# Two principles of decoupling

Jianhui Li and Tongou Yang  Department of Mathematics, Northwestern University
Evanston, IL 60208, United States<EMAIL_ADDRESS>Department of
Mathematics, University of California
Los Angeles, CA 90095, United States<EMAIL_ADDRESS>
###### 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (1)

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

  2. (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. (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. (1)

$r(s)$ is affine, or

  2. (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. (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. (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. (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. (1)

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

  2. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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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