

(Here we have omitted the unimportant implicit constants.)

3.6.1. Iteration

Iterating inequality (3.2) for $N$ times, we obtain

\displaystyle D(\delta)

\displaystyle\leq C_{\varepsilon}\delta^{-\varepsilon}(1+\delta^{-\varepsilon^{2}}+\delta^{-2\varepsilon^{2}}+\cdots+\delta^{-N\varepsilon^{2}})+\delta^{-N\varepsilon^{2}}D(K^{N\beta}\delta).

We will stop this iteration once we reach $K^{N\beta}\delta\sim 1$ , that is, when

(3.3)

N\sim(\beta\varepsilon)^{-1}.

In this case we have $D(K^{\beta N}\delta)\sim 1$ , and

1+\delta^{-\varepsilon^{2}}+\delta^{-2\varepsilon^{2}}+\cdots+\delta^{-N\varepsilon^{2}}+\delta^{-N\varepsilon^{2}}D(K^{N\beta}\delta)\lesssim\delta^{-O(\varepsilon)},

which finishes the proof of the decoupling inequality.

3.7. Analysis of overlap

There are two main sources of overlaps between parallelograms, namely, overlaps between parallelograms created within one iteration, and overlaps between different iterations.

The former kind of overlap is guaranteed by the overlap condition in the assumption of Theorem 3.5. For the second kind of overlap, now we can see that the choice of $K=\delta^{-\varepsilon}$ leads to $N=O(1/\varepsilon)$ , and so we may just roughly bound the overlapping number by $O_{\varepsilon}(1)$ .

We remark that this is a slight improvement of the corresponding argument in [LY23], where we were only able to obtain $O(\delta^{-\varepsilon})$ overlap.

Moreover, if the rescaling in Subsection 3.5.3 does not involve rotation, the final collection $\mathcal{R}(\phi)$ will have non-overlapping interior if every decoupling steps generates families of parallelograms with non-overlapping interior.

4. Appendix

4.1. Facts about parallelograms

We also often need to transform many geometric objects into parallelograms, which are much easier to handle.

Proposition 4.1.

The following statements are true.

(1)

If $P$ is a convex body in $\mathbb{R}^{n}$ , then there exist a parallelogram $T$ and a constant $C=C_{n}$ such that $C^{-1}T\subseteq P\subseteq CT$ .
(2)

Let $T_{1},T_{2}$ be parallelograms in $\mathbb{R}^{n}$ with nonempty intersection. Then there exists some degree constant $C=10n$ and some parallelogram $T$ such that

$T_{1}\cap T_{2}\subseteq T\subseteq CT_{1}\cap CT_{2},$

Here $CT$ means the dilation of $T$ with respect to its centre by a factor of $C$ .

This follows from an easy application of the John Ellipsoid Theorem [Joh14]. Alternatively, the reader could read Proposition 4.10 of [LY23] for an elementary proof in the case $n=2$ .

4.2. Facts about polynomials

The following facts about polynomials will be used extensively, and are the key to many of our analysis of polynomials.

Proposition 4.2.

For any $n$ -variate real polynomial $P=\sum_{\alpha}c_{\alpha}x^{\alpha}$ of degree at most $d$ , we have the following relation:

\sup_{[-1,1]^{n}}|P|\sim_{n,d}\max_{\alpha}|c_{\alpha}|.

As a result, we also have the following relations:

	$\displaystyle\sup_{[-1,1]^{n}}\|P\|\sim_{n,d}\sum_{\alpha}\|c_{\alpha}\|$
	$\displaystyle\sup_{[-C,C]^{n}}\|P\|\sim_{n,d}\sup_{B^{n}(0,C)}\|P\|\lesssim_{n,d}C^{d}\sup_{[-1,1]^{n}}\|P\|,$

for any constant $C\geq 1$ .

For a proof of this proposition, the reader may consult [Kel28].

Corollary 4.3.

For any parallelogram $T\subset\mathbb{R}^{n}$ and any constant $C\geq 1$ ,

\sup_{T}|P|\leq\sup_{CT}|P|\lesssim_{n,d}C^{d}\sup_{T}|P|.

Proof.

Let $l$ be an affine transformation that maps $[-1,1]^{n}$ to $T$ . Apply Proposition 4.2 to $P\circ l$ to get

\sup_{[-1,1]^{n}}|P\circ l|\lesssim_{n,d}\sup_{[-C,C]^{n}}C^{d}|P\circ l|,

which implies the desired relation. ∎

4.3. Flatness of functions

Since we only need the different notions of flatness in Section 2, we just focus on flatness in dimension $n-1$ , namely, for hypersurfaces.

