The square root cancellation heuristic, briefly mentioned in the preceding set of notes, predicts that if a collection ${z_1,\dots,z_n}$ of complex numbers have phases that are sufficiently “independent” of each other, then

$\displaystyle |\sum_{j=1}^n z_j| \approx (\sum_{j=1}^n |z_j|^2)^{1/2};$

similarly, if ${f_1,\dots,f_n}$ are a collection of functions in a Lebesgue space ${L^p(X,\mu)}$ that oscillate “independently” of each other, then we expect

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p(X,\mu)} \approx \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p(X,\mu)}.$

We have already seen one instance in which this heuristic can be made precise, namely when the phases of ${z_j,f_j}$ are randomised by a random sign, so that Khintchine’s inequality (Lemma 4 from Notes 1) can be applied. There are other contexts in which a square function estimate

$\displaystyle \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p(X,\mu)} \lesssim \| \sum_{j=1}^n f_j \|_{L^p(X,\mu)}$

or a reverse square function estimate

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p(X,\mu)} \lesssim \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p(X,\mu)}$

(or both) are known or conjectured to hold. For instance, the useful Littlewood-Paley inequality implies (among other things) that for any ${1 < p < \infty}$, we have the reverse square function estimate

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)} \lesssim_{p,d} \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p({\bf R}^d)}, \ \ \ \ \ (1)$

whenever the Fourier transforms ${\hat f_j}$ of the ${f_j}$ are supported on disjoint annuli ${\{ \xi \in {\bf R}^d: 2^{k_j} \leq |\xi| < 2^{k_j+1} \}}$, and we also have the matching square function estimate

$\displaystyle \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p({\bf R}^d)} \lesssim_{p,d} \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)}$

if there is some separation between the annuli (for instance if the ${k_j}$ are ${2}$-separated). We recall the proofs of these facts below the fold. In the ${p=2}$ case, we of course have Pythagoras’ theorem, which tells us that if the ${f_j}$ are all orthogonal elements of ${L^2(X,\mu)}$, then

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^2(X,\mu)} = (\sum_{j=1}^n \| f_j \|_{L^2(X,\mu)}^2)^{1/2} = \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^2(X,\mu)}.$

In particular, this identity holds if the ${f_j \in L^2({\bf R}^d)}$ have disjoint Fourier supports in the sense that their Fourier transforms ${\hat f_j}$ are supported on disjoint sets. For ${p=4}$, the technique of bi-orthogonality can also give square function and reverse square function estimates in some cases, as we shall also see below the fold.
In recent years, it has begun to be realised that in the regime ${p > 2}$, a variant of reverse square function estimates such as (1) is also useful, namely decoupling estimates such as

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)} \lesssim_{p,d} (\sum_{j=1}^n \|f_j\|_{L^p({\bf R}^d)}^2)^{1/2} \ \ \ \ \ (2)$

(actually in practice we often permit small losses such as ${n^\varepsilon}$ on the right-hand side). An estimate such as (2) is weaker than (1) when ${p\geq 2}$ (or equal when ${p=2}$), as can be seen by starting with the triangle inequality

$\displaystyle \| \sum_{j=1}^n |f_j|^2 \|_{L^{p/2}({\bf R}^d)} \leq \sum_{j=1}^n \| |f_j|^2 \|_{L^{p/2}({\bf R}^d)},$

and taking the square root of both side to conclude that

$\displaystyle \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p({\bf R}^d)} \leq (\sum_{j=1}^n \|f_j\|_{L^p({\bf R}^d)}^2)^{1/2}. \ \ \ \ \ (3)$

However, the flip side of this weakness is that (2) can be easier to prove. One key reason for this is the ability to iterate decoupling estimates such as (2), in a way that does not seem to be possible with reverse square function estimates such as (1). For instance, suppose that one has a decoupling inequality such as (2), and furthermore each ${f_j}$ can be split further into components ${f_j= \sum_{k=1}^m f_{j,k}}$ for which one has the decoupling inequalities

$\displaystyle \| \sum_{k=1}^m f_{j,k} \|_{L^p({\bf R}^d)} \lesssim_{p,d} (\sum_{k=1}^m \|f_{j,k}\|_{L^p({\bf R}^d)}^2)^{1/2}.$

Then by inserting these bounds back into (2) we see that we have the combined decoupling inequality

$\displaystyle \| \sum_{j=1}^n\sum_{k=1}^m f_{j,k} \|_{L^p({\bf R}^d)} \lesssim_{p,d} (\sum_{j=1}^n \sum_{k=1}^m \|f_{j,k}\|_{L^p({\bf R}^d)}^2)^{1/2}.$

This iterative feature of decoupling inequalities means that such inequalities work well with the method of induction on scales, that we introduced in the previous set of notes.
In fact, decoupling estimates share many features in common with restriction theorems; in addition to induction on scales, there are several other techniques that first emerged in the restriction theory literature, such as wave packet decompositions, rescaling, and bilinear or multilinear reductions, that turned out to also be well suited to proving decoupling estimates. As with restriction, the curvature or transversality of the different Fourier supports of the ${f_j}$ will be crucial in obtaining non-trivial estimates.
Strikingly, in many important model cases, the optimal decoupling inequalities (except possibly for epsilon losses in the exponents) are now known. These estimates have in turn had a number of important applications, such as establishing certain discrete analogues of the restriction conjecture, or the first proof of the main conjecture for Vinogradov mean value theorems in analytic number theory.
These notes only serve as a brief introduction to decoupling. A systematic exploration of this topic can be found in this recent text of Demeter.

— 1. Square function and reverse square function estimates —

We begin with a form of the Littlewood-Paley inequalities. Given a region ${\Omega \in {\bf R}^d}$, we say that a tempered distribution ${f}$ on ${{\bf R}^d}$ has Fourier support in ${\Omega}$ if its distributional Fourier transform ${\hat f}$ is supported in (the closure of) ${\Omega}$.

Theorem 1 (Littlewood-Paley inequalities) Let ${d \geq 1}$, let ${k_1,\dots,k_n}$ be distinct integers, let ${1 < p < \infty}$, and for each ${1 \leq j \leq n}$ let ${f_j \in L^p({\bf R}^d)}$ be a function with Fourier support in the annulus ${\{ \xi \in {\bf R}^d: 2^{k_j} \leq |\xi| \leq 2^{k_j+1}\}}$.

• (i) (Reverse square function inequality) One has

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)} \lesssim_{p,d} \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p({\bf R}^d)}.$

• (ii) (Square function inequality) If the ${k_j}$ are ${2}$-separated (thus ${|k_j-k_{j'}| \geq 2}$ for any ${1 \leq j < j' \leq n}$) then

$\displaystyle \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p({\bf R}^d)} \lesssim_{p,d} \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)}.$

Proof: We begin with (ii). We use a randomisation argument. Let ${\psi \in C^\infty_c({\bf R}^d)}$ be a bump function supported on the annulus ${\{ \xi: 0.9 \leq |\xi| \leq 2.1\}}$ that equals one on ${\{ \xi: 1 \leq |\xi| \leq 2 \}}$, and for each ${j=1,\dots,n}$ let ${P_{k_j}}$ be the Fourier multiplier defined by

$\displaystyle \widehat{P_{k_j} f}(\xi) := \psi(\xi/2^{k_j}) \hat f(\xi),$

at least for functions ${f}$ in the Schwartz class. Clearly the operator ${P_{k_j}}$ is given by convolution with a (${k_j}$-dependent) Schwartz function, so this multiplier is bounded on every ${L^p}$ space. Writing ${f := \sum_{j=1}^n f_j}$, we see from the separation property of the ${k_j}$ that we have the reproducing formula

$\displaystyle f_j = P_{k_j} f.$

Now let ${\epsilon_1,\dots,\epsilon_n \in \{-1,+1\}}$ be random signs drawn uniformly and independently at random, thus

$\displaystyle \sum_{j=1}^n \epsilon_j f_j = (\sum_{j=1}^n \epsilon_j P_{k_j}) f.$

The operator ${\sum_{j=1}^n \epsilon_j P_{k_j}}$ is a Fourier multiplier with symbol ${\sum_{j=1}^n \epsilon_n \psi(\xi/2^{k_j})}$. This symbol obeys the hypotheses of the Hörmander-Miklin multiplier theorem, uniformly in the choice of signs; since we are in the non-endpoint case ${1 < p < \infty}$, we thus have

$\displaystyle \| \sum_{j=1}^n \epsilon_j f_j \|_{L^p({\bf R}^d)} \lesssim_{p,d} \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)}$

uniformly in the ${\epsilon_j}$. Taking ${p^{th}}$ power means of this estimate using Khintchine’s inequality (Lemma 4 from Notes 1), we obtain (ii) as desired.
Now we turn to (i). By treating the ${k_j}$ even and ${k_j}$ odd cases separately and using the triangle inequality, we may assume without loss of generality that the ${k_j}$ all have the same parity, so in particular are ${2}$-separated. (Why are we permitted to use this reduction for part (i) but not for part (ii)?) Now we use the projections ${P_{k_j}}$ from before in a slightly different way, noting that

$\displaystyle f_j = \epsilon_j P_{k_j} \sum_{j'=1}^n \epsilon_{j'} f_{j'},$

for any ${j=1,\dots,n}$, and hence

$\displaystyle \sum_{j=1}^n f_j = (\sum_{j=1}^n \epsilon_j P_{k_j}) \sum_{j'=1}^n \epsilon_{j'} f_{j'}.$

Applying the Hörmander-Mikhlin theorem as before, we conclude that

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)} \lesssim_{p,d} \| \sum_{j=1}^n \epsilon_j f_j \|_{L^p({\bf R}^d)}$

and on taking ${p^{th}}$ power means and using Khintchine’s inequality as before we conclude (i). $\Box$

Exercise 2 (Smooth Littlewood-Paley estimate) Let ${d \geq 1}$ and ${1 < p < \infty}$, and let ${\phi\in C^\infty_c({\bf R}^d)}$ be a bump function supported on ${B(0,1)}$ that equals ${1}$ on ${B(0,1/2)}$. For any integer ${j}$, let ${P_j}$ denote the Fourier multiplier, defined on Schwartz functions ${f}$ by

$\displaystyle \widehat{P_j f}(\xi) := (\phi(\xi/2^j) - \phi(\xi/2^{j-1})) \hat f(\xi)$

and extended to ${L^p({\bf R}^d)}$ functions by continuity. Show that for any ${f \in L^p({\bf R}^d)}$, one has

$\displaystyle \| (\sum_{j \in {\bf Z}} |P_j f|^2)^{1/2} \|_{L^p({\bf R}^d)} \sim_{p,d} \|f\|_{L^p({\bf R}^d)}$

(in particular, the left-hand side is finite).

We remark that when ${d=1}$, the condition that the ${k_j}$ be ${2}$-separated can be removed from Theorem 1(ii), by using the Marcinkiewicz multiplier theorem in place of the Hörmander-Mikhlin multiplier theorem. But, perhaps surprisingly, the condition cannot be removed in higher dimensions, as a consequence of Fefferman’s surprising result on the unboundedness of the disk multiplier.

Exercise 3 (Unboundedness of the disc multiplier) Let ${D}$ denote either the disk ${\{ \xi: |\xi| \leq 1 \}}$ or the annulus ${\{ \xi: 1 \leq |\xi| \leq 2 \}}$. Let ${P_\Omega}$ denote the Fourier multiplier defined on Schwartz functions ${f \in {\mathcal S}({\bf R}^2)}$ by

$\displaystyle \widehat{P_\Omega f}(\xi) = 1_\Omega(\xi) \hat f(\xi).$

Suppose that ${1 < p < \infty}$ is such that

$\displaystyle \| P_D f \|_{L^p({\bf R}^2)} \lesssim_p \| f \|_{L^p({\bf R}^2)} \ \ \ \ \ (4)$

for all ${f \in {\mathcal S}({\bf R}^2)}$.

• (i) Show that for any collection ${H_1,\dots,H_n}$ of half-planes in ${{\bf R}^2}$, and any functions ${f_1,\dots,f_n \in {\mathcal S}({\bf R}^2)}$, that

$\displaystyle \| (\sum_{j=1}^n |P_{H_j} f_j|^2)^{1/2} \|_{L^p({\bf R}^2)} \lesssim_p \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^p({\bf R}^2)}.$

(Hint: first rescale the set ${D}$ by a large scaling factor ${R}$, apply the Marcinkiewicz-Zygmund theorem (Exercise 7 from Notes 1), exploit the symmetries of the Fourier transform, then take a limit as ${R \rightarrow \infty}$.)

• (ii) Let ${R_1,\dots,R_n}$ be a collection of ${\delta \times 1}$ rectangles for some ${\delta < 1}$, and for each ${j=1,\dots,n}$, let ${\tilde R_j}$ be a rectangle formed from ${R_j}$ by translating by a distance ${2}$ in the direction of the long axis of ${R_j}$. Use (i) to show that

$\displaystyle \| \sum_{j=1}^n 1_{\tilde R_j} \|_{L^{p/2}({\bf R}^2)} \sim_p \| \sum_{j=1}^n 1_{R_j} \|_{L^{p/2}({\bf R}^2)}.$

(Hint: a direct application of (i) will give just one side of this estimate, but then one can use symmetry to obtain the other side.)

• (iii) In Fefferman’s paper, modifying a classic construction of a Besicovitch set, it was shown that for any ${\varepsilon>0}$, there exists a collection of ${1 \times \delta}$ rectangles ${R_1,\dots,R_n}$ for some ${0 < \delta < 1}$ with ${n \sim 1/\delta}$ such that all the rectangles ${\tilde R_j}$ are disjoint, but such that ${\bigcup_{j=1}^n R_j}$ has measure ${O(\varepsilon)}$. Assuming this fact, conclude that the multiplier estimate (4) fails unless ${p=2}$.
• (iv) Show that Theorem 1(ii) fails when ${d=2}$ and the requirement that the ${k_j}$ be ${2}$-separated is removed.