Let $\phi:[-1,1]^{n-1}\to\mathbb{R}$ be a continuous function. Let $\Omega$ be a parallelogram contained in $[-1,1]^{n}$ and $\delta>0$ . We say $\Omega$ is $(\phi,\delta)$ -flat

(1)

in the first sense, if there exists an affine function $\lambda:\mathbb{R}^{n}\to\mathbb{R}$ such that

(4.1) $\sup_{x\in\Omega}|\phi(x)-\lambda(x)|\leq\delta;$
(2)

in the second sense, if $\phi$ is $C^{1}$ , and

(4.2) $\sup_{x,y\in\Omega}|\phi(y)-\phi(x)-\nabla\phi(x)\cdot(y-x)|\leq\delta;$

(3)

in the third sense, if $\phi$ is $C^{2}$ , and for each $v\in\mathbb{S}^{n-1}$ we have

(4.3)

\sup_{\begin{subarray}{c}x\in\Omega,v\in\mathbb{S}^{n-1}\\ t\in\mathbb{R}:x+tv\in\Omega\end{subarray}}\left|v^{T}D^{2}\phi(x)v\right||t|^{2}\leq\delta.

Equivalently, this means that given any line segment $\mathcal{L}$ in the direction of $v$ and contained $\Omega$ , we have

(4.4)

\sup_{x\in\mathcal{L}}\left|v^{T}D^{2}\phi(x)v\right||\Lambda|^{2}\leq\delta.

Note that all the above flatness are invariant under affine transformations. Namely, if $L:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is an affine transformation, then a set $\Omega$ is $(\phi,\delta)$ -flat if and only if $L\Omega$ is $(\phi\circ L,\delta)$ -flat, in all the three senses above. Thus in particular, we have

Proposition 4.4.

If $\phi$ is $\delta$ -flat over $[-1,1]^{n}$ in the third sense, then we have $\sup_{x\in[-1,1]^{n}}|D^{2}\phi(x)|\lesssim_{n}\delta$ .

Fix $\phi,\Omega,\delta$ . Below we denote by $F1(\delta),F2(\delta),F3(\delta)$ the property that $\phi$ is $\delta$ -flat over $\Omega$ in the first, second, third sense, respectively.

Proposition 4.5.

We have $F3(\delta)\implies F2(\delta)\implies F1(\delta)$ . Conversely, if $\phi$ is a polynomial of degree $d$ and $\Omega$ is a parallelogram, then we also have $F1(\delta)\implies F3(C_{d,n}\delta)$ , where $C_{d,n}$ is a constant depending only on $d,n$ . However, $F1(\delta)\implies F2(C\delta)$ and $F2(\delta)\implies F3(C\delta)$ are both false, for any fixed constant $C=C_{n}$ depending on $n$ only.

Proof.

$F3(\delta)\implies F2(\delta)\implies F1(\delta)$ is trivial. For the implication $F1(\delta)\implies F3(C\delta)$ , we let $\Lambda:[-1,1]^{n}\to\Omega$ be an affine bijection. Then by the assumption,

(4.5)

\sup_{x\in[-1,1]^{n}}|\phi(\Lambda(x))-\lambda(\Lambda(x))|\lesssim_{d,n}\delta.

Using Proposition 4.2, we see all coefficients of $\phi(\Lambda(x))-\lambda(\Lambda(x))$ are $O_{d,n}(\delta)$ . Thus, for any direction $v\in\mathbb{S}^{n-1}$ , we have

\sup_{x\in[-1,1]^{n}}|v^{T}D^{2}(\phi\circ\Lambda)(x)v|\lesssim_{d,n}\delta.

Rescaling back, this gives $F3(C_{d,n}\delta)$ .

Lastly, using the simple example $\phi(t)=|t|^{2d}$ when $d$ can be arbitrarily large, we observe that $F1(\delta)\implies F2(C\delta)$ and $F2(\delta)\implies F3(C\delta)$ are both false, for any fixed constant $C=C_{n}$ depending on $n$ only. ∎

4.4. Flatness of parametrised surfaces

Definition 4.6.

We say a subset $\Omega\subseteq\mathbb{R}^{n}$ is $\delta$ -flat if there exists a hyperplane $H$ such that $\Omega\subseteq N_{\delta}(H)$ .

Proposition 4.7.

Let $r,\psi$ be continuous functions, and let $\delta>0$ . Assume $S$ is $(r,\delta)$ -flat in sense of (4.1). Assume also $T$ is $(\psi,\delta)$ -flat in sense of (4.1). Then the subsets

	$\displaystyle\{(s,r(s)t,r(s)\psi(t):s\in S,t\in T\}$
	$\displaystyle\{(s,t,r(s)+\psi(t)):s\in S,t\in T\}$

are $O(\delta)$ -flat in the sense of Definition 4.6. Here the implicit constant depends only on the sup norms of $r,\psi$ .

Proof.

By flatness in sense of (4.1), we may assume $r,\psi$ are affine functions. For the first subset, we need to find a hyperplane $H$ such that

\{(s,r(s)t,r(s)\psi(t):s\in S,t\in T\}\subseteq N_{O(\delta)}(H).

Write $\psi(t)=At+b$ for a matrix $A$ and a vector $b$ . Defining $t^{\prime}=r(s)t$ , we compute directly that

r(s)\psi\left(\frac{t^{\prime}}{r(s)}\right)=At^{\prime}+r(s)b,

which is an affine function in $s,t^{\prime}$ . Thus we may just take the hyperplane $H$ to be parametrised by

(4.6)

(s,t^{\prime},At^{\prime}+r(s)b).

The argument for the latter set is even easier. ∎

4.5. A more general small Hessian proposition

We provide a slight generalisation of Theorem 3.2 of [LY23].

Proposition 4.8.

Let $P$ be a bivariate polynomial of degree at most $d$ and coefficients bounded by $1$ . Let $R\subseteq[-1,1]^{2}$ be a parallelogram, and suppose for some $\sigma\in(0,1]$ we have $|\det D^{2}P|\leq\sigma$ on $R$ . Then there exists another bivariate polynomial $Q$ of degree at most $d$ with coefficients bounded by $O_{d}(1)$ , such that $\det D^{2}Q\equiv 0$ , and $\left\lVert P-Q\right\rVert_{L^{\infty}(R)}\lesssim_{d}\sigma^{\beta}$ , where $\beta=\beta(d)\in(0,1]$ . Moreover, $\nabla_{u}Q\equiv\vec{c}$ for some $|u|=1$ .

The case $R=[-1,1]^{2}$ is equivalent to the statement of Theorem 3.2 of [LY23].

Proof.

Let $\alpha=\alpha(d)\in(0,1]$ be the exponent given in Theorem 3.2 of [LY23]. We consider two cases according to whether $|R|\leq\sigma^{\frac{\alpha}{d}}$ or not.

If $|R|\leq\sigma^{\frac{\alpha}{d}}$ , then the shorter dimension of $R$ is $\leq\sigma^{\frac{\alpha}{2d}}$ . Let $L$ be the unbounded straight line along one of the longer edges of $R$ , and let $\pi_{L}$ be the orthogonal projection from $\mathbb{R}^{2}$ onto $L$ . Then we simply define $Q(x)=P(\pi_{L}x)$ .

Then $\det D^{2}Q\equiv 0$ , $Q$ has all coefficients bounded by $O_{d}(1)$ , and $\left\lVert P-Q\right\rVert_{L^{\infty}(R)}\lesssim_{d}\sigma^{\frac{\alpha}{2d}}$ .

If $|R|>\sigma^{\frac{\alpha}{d}}$ , then both dimensions of $R$ must be at least $\sigma^{\frac{\alpha}{d}}/10$ . Let $\Lambda:[-1,1]^{2}\to R$ be an affine bijection. With this, denote

P_{R}(x):=P(\Lambda x)-P(\Lambda 0)-\nabla P(\Lambda 0)\cdot x.

Then $P_{R}$ has degree at most $d$ with coefficients bounded by $O_{d}(1)$ , and $|\det D^{2}P_{R}|\leq\sigma$ on $[-1,1]^{2}$ . Apply Theorem 3.2 of [LY23] to find a rotation $\rho:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that

(P_{R}\circ\rho)(x)=A(\pi x)+\sigma^{\alpha}B(x_{1},x_{2}),

where $\pi x:=x_{1}$ and $A,B$ are polynomials of degree at most $d$ and coefficients bounded by $O_{d}(1)$ . This means $P_{R}(x)=A(\rho^{-1}\pi x)+\sigma^{\alpha}B(\rho^{-1}x)$ , and so

	$\displaystyle P(x)$	$\displaystyle=P_{R}(\Lambda^{-1}x)+P(\Lambda 0)+\nabla P(\Lambda 0)\cdot(\Lambda^{-1}x)$
		$\displaystyle=A(\rho^{-1}\pi\Lambda^{-1}x)+\sigma^{\alpha}B(\rho^{-1}\Lambda^{-1}x)+P(\Lambda 0)+\nabla P(\Lambda 0)\cdot(\Lambda^{-1}x).$

Let

Q(x)=A(\rho^{-1}\pi\Lambda^{-1}x)+P(\Lambda 0)+\nabla P(\Lambda 0)\cdot(\Lambda^{-1}x).