Exercise 4 Let ${\varphi \in C^\infty_c({\bf R})}$ be a bump function supported on ${[-1,2]}$ that equals one on ${[0,1]}$. For each integer ${n}$, let ${\tilde P_n}$ be the Fourier multiplier defined for ${f \in {\mathcal S}({\bf R})}$ by

$\displaystyle \widehat{\tilde P_n f}(\xi) := \varphi(\xi - n) \hat f(\xi)$

and also define

$\displaystyle \widehat{P_{[n,n+1]} f}(\xi) := 1_{[n,n+1]}(\xi) \hat f(\xi).$

• (i) For any ${2 \leq p \leq \infty}$, establish the square function estimate

$\displaystyle \| (\sum_{n \in {\bf Z}} |\tilde P_n f|^2)^{1/2} \|_{L^p({\bf R})} \lesssim \|f\|_{L^p({\bf R})}$

for ${f \in {\mathcal S}({\bf R})}$. (Hint: interpolate between the ${p=2,p=\infty}$ cases, and for the latter use Plancherel’s theorem for Fourier series.)

• (ii) For any ${2 \leq p < \infty}$, establish the square function estimate

$\displaystyle \| (\sum_{n \in {\bf Z}} |P_{[n,n+1]} f|^2)^{1/2} \|_{L^p({\bf R})} \lesssim_p \|f\|_{L^p({\bf R})}$

for ${f \in {\mathcal S}({\bf R})}$. (Hint: from the boundedness of the Hilbert transform, ${P_{[0,1]}}$ is bounded in ${L^p}$. Combine this with the Marcinkiewicz-Zygmund theorem (Exercise 7 from Notes 1), then use the symmetries of the Fourier transform, part (i), and the identity ${P_{[n,n+1]} = P_{[n,n+1]} \tilde P_n}$.)

• (iii) For any ${1 < p \leq 2}$, establish the reverse square function estimate

$\displaystyle \|f\|_{L^p({\bf R})} \lesssim_p \| (\sum_{n \in {\bf Z}} |P_{[n,n+1]} f|^2)^{1/2} \|_{L^p({\bf R})}$

for ${f \in {\mathcal S}({\bf R})}$. (Hint: use duality as in the solution to Exercise 2 in this set of notes, or Exercise 11 in Notes 1, and part (ii).)

• (iv) Show that the estimate (ii) fails for ${1, and similarly the estimate (iii) fails for ${2.

Remark 5 The inequalities in Exercise 4 have been generalised by replacing the partition ${{\bf R} = \bigcup_n [n,n+1]}$ with an arbitrary partition of the real line into intervals; see this paper of Rubio de Francia.

If ${f_1,\dots,f_n \in {\mathcal S}({\bf R}^d)}$ are functions with disjoint Fourier supports, then as mentioned in the introduction, we have from Pythagoras’ theorem that

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^2({\bf R}^d)} = \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^2({\bf R}^d)}.$

We have the following variants of this claim:

Lemma 6 (${L^2}$ and ${L^4}$ reverse square function estimates) Let ${f_1,\dots,f_n \in {\mathcal S}({\bf R}^d)}$ have Fourier transforms supported on the sets ${\Omega_1,\dots,\Omega_n \subset {\bf R}^d}$ respectively.

• (i) (Almost orthogonality) If the sets ${\Omega_1,\dots,\Omega_n}$ have overlap at most ${A_2}$ (i.e., every ${\xi \in {\bf R}^d}$ lies in at most ${A_2}$ of the ${\Omega_i}$) for some ${A_2>0}$, then

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^2({\bf R}^d)} \leq A_2^{1/2} \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^2({\bf R}^d)}.$

• (ii) (Almost bi-orthogonality) If the sets ${\Omega_i+\Omega_j := \{ \xi_i + \xi_j: \xi_i \in \Omega_i, \xi_j \in \Omega_j\}}$ with ${1 \leq i,j \leq n}$ have overlap at most ${A_4}$ for some ${A_4>0}$, then

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^4({\bf R}^d)} \leq A_4^{1/4} \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^4({\bf R}^d)}.$

Proof: For (i), observe from Plancherel’s theorem that

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^2({\bf R}^d)} = \| \sum_{j=1}^n \hat f_j \|_{L^2({\bf R}^d)}.$

By hypothesis, for each frequency ${\xi \in {\bf R}^d}$ at most ${A_2}$ of the ${\hat f_j(\xi)}$ are non-zero, thus by Cauchy-Schwarz we have the pointwise estimate

$\displaystyle |\sum_{j=1}^n \hat f_j(\xi)| \leq A_2^{1/2} (\sum_{j=1}^n |\hat f_j(\xi)|^2)^{1/2}$

and hence by Fubini’s theorem

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^2({\bf R}^d)} \leq A_2^{1/2} (\sum_{j=1}^n \|\hat f_j \|_{L^2({\bf R}^d)}^2)^{1/2}.$

The claim then follows by a further application of Plancherel’s theorem and Fubini’s theorem. For (ii), we observe that

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^4({\bf R}^d)}^2 = \| \sum_{i=1}^n \sum_{j=1}^n f_i f_j \|_{L^2({\bf R}^2)}$

and

$\displaystyle \| (\sum_{j=1}^n |f_j|^2)^{1/2} \|_{L^4({\bf R}^d)}^2 = \| (\sum_{i=1}^n \sum_{j=1}^n |f_i f_j|^2)^{1/2} \|_{L^2({\bf R}^2)}.$

Since ${f_i f_j}$ has Fourier support in ${\Omega_i + \Omega_j}$, the claim (ii) now follows from (i). $\Box$

Remark 7 By using ${f_i \overline{f_j}}$ in place of ${f_i f_j}$, one can also establish a variant of Lemma 6(ii) in which the sum set ${\Omega_i + \Omega_j}$ is replaced by the difference set ${\Omega_i - \Omega_j := \{ \xi_i - \xi_j: \xi_i \in \Omega_i, \xi_j \in \Omega_j \}}$. It is also clear how to extend the lemma to other even exponent Lebesgue spaces such as ${L^6}$; see for instance this recent paper of Gressman, Guo, Pierce, Roos, and Yung. However, we will not use these variants here.

We can use this lemma to establish the following ${L^4}$ reverse square function estimate for the circle:

Exercise 8 (Square function estimate for circle and parabola) Let ${0 < \delta \leq 1}$, let ${\Omega}$ be a ${\gtrsim \delta}$-separated subset of the unit circle ${S^1}$, and for each ${\omega \in \Omega}$, let ${f_\omega \in {\mathcal S}({\bf R}^2)}$ have Fourier support in the rectangle

$\displaystyle R_\omega := \{ \xi \in {\bf R}^2: |\xi \cdot \omega - 1| < \delta^2; |\xi - (\xi \cdot \omega) \omega| < \delta \}.$

• (i) Use Lemma 6(ii) to establish the reverse square function estimate

$\displaystyle \|\sum_{\omega \in \Omega} f_\omega \|_{L^4({\bf R}^2)} \lesssim \| (\sum_{\omega \in \Omega} |f_\omega|^2)^{1/2} \|_{L^4({\bf R}^2)}.$

• (ii) If the elements of ${\Omega}$ are ${C\delta}$-separated for a sufficiently large absolute constant ${C}$, establish the matching square function estimate

$\displaystyle \|\sum_{\omega \in \Omega} f_\omega \|_{L^4({\bf R}^2)} \gtrsim \| (\sum_{\omega \in \Omega} |f_\omega|^2)^{1/2} \|_{L^4({\bf R}^2)}.$

• (iii) Obtain analogous claims to (i), (ii) in which ${\Omega = \phi(\Sigma)}$ for some ${\delta}$-separated subset ${\Sigma}$ of ${[-1,1]}$, where ${\phi: {\bf R} \rightarrow {\bf R}^2}$ is the graphing function ${\phi(\xi) :=(\xi,\xi^2): \xi = O(1) \}}$, and to each ${\omega = (\omega_1,\omega_1^2) \in \Omega}$ one uses the parallelogram

$\displaystyle \tilde R_\omega := \{ (\xi_1,\xi_2) \in {\bf R}^2: |\xi_1 - \omega_1| < \delta; \ \ \ \ \ (5)$

$\displaystyle |\xi_2 - \omega_1^2 - 2\omega_1(\xi_1-\omega_1)| < \delta^2 \}$

in place of ${R_\omega}$.