Then $Q$ is a polynomial of degree at most $d$ , $\det D^{2}Q\equiv 0$ , and $\left\lVert P-Q\right\rVert_{L^{\infty}(R)}\lesssim_{d}\sigma^{\alpha}$ . It remains to show that all coefficients of $Q$ are bounded by $O_{d}(1)$ . But since $P$ has coefficients bounded by $1$ , it remains to show all coefficients of $\sigma^{\alpha}B(\rho^{-1}\Lambda^{-1}x)$ are bounded above by $O_{d}(1)$ . But this follows from the assumption that both dimensions of $R$ are at least $\sigma^{\frac{\alpha}{d}}/10$ .

Thus, we see that taking $\beta=\frac{\alpha}{2d}$ suffices. ∎

4.6. Combining decoupling inequalities

We provide a proof of how we can combine different $\ell^{q}(L^{p})$ decoupling inequalities at different levels. For simplicity of presentation, we only provide the case when we have strict partitions, as the case of boundedly overlapping covers is similar.

Proposition 4.9.

Let $\mathcal{T}$ be a partition of a parallelogram $P_{0}$ by parallelograms $T$ , and for each $T$ , let $\mathcal{S}(T)$ be a partition of $T$ by parallelograms $S$ . Denote $\mathcal{S}=\cup_{T\in\mathcal{T}}\mathcal{S}(T)$ , which is a partition of $P_{0}$ at a finer level. Then for all $p,q,\alpha$ we have

\mathrm{Dec}(P_{0},\mathcal{S},p,q,\alpha)\lesssim\log(\#\mathcal{S})\mathrm{Dec}(P_{0},\mathcal{T},p,q,\alpha)\sup_{T\in\mathcal{T}}\mathrm{Dec}(T,\mathcal{S}(T),p,q,\alpha).

Proof.

For each $k\geq 1$ , denote

\mathcal{T}_{k}:=\{T\in\mathcal{T}:\#\mathcal{S}(T)\in[2^{k-1},2^{k})\}.

Then there are $O(\log(\#\mathcal{S}))$ many such $k$ ’s. Given test functions $f_{S}$ each Fourier supported on $S$ , and denote $f_{T}:=\sum_{S\in\mathcal{S}(T)}f_{S}$ , $f:=\sum_{T\in\mathcal{T}}f_{T}$ . Then we have

	$\displaystyle\left\lVert f\right\rVert_{p}$	$\displaystyle\leq\sum_{k}\left\lVert f_{k}\right\rVert_{p}$
		$\displaystyle\leq\sum_{k}\mathrm{Dec}(P_{0},\mathcal{T},p,q,\alpha)(\#\mathcal{T}_{k})^{\alpha}\left\lVert\left\lVert f_{T}\right\rVert_{p}\right\rVert_{\ell^{q}(T\in\mathcal{T}_{k})}$
		$\displaystyle\leq\sum_{k}\mathrm{Dec}(P_{0},\mathcal{T},p,q,\alpha)(\#\mathcal{T}_{k})^{\alpha}2^{k\alpha}\sup_{T\in\mathcal{T}}\mathrm{Dec}(T,\mathcal{S}(T),p,q,\alpha)\left\lVert\left\lVert f_{S}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S})}$
		$\displaystyle\leq\sum_{k}\mathrm{Dec}(P_{0},\mathcal{T},p,q,\alpha)(\#\mathcal{S})^{\alpha}\sup_{T\in\mathcal{T}}\mathrm{Dec}(T,\mathcal{S}(T),p,q,\alpha)\left\lVert\left\lVert f_{S}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S})}$
		$\displaystyle\lesssim\log(\#\mathcal{S})\mathrm{Dec}(P_{0},\mathcal{T},p,q,\alpha)(\#\mathcal{S})^{\alpha}\sup_{T\in\mathcal{T}}\mathrm{Dec}(T,\mathcal{S}(T),p,q,\alpha)\left\lVert\left\lVert f_{S}\right\rVert_{p}\right\rVert_{\ell^{q}(S\in\mathcal{S})},$

from which the result follows. ∎

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	$\displaystyle\|(\tau-t)\cdot v\|$	$\displaystyle=\|(r(s)-1)t\cdot v+E(s)c\cdot v\|$
		$\displaystyle\lesssim\delta^{\varepsilon}\|\mathrm{proj}_{v}(T^{\prime})\|+\sigma$
		$\displaystyle\sim\delta^{\varepsilon}\|\mathrm{proj}_{v}(T^{\prime})\|,$