• (iv) (Optional, as it was added after this exercise was first assigned as homework) Show that in (i) one cannot replace the ${L^4}$ norms on both sides by ${L^p}$ for any given ${4 < p < \infty}$. (Hint: use a Knapp type example for each ${f_\omega}$ and ensure that there is enough constructive interference in ${\sum_{\omega \in \Omega} f_\omega}$ near the origin.) On the other hand, using Exercise 4 show that the ${L^4}$ norm in (ii) can be replaced by an ${L^p}$ norm for any ${2 \leq p \leq \infty}$.

For a more sophisticated estimate along these lines, using sectors of the plane rather than rectangles near the unit circle, see this paper of Cordóba. An analogous reverse square function estimate is also conjectured in higher dimensions (with ${L^4}$ replaced by the endpoint restriction exponent ${L^{2d/(d-1)}}$), but this remains open, and in fact is at least as hard as the restriction and Kakeya conjectures; see this paper of Carbery.

— 2. Decoupling estimates —

We now turn to decoupling estimates. We begin with a general definition.

Definition 9 (Decoupling constant) Let ${{\mathcal U} = \{U_1,\dots,U_n\}}$ be a finite collection of non-empty open subsets of ${{\bf R}^d}$ for some ${n \geq 1}$ (we permit repetitions, so ${{\mathcal U}}$ may be a multi-set rather than a set), and let ${1 \leq p \leq \infty}$. We define the decoupling constant ${\mathrm{Dec}_p({\mathcal U})}$ to be the smallest constant for which one has the inequality

$\displaystyle \| \sum_{j=1}^n f_j \|_{L^p({\bf R}^d)} \leq \mathrm{Dec}_p({\mathcal U}) (\sum_{j=1}^n \|f_j\|_{L^p({\bf R}^d)}^2)^{1/2} \ \ \ \ \ (6)$

whenever ${f_j \in {\mathcal S}({\bf R}^d)}$ has Fourier support in ${U_j}$ for ${j=1,\dots,n}$.

We have the trivial upper and lower bounds

$\displaystyle 1 \leq \mathrm{Dec}_p({\mathcal U}) \leq n^{1/2}, \ \ \ \ \ (7)$

with the lower bound arising from restricting to the case when all but one of the ${f_j}$ vanish, and the upper bound following from the triangle inequality and Cauchy-Schwarz. In the literature, decoupling inequalities are also considered with the ${\ell^2}$ summation of the ${\|f_j\|_{L^p({\bf R}^d)}}$ norms replaced by other summations (for instance, the original decoupling inequality of Wolff used ${\ell^p}$ norms) but we will focus only on ${\ell^2}$ decoupling estimates in this post. In the literature it is common to restrict attention to the case when the sets ${U_1,\dots,U_n}$ are disjoint, but for minor technical reasons we will not impose this extra condition in our definition.

Exercise 10 (Elementary properties of decoupling constants) Let ${1 \leq p \leq \infty}$ and ${d \geq 1}$.

• (i) (Monotonicity) Show that

$\displaystyle \mathrm{Dec}_p( \{ U_1,\dots,U_n\}) \leq \mathrm{Dec}_p( \{ U'_1,\dots,U'_n\})$

whenever ${U_j,U'_j}$ are non-empty open subsets of ${{\bf R}^d}$ with ${U_j \subset U'_j}$ for ${j=1,\dots,n}$.

• (ii) (Triangle inequality) Show that

$\displaystyle \mathrm{Dec}_p( {\mathcal U} ), \mathrm{Dec}_p( {\mathcal U}' ) \leq \mathrm{Dec}_p( {\mathcal U} \cup {\mathcal U}' )$

$\displaystyle \leq (\mathrm{Dec}_p( {\mathcal U} )^2 + \mathrm{Dec}_p( {\mathcal U}' )^2)^{1/2}$

for any finite non-empty collections ${{\mathcal U}, {\mathcal U}'}$ of open non-empty subsets of ${{\bf R}^d}$.

• (iii) (Affine invariance) Show that

$\displaystyle \mathrm{Dec}_p( \{ LU_1,\dots,LU_n\}) = \mathrm{Dec}_p( \{ U_1,\dots,U_n\})$

whenever ${U_1,\dots,U_n \subset {\bf R}^d}$ are open non-empty and ${L: {\bf R}^d \rightarrow {\bf R}^d}$ is an invertible affine transformation.

• (iv) (Interpolation) Suppose that ${\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}}$ for some ${1 \leq p_0 \leq p \leq p_1 \leq \infty}$ and ${0 \leq \theta \leq 1}$, and suppose also that ${{\mathcal U} = \{U_1,\dots,U_n\}}$ is a non-empty collection of open non-empty subsets of ${{\bf R}^d}$ for which one has the projection bounds

$\displaystyle \| P_{U_j} f \|_{L^{p_i}({\bf R}^d)} \lesssim_{p_i,d} \|f\|_{L^{p_i}({\bf R}^d)}$

for all ${f \in {\mathcal S}({\bf R}^d)}$, ${i=0,1}$, and ${j=1,\dots,n}$, where the Fourier multiplier ${P_{U_j}}$ is defined by

$\displaystyle \widehat{P_{U_j} f}(\xi) = 1_{U_j}(\xi) \hat f(\xi).$

Show that

$\displaystyle \mathrm{Dec}_p({\mathcal U}) \lesssim_{p_0,p_1,d,\theta} \mathrm{Dec}_{p_0}({\mathcal U})^{1-\theta} \mathrm{Dec}_{p_1}({\mathcal U})^{\theta}.$

• (v) (Multiplicativity) Suppose that ${{\mathcal U} = \{ U_1,\dots,U_n\}}$ is a family of open non-empty subsets of ${{\bf R}^d}$, with each ${U_j, j=1,\dots,n}$ containing further open non-empty subsets ${U_{j,i}}$ for ${i=1,\dots,m}$. Show that

$\displaystyle \mathrm{Dec}_p( \{ U_{j,i}: 1 \leq j \leq n, 1 \leq i \leq m \} ) \leq \mathrm{Dec}_p({\mathcal U})$

$\displaystyle \times \sup_{1 \leq j \leq n} \mathrm{Dec}_p( \{ U_{j,1},\dots,U_{j,m}\}).$

• (vi) (Adding dimensions) Suppose that ${\{ U_1,\dots,U_n\}}$ is a family of disjoint open non-empty subsets of ${{\bf R}^d}$ and that ${p \geq 2}$. Show that for any ${d' \geq 1}$, one has

$\displaystyle \mathrm{Dec}_p( \{ U_1,\dots,U_n\} ) = \mathrm{Dec}_p( \{ U_1 \times {\bf R}^{d'},\dots,U_n \times {\bf R}^{d'}\} )$

where the right-hand side is a decoupling constant in ${{\bf R}^d \times {\bf R}^{d'} = {\bf R}^{d+d'}}$.