	$\displaystyle\|Q(t)-Q(\tau)\|$	$\displaystyle=\|Q^{\prime}(t^{\prime})-Q^{\prime}(\tau^{\prime})\|$
		$\displaystyle\lesssim\sup_{v\in\mathbb{S}^{l-1}}\sup_{x\in[-1,1]^{l}}\|v^{T}D^{2}Q^{\prime}(x)v\|\|t^{\prime}-\tau^{\prime}\|$
		$\displaystyle\lesssim\sigma\delta^{\varepsilon}=\delta.$

	$\displaystyle\|Q(t)-Q(\tau)\|$	$\displaystyle=\left\|\int_{0}^{1}\nabla Q((1-u)\tau+ut)\cdot(t-\tau)du\right\|$
		$\displaystyle=\left\|\int_{0}^{1}\nabla Q((1-u)\tau+ut)\cdot(t-\tau)du-\int_{0}^{1}\nabla Q(0)\cdot(t-\tau)du\right\|$
		$\displaystyle\leq\|t-\tau\|\int_{0}^{1}\|\nabla Q((1-u)\tau+ut)-\nabla Q(0)\|du$
		$\displaystyle\lesssim\sup_{t^{\prime}\in T}\|v^{T}D^{2}Q(t^{\prime})v\|\|t-\tau\|(\|t\|+\|\tau\|).$

Two principles of decoupling

Abstract.

1. Introduction

1.1. Decoupling inequalities

Definition 1.1.

1.1.1. Decoupling for manifolds in ℝn\mathbb{R}^{n}

Definition 1.2 (δ\delta-flat subsets).

Definition 1.3.

1.2. A radial principle of decoupling

1.3. A degeneracy locating principle of decoupling

1.4. Two different scales of induction

1.5. Corollaries

Corollary 1.4 (Decoupling for radial surfaces).

Proof of Corollary 1.4 using the simplified versions of the two principles.

Corollary 1.5 (Decoupling for graphs of additively separable functions).

Proof of Corollary 1.5 using the simplified version of degeneracy locating principle.

Corollary 1.6 (Decoupling for analytic curves in ℝn\mathbb{R}^{n}).

Proof of Corollary 1.6 using the simplified version of degeneracy locating principle..

1.6. Outline of the paper

1.7. Notation

1.7.1. Equivalence of geometric objects

1.7.2. Enlarged overlap

Definition 1.7.

1.8. Acknowledgement

2. The radial principle

2.1. Statement of theorem

Definition 2.1 (Dyadic mesh).

Theorem 2.2 (Radial principle of decoupling).

2.2. Preliminary reductions

2.3. Intermediate scale decoupling

2.4. Approximation

2.4.1. Affine invariance in ϕ\phi

2.4.2. Affine invariance in rr

2.4.3. Approximation

Lemma 2.3.

Proof of lemma.

2.5. Applying decoupling assumption

3. The degeneracy locating principle

3.1. The setup

Definition 3.1 (Affine equivalence).

Lemma 3.2.

Definition 3.3 (Rescaling invariance).

Definition 3.4 (Degeneracy determinant).

3.2. Statement of theorem

Theorem 3.5.

3.3. The base case

3.4. Degenerate and nondegenerate parts

3.5. The degenerate subset

3.5.1. Sublevel set decoupling

3.5.2. Approximation and Degenerate decoupling

3.5.3. Rescaling

3.6. Decoupling inequality

3.6.1. Iteration

3.7. Analysis of overlap

4. Appendix

4.1. Facts about parallelograms

Proposition 4.1.

4.2. Facts about polynomials

Proposition 4.2.

Corollary 4.3.

Proof.

4.3. Flatness of functions

Proposition 4.4.

Proposition 4.5.

Proof.

4.4. Flatness of parametrised surfaces

Definition 4.6.

Proposition 4.7.

Proof.

4.5. A more general small Hessian proposition

Proposition 4.8.

Proof.

4.6. Combining decoupling inequalities

Proposition 4.9.

Proof.

References

1.1.1. Decoupling for manifolds in $\mathbb{R}^{n}$

Definition 1.2 ( $\delta$ -flat subsets).

Corollary 1.6 (Decoupling for analytic curves in $\mathbb{R}^{n}$ ).

2.4.1. Affine invariance in $\phi$

2.4.2. Affine invariance in $r$