The most useful decoupling inequalites in practice turn out to be those where the decoupling constant ${\mathrm{Dec}_p({\mathcal U})}$ is close to the lower bound of ${1}$, for instance if one has the sub-polynomial bounds

$\displaystyle \mathrm{Dec}_p({\mathcal U}) \lesssim_{p,d,\varepsilon} n^\varepsilon$

for every ${\varepsilon>0}$. We informally say that the collection ${{\mathcal U}}$ of sets exhibits decoupling in ${L^p}$ when this is the case.
For ${p=2,4}$, Lemma 6 (and (3)) already gives some decoupling estimates: one has

$\displaystyle \mathrm{Dec}_2({\mathcal U}) \leq A_2^{1/2} \ \ \ \ \ (8)$

if the sets ${U_1,\dots,U_n}$ have an overlap of at most ${A_2}$, and similarly

$\displaystyle \mathrm{Dec}_4({\mathcal U}) \leq A_4^{1/4}$

when the sets ${U_i+U_j}$, ${1 \leq i,j \leq n}$ have an overlap of at most ${A_4}$.
For ${p<2}$, it is not possible to exhibit decoupling in the limit ${n \rightarrow \infty}$:

Exercise 11 If ${{\mathcal U} = \{U_1,\dots,U_n\}}$ is a collection of non-empty open subsets of ${{\bf R}^d}$, show that

$\displaystyle \mathrm{Dec}_p({\mathcal U}) \gtrsim_{p,d} n^{\frac{1}{p}-\frac{1}{2}}$

for any ${1 \leq p \leq 2}$. (Hint: select the ${f_1,\dots,f_n}$ to be concentrated in widely separated large balls.)

Henceforth we now focus on the regime ${p \geq 2}$. By (8), ${L^2}$ decoupling is easily obtained if the regions ${U_j}$ are of bounded overlap. For ${p}$ larger than ${2}$, bounded overlap is insufficient by itself; the arrangement of the regions ${U_j}$ must also exhibit some “curvature”, as the following example shows.

Exercise 12 If ${{\mathcal U} = \{ (i,i+1): 0 \leq i < n \}}$, and ${2 \leq p \leq \infty}$, show that

$\displaystyle \mathrm{Dec}_p({\mathcal U}) \sim_p n^{\frac{1}{2} - \frac{1}{p}}.$

(Hint: for the upper bound, use a variant of Exercise 24(ii) from Notes 1, or adapt the interpolation arguemnt used to establish that exercise.)

Now we establish a significantly more non-trivial decoupling theorem:

Theorem 13 (Decoupling for the parabola) Let ${0 < \delta \leq 1}$, let ${\Omega = \phi(\Sigma)}$ for some ${\delta}$-separated subset ${\Sigma}$ of ${[-1,1]}$, where ${\phi(\xi) := (\xi,\xi^2)}$, and to each ${\omega \in \Omega}$ let ${\tilde R_{\omega,\delta} = \tilde R_{\omega}}$ be the parallelogram (5). Then

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\omega,\delta}: \omega \in \Omega \} ) \lesssim_\varepsilon \delta^{-\varepsilon}$

for any ${\varepsilon>0}$.

This result was first established by Bourgain and Demeter; our arguments here will loosely follow an argument of Li, that is based in turn on the efficient congruencing methods of Wooley, as recounted for instance in this exposition of Pierce.
We first explain the significance of the exponent ${6}$ in Theorem 13. Let ${\Sigma}$ be a maximal ${\delta}$-separated subset ${[-1,1]}$ for some small ${0 < \delta < 1}$, so that ${\Sigma}$ has cardinality ${\sim \delta^{-1}}$. For each ${\omega \in \phi(\Sigma)}$, choose ${f_\omega}$ so that ${\hat f_\omega}$ is a non-negative bump function (not identically zero) adapted to the parallelogram ${\tilde R_{\omega,\delta}}$, which is comparable to a ${\delta \times \delta^2}$ rectangle. From the Fourier inversion formula, ${f_\omega}$ will then have magnitude ${O(\delta^3)}$ on a dual rectangle of dimensions comparable to ${1/\delta \times 1/\delta^2}$, and is rapidly decreasing away from that rectangle, so we have

$\displaystyle \|f_\omega\|_{L^p({\bf R}^2)} \lesssim_p \delta^3 \delta^{-3/p}$

for all ${1 \leq p \leq \infty}$ and ${\omega \in \phi(\Sigma)}$. In particular

$\displaystyle ( \sum_{\omega \in \phi(\Sigma)} \|f_\omega\|_{L^p({\bf R}^2)}^2)^{1/2} \sim_p \delta^{\frac{5}{2} - \frac{3}{p}}.$

On the other hand, we have ${\mathrm{Re} f_\omega(x) \gtrsim \delta^3}$ for ${|x| \leq c}$ if ${c>0}$ is a sufficiently small absolute constant, hence

$\displaystyle \|\sum_{\omega \in \phi(\Sigma)} f_\omega\|_{L^p({\bf R}^2)} \gtrsim_p \delta^2.$

Comparing this with (6), we conclude that

$\displaystyle \mathrm{Dec}_p( \{ \tilde R_\omega: \omega \in \phi(\Sigma) \} ) \gtrsim_p \delta^{\frac{3}{p} - \frac{1}{2}}$

so Theorem 13 cannot hold if the exponent ${6}$ is replaced by any larger exponent. On the other direction, by using Exercise 10(iv) and the trivial ${p=2}$ decoupling from (8), we see that we also have decoupling in ${L^p}$ for any ${2 \leq p \leq 6}$. (Note from the boundedness of the Hilbert transform that a Fourier projection to any polygon of boundedly many sides will be bounded in ${L^p}$ for any ${1 < p < \infty}$ with norm ${O_p(1)}$.) Note that reverse square function estimates in Exercise 8 only give decoupling in the smaller range ${2 \leq p \leq 4}$; the ${L^6}$ version of Lemma 6 is not strong enough to extend the decoupling estimates to larger ranges because the triple sums ${\tilde R_{\omega_1} + \tilde R_{\omega_2} + \tilde R_{\omega_3}}$ have too much overlap.
For any ${0 < \delta \leq 1}$, let ${D(\delta)}$ denote the supremum of the decoupling constants ${\mathrm{Dec}_6( \{ \tilde R_{\omega,\delta}: \omega \in \phi(\Sigma) \} )}$ over all ${\delta}$-separated subsets ${\Sigma}$ of ${[-1,1]}$. From (7) we have the trivial bound

$\displaystyle 1 \leq D(\delta) \lesssim \delta^{-1/2}$

and our task is to show that

$\displaystyle D(\delta) \lesssim_\varepsilon \delta^{-\varepsilon} \ \ \ \ \ (9)$

for any ${\varepsilon>0}$.
We first make a minor observation on the stability of ${D}$ that is not absolutely essential for the arguments, but is convenient for cleaning up the notation slightly (otherwise we would have to replace various scales ${\delta}$ that appear in later arguments by comparable scales ${\delta' \sim \delta}$).

Lemma 14 (Stability of ${D}$) If ${0 < \delta,\delta' \leq 1}$ are such that ${\delta \sim \delta'}$ then ${D(\delta) \sim D(\delta')}$.

Proof: Without loss of generality we may assume that ${\delta' \geq \delta}$. We first show that ${D(\delta) \lesssim D(\delta')}$. We need to show that

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\omega,\delta}: \omega \in \phi(\Sigma)\} ) \lesssim D(\delta')$

whenever ${\Sigma}$ is a ${\delta}$-separated subset of the ${[-1,1]}$. By partitioning ${\Sigma}$ into ${O(1)}$ pieces and using Exercise 10(ii) we may assume without loss of generality that ${\Sigma}$ is in fact ${\delta'}$-separated. In particular

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\omega,\delta'}: \omega \in \phi(\Sigma) \} ) \leq D(\delta').$

The claim now follows from Exercise 10(i) and the inclusion

$\displaystyle \tilde R_{\omega,\delta} \subset \tilde R_{\omega,\delta'}.$

Conversely, we need to show that

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\omega',\delta'}: \omega' \in \phi(\Sigma') \} ) \lesssim D(\delta)$

whenever ${\Sigma' \subset [-1,1]}$ is ${\delta'}$-separated, or equivalently that

$\displaystyle \| \sum_{\omega' \in \phi(\Sigma')} f_{\omega'} \|_{L^6({\bf R}^2)} \lesssim D(\delta) (\sum_{\omega' \in \phi(\Sigma')} \|f_{\omega'}\|_{L^6({\bf R}^2)}^2)^{1/2}$

when ${f_{\omega'} \in L^6({\bf R}^2)}$ (we can extend from ${{\mathcal S}({\bf R}^2)}$ to all of ${L^6({\bf R}^2)}$ by a limiting argument). From elementary geometry we see that for each ${\xi' \in \Sigma'}$ we can find a subset ${\Sigma_{\xi'}}$ of ${[\xi'-\delta',\xi'+\delta']}$ of cardinality ${O(1)}$, such that the parallelograms ${\tilde R_{\phi(\xi),\delta} + (0, jc\delta^2)}$ with ${j=O(1)}$ an integer, ${\xi \in \Sigma_{\xi'}}$, and ${c>0}$ a sufficiently small absolute constant, cover ${R_{\phi(\xi'),\delta'}}$. In particular, using Fourier projections to polygons with ${O(1)}$ sides, one can split

$\displaystyle f_{\phi(\xi')} = \sum_{\xi \in \Sigma_{\xi'}} \sum_{j=O(1)} g_{\phi(\xi),j}, \ \ \ \ \ (10)$

where each ${g_{\phi(\xi),j} \in L^6({\bf R}^2)}$ has Fourier support in ${\tilde R_{\phi(\xi),\delta} + (0,jc\delta^2)}$ and

$\displaystyle \|g_{\phi(\xi),j} \|_{L^6({\bf R}^2)} \lesssim \|f_{\phi(\xi')} \|_{L^6({\bf R}^2)}. \ \ \ \ \ (11)$

Now the collection ${\bigcup_{\xi' \in \Sigma'} \Sigma_{\xi'}}$ can be partitioned into ${O(1)}$ subcollections, each of which is ${\delta}$-separated. From this and Exercise 10(ii), (iii) we see that

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\phi(\xi),\delta} + (0, jc\delta^2): \xi \in \bigcup_{\xi' \in \Sigma'} \Sigma_{\xi'}; j = O(1) \} ) \lesssim D(\delta)$

and thus

$\displaystyle \| \sum_{\xi' \in\Sigma'} \sum_{\xi \in \Sigma_{\xi'}} \sum_{j = O(1)} g_{\phi(\xi),j} \|_{L^6({\bf R}^2)} \lesssim D(\delta) (\sum_{\xi' \in \Sigma'} \sum_{\xi \in \Sigma_{\xi'}} \sum_{j=O(1)} \|g_{\phi(\xi),j}\|_{L^6({\bf R}^2)}^2)^{1/2}.$

Applying (10), (11) we obtain the claim. $\Box$
More importantly, we can use the symmetries of the parabola to control decoupling constants for parallelograms ${R_{\omega,\delta}}$ in a set of diameter ${O(\delta_0)}$ in terms of a coarser scale decoupling constant ${D(\delta/\delta_0)}$:

Proposition 15 (Parabolic rescaling) Let ${0 < \delta \leq \delta_0 \leq 1}$, and let ${\Sigma}$ be a ${\delta}$-separated subset of an interval ${I \subset [-1,1]}$ of length ${2\delta_0}$. Then

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\omega,\delta}: \omega \in \phi(\Sigma) \} ) \lesssim D(\delta/\delta_0). \ \ \ \ \ (12)$

Proof: We can assume that ${\delta_0 for a small absolute constant ${c}$, since when ${c \leq \delta_0 \leq 1}$ the claim follows from Lemma 14. Write ${I = [\xi_0-\delta_0,\xi_0+\delta_0]}$. Applying the Galilean transform

$\displaystyle G: (\xi_1, \xi_2) \mapsto (\xi_1 - \xi_{0}, \xi_2 - 2 \xi_{0} \xi_1 + \xi_{0}^2)$

(which preserves the parabola, and maps parallelograms ${\tilde R_{\omega,\delta}}$ to ${\tilde R_{G(\omega),\delta}}$) and Exercise 10(iii), we may normalise ${\xi_0=0}$, so ${\Sigma \subset [-\delta_0,\delta_0]}$.
Now let ${T: {\bf R}^2 \rightarrow {\bf R}^2}$ be the parabolic rescaling map

$\displaystyle T: (\xi_1,\xi_2) \mapsto (\xi_1/\delta_0, \xi_2/\delta^2_0).$

Observe that ${T}$ maps ${\tilde R_{\phi(\xi),\delta}}$ to ${\tilde R_{\phi(\xi/\delta_0),\delta/\delta_0}}$ for any ${\xi}$. From Exercise 10(iii) again, we can write the left-hand side of (12) as

$\displaystyle \mathrm{Dec}_6( \{ \tilde R_{\omega,\delta/\delta}: \omega \in \phi(\Sigma/\delta_0)\} );$

since ${\Sigma/\delta_0}$ is ${\delta_0}$-separated, the claim then follows. $\Box$

Exercise 16 (Multiplicativity) Show that ${D(\delta_1 \delta_2) \lesssim D(\delta_1) D(\delta_2)}$ for all ${0 < \delta_1,\delta_2 \leq 1}$.

The multiplicativity property in Exercise 16 suggests that an induction on scales approach could be fruitful to establish (9). Interestingly, it does not seem possible to induct directly on ${D(\delta)}$; all the known proofs of this decoupling estimate proceed by introducing some auxiliary variant of ${D(\delta)}$ that looks more complicated (in particular, involving additional scale parameters than just the base scale ${\delta}$), but which obey some inequalities somewhat reminiscent of the one in Exercise 16 for which an induction on scale argument can be profitably executed. It is yet not well understood exactly what choices of auxiliary quantity work best, but we will use the following choice of Li of a certain “asymmetric bilinear” variant of the decoupling constant:

Definition 17 (Bilinear decoupling constant) Let ${0 \leq \delta \leq \rho_1,\rho_2 \leq \nu \leq 1}$. Define ${M_{2,4}(\delta,\nu, \rho_1,\rho_2)}$ to be the best constant for which one has the estimate

$\displaystyle \int_{{\bf R}^2} |\sum_{\omega_1 \in \phi(\Sigma_1)} f_{\omega_1}|^2 |\sum_{\omega_2 \in \phi(\Sigma_2)} g_{\omega_2}|^4 \leq M_{2,4}(\delta,\nu,\rho_1,\rho_2)^6 \ \ \ \ \ (13)$

$\displaystyle \times (\sum_{\omega_1 \in \phi(\Sigma_1)} \|f_{\omega_1}\|_{L^6({\bf R}^2)}^2) (\sum_{\omega_2 \in \phi(\Sigma_2)} \|g_{\omega_2}\|_{L^6({\bf R}^2)}^2)^2$

whenever ${\Sigma_1,\Sigma_2}$ are ${\delta}$-separated subsets of intervals ${I_1, I_2 \subset [0,1]}$ of length ${\rho_1, \rho_2}$ respectively with ${\mathrm{dist}(I_1,I_2) \geq \nu}$, and for each ${\omega_1 \in \phi(\Omega_1)}$, ${f_{\omega_1} \in {\mathcal S}({\bf R})}$ has Fourier support in ${\tilde R_{\omega_1,\delta}}$, and similarly for each ${\omega_2 \in \phi(\Omega_2)}$, ${g_{\omega_2} \in {\mathcal S}({\bf R})}$ has Fourier support in ${\tilde R_{\omega_2,\delta}}$.

The scale ${\nu}$ is present for technical reasons and the reader may wish to think of it as essentially being comparable to ${1}$. Rather than inducting in ${\delta}$, we shall mostly keep ${\delta}$ fixed and primarily induct instead on ${\rho_1,\rho_2}$. As we shall see later, the asymmetric splitting of the sixth power exponent as ${6=2+4}$ is in order to exploit ${L^2}$ orthogonality in the first factor.
From Hölder’s inequality, the left-hand side of (13) is bounded by

$\displaystyle \| \sum_{\omega_1 \in \phi(\Sigma_1)} f_{\omega_1}\|_{L^6({\bf R}^2)}^2 \| \sum_{\omega_1 \in \phi(\Sigma_1)} g_{\omega_1}\|_{L^6({\bf R}^2)}^4$

from which we conclude the bound

$\displaystyle M_{2,4}(\delta,\nu,\rho_1,\rho_2) \leq D(\delta). \ \ \ \ \ (14)$

When ${\rho_1,\rho_2}$ are at their maximal size ${\rho_1=\rho_2=\nu}$ we can use these bilinear decoupling constants ${M_{2,4}(\delta,\nu,\rho_1,\rho_2)}$ to recover control on the decoupling constants ${D(\delta)}$, thanks to parabolic rescaling:

Proposition 18 (Bilinear reduction) If ${0 \leq \delta \leq \nu \leq 1}$, then

$\displaystyle D(\delta) \lesssim \nu^{-O(1)} M_{2,4}(\delta, \nu, \nu, \nu) + D(\delta/\nu).$

Proof: Let ${\Sigma}$ be a ${\delta}$-separated subset of ${[-1,1]}$, and for each ${\omega \in \phi(\Sigma)}$ let ${f_\omega \in L^6({\bf R}^2)}$ be Fourier supported in ${\tilde R_{\omega,\delta}}$. We may normalise ${\sum_{\omega \in \phi(\Sigma)} \| f_\omega\|_{L^6({\bf R}^2)}^2 = 1}$. It will then suffice to show that

$\displaystyle \int_{{\bf R}^2} |\sum_{\omega \in \phi(\Sigma)} f_\omega|^6 \lesssim \nu^{-O(1)} M_{2,4}(\delta, \nu, \nu, \nu)^6 + D(\delta/\nu)^6. \ \ \ \ \ (15)$

We partition ${\Sigma}$ into disjoint components ${\Sigma_I}$, each of which is supported in a subinterval ${I}$ of ${[-1,1]}$ of length ${\nu}$, with the family ${{\mathcal I}}$ of intervals ${I}$ having bounded overlap, so in particular ${{\mathcal I}}$ has cardinality ${O(\nu^{-1})}$. Then for any ${x \in {\bf R}^2}$, we of course have

$\displaystyle |\sum_{\omega \in \phi(\Sigma)} f_\omega(x)| \leq \sum_{I \in {\mathcal I}} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega(x)|.$

From the pigeonhole principle, this implies at least one of the following statements needs to hold for each given ${x}$:

• (i) (Narrow case) There exists ${I \in {\mathcal I}}$ such that

$\displaystyle |\sum_{\omega \in \phi(\Sigma)} f_\omega(x)| \lesssim |\sum_{\omega \in \phi(\Sigma_I)} f_\omega(x)|.$

• (ii) (Broad case) There exist distinct intervals ${I,J \in {\mathcal I}}$ with ${\mathrm{dist}(I,J) \geq \nu}$ such that

$\displaystyle |\sum_{\omega \in \phi(\Sigma)} f_\omega(x)| \lesssim \nu^{-O(1)} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega(x)|, \nu^{-O(1)} |\sum_{\omega \in \phi(\Sigma_J)} f_\omega(x)|.$

(The reason for this is as follows. Write ${A := |\sum_{\omega \in \phi(\Sigma)} f_\omega(x)|}$ and ${A_I := |\sum_{\omega \in \phi(\Sigma_I)} f_\omega(x)|}$, then ${A \leq \sum_I A_I}$. Let ${N = O(\nu^{-1})}$ be the number of intervals ${I}$, then ${\sum_{I: A_I \leq A/2N} A_I \leq A/2}$, hence ${\sum_{I: A_I > A/2N} A_I > A/2}$. If there are only ${O(1)}$ intervals ${I}$ for which ${A_I > A/2N}$, then by the pigeonhole principle we have ${A_I \gtrsim A}$ for one of these ${I}$ and we are in the narrow case (i); otherwise, if there are sufficiently many ${I}$ for which ${A_I > A/2N}$, one can find two such ${I,J}$ with ${\mathrm{dist}(I,J) \geq \nu}$, and we are in the broad case (ii).) This implies the pointwise bound

$\displaystyle |\sum_{\omega \in \phi(\Sigma)} f_\omega|^6 \lesssim \nu^{-O(1)} \sum_{I,J \in {\mathcal I}: \mathrm{dist}(I,J) \geq \nu} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega|^2 |\sum_{\omega \in \phi(\Sigma_J)} f_\omega|^4$

$\displaystyle + \sum_{I \in {\mathcal I}} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega|^6.$

(We remark that more advanced versions of this “narrow-broad decomposition” in higher dimensions, taking into account more of the geometry of the various frequencies ${\omega}$ that arise in such sums, are useful in both restriction and decoupling theory; see this paper of Guth for more discussion.) From (13) we have

$\displaystyle \int_{{\bf R}^2} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega|^2 |\sum_{\omega \in \phi(\Sigma_J)} f_\omega|^4 \lesssim M_{2,4}(\delta,\nu,\nu,\nu)^6$

while from Proposition 15 we have

$\displaystyle \int_{{\bf R}^2} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega|^6 \lesssim D(\delta/\nu)^6 (\sum_{\omega \in \phi(\Sigma_I)} \| f_\omega\|_{L^6({\bf R}^2)}^2)^3$

and hence

$\displaystyle \sum_{I \in {\mathcal I}} \int_{{\bf R}^2} |\sum_{\omega \in \phi(\Sigma_I)} f_\omega|^6 \lesssim D(\delta/\nu)^6 (\sum_{\omega \in \phi(\Sigma)} \| f_\omega\|_{L^6({\bf R}^2)}^2)^3$

$\displaystyle = D(\delta/\nu)^6.$

Combining all these estimates, we obtain the claim. $\Box$
In practice the ${D(\delta/\nu)}$ term here will be negligible as long as ${\nu}$ is just slightly smaller than ${1}$ (e.g. ${\nu = \delta^\varepsilon}$ for some small ${\varepsilon>0}$). Thus, the above bilinear reduction is asserting that up to powers of ${\nu}$ (which will be an acceptable loss in practice), the quantity ${D(\delta)}$ is basically comparable to ${M_{2,4}(\delta,\nu,\nu,\nu)}$.
If we immediately apply insert (14) into the above lemma, we obtain a useless inequality due to the loss of ${\nu^{-O(1)})}$ in the main term on the right-hand side. To get an improved estimate, we will need a recursive inequality that allows one to slowly gain additional powers of ${\nu}$ at the cost of decreasing the size of ${\rho_1,\rho_2}$ factors (but as long as ${\nu}$ is much larger than ${\delta}$, we will have enough “room” to iterate this inequality repeatedly). The key tool for doing this (and the main reason why we make the rather odd choice of splitting the exponents as ${6=2+4}$) is

Proposition 19 (Key estimate) If ${0 \leq \delta \leq \rho'_1,\rho_1,\rho_2 \leq \nu \leq 1}$ with ${\rho_2^2 \leq \rho'_1 \leq \rho_1}$ then

$\displaystyle M_{2,4}(\delta,\nu,\rho_1,\rho_2) \lesssim \nu^{-O(1)} M_{2,4}(\delta,\nu,\rho'_1,\rho_2).$

Proof: It will suffice to show that

$\displaystyle \int_{{\bf R}^2} |\sum_{\omega_1 \in \phi(\Sigma_1)} f_{\omega_1}|^2 |\sum_{\omega_2 \in \phi(\Sigma_2)} g_{\omega_2}|^4 \lesssim \nu^{-O(1)} M_{2,4}(\delta,\nu,\rho'_1,\rho_2)^6 \ \ \ \ \ (16)$

whenever ${\Sigma_1, \Sigma_2}$ are ${\delta}$-separated subsets of intervals ${I, J \subset [-1,1]}$ of length ${\rho_1, \rho_2}$ respectively with ${\mathrm{dist}(I,J) \geq \nu}$, and ${f_{\omega_1}, g_{\omega_2}}$ have Fourier support on ${\tilde R_{\omega_1,\delta}, \tilde R_{\omega_2,\delta}}$ respectvely for ${\omega_1 \in \phi(\Sigma_1), \omega_2 \in \phi(\Sigma_2)}$, and we have the normalisation

$\displaystyle \sum_{\omega_1 \in \phi(\Sigma_1)} \|f_{\omega_1}\|_{L^6({\bf R}^2)}^2 = \sum_{\omega_2 \in \phi(\Sigma_2)} \|g_{\omega_2}\|_{L^6({\bf R}^2)}^2 = 1.$

We can partition ${\Sigma_1}$ as ${\Sigma_1 = \biguplus_{I' \in {\mathcal I}'} \Sigma_{1,I'}}$, where ${{\mathcal I}'}$ is a collection of intervals ${I' \subset I}$ of length ${\rho'_1}$ that have bounded overlap, and ${\Sigma_{1,I'}}$ is a ${\delta}$-separated subset of ${I'}$. We can then rewrite the left-hand side of (16) as

$\displaystyle \| \sum_{I' \in {\mathcal I}'} F_{I'} G^2 \|_{L^2({\bf R}^2)}^2$

where

$\displaystyle F_{I'} := \sum_{\omega_1 \in \phi(\Sigma_{1,I'})} f_{\omega_1}$

and

$\displaystyle G := \sum_{\omega_2 \in \phi(\Sigma_2)} g_{\omega_2}.$

From (13) we have

$\displaystyle \| F_{I'} G^2 \|_{L^2({\bf R}^2)}^2 \leq M_{2,4}(\delta,\nu,\rho'_1,\rho_2)^6 \sum_{\omega_1 \in \phi(\Sigma_{1,I'})} \|f_{\omega_1}\|_{L^6({\bf R}^2)}^2$

so it will suffice to prove the almost orthogonality estimate

$\displaystyle \| \sum_{I' \in {\mathcal I}'} F_{I'} G^2 \|_{L^2({\bf R}^2)}^2 \lesssim \nu^{-O(1)} \sum_{I' \in {\mathcal I}'} \| F_{I'} G^2 \|_{L^2({\bf R}^2)}^2.$

By Lemma 6(i), it suffices to show that the Fourier supports of ${F_{I'} G^2}$ have overlap ${O(\nu^{-O(1)})}$.
Applying a Galilean transformation, we may normalise the interval ${J}$ to be centered at the origin, thus ${J = [-\rho_2/2,\rho_2/2]}$, and ${I}$ is now at a distance ${\gtrsim \nu}$ from the origin. (Strictly speaking this may push ${I, I'}$ out to now lie in ${[-2,2]}$ rather than ${[-1,1]}$, but this will not make a significant impact to the arguments.) In particular, all the rectangles ${\tilde R_{\omega_2,\delta}}$, ${\omega_2 \in \Sigma_2}$, now lie in a rectangle of the form ${\{ (\xi_1,\xi_2): \xi_1 = O(\rho_2), \xi_2 = O(\rho_2^2 + \delta) \}}$, and hence ${G}$ and ${G^2}$ have Fourier support in such a rectangle also (after enlarging the implied constants in the ${O()}$ notation appropriately). Meanwhile, if ${I' \in {\mathcal I}}$ is centered at ${\xi_{I'}}$, then (since the map ${\xi \mapsto \xi^2}$ has Lipschitz constant ${O(1)}$ when ${\xi \in [-2,2]}$ and ${\delta \leq \rho'_1}$) the parallelogram ${\tilde R_{\omega_1,\delta}}$ is supported in the strip ${\{ (\xi_1,\xi_2): \xi_2 = \xi_{I'}^2 + O( \rho'_1 ) \}}$ for any ${\omega_1 \in \Sigma_{1,I'}}$, hence ${F_{I'}}$ will also be supported in such a strip. Since ${\rho'_1 \geq \rho_2^2, \delta}$, ${F_{I'} G^2}$ is supported in a similar strip (with slightly different implied constants in the ${O()}$ notation). Thus, if ${F_{I'} G^2}$ and ${F_{J'} G^2}$ have overlapping Fourier supports for ${I',J' \in {\mathcal I}'}$, then ${\xi_{I'}^2 = \xi_{J'}^2 + O(\rho'_1)}$, hence (since ${|\xi_{I'}| \gtrsim \nu}$) ${\xi_{I'} = \pm \xi_{J'} + O(\nu^{-O(1)} \rho'_1)}$. Since the intervals ${J'}$ have length ${\rho'_1}$ and bounded overlap, we thus see that each ${I'}$ has at most ${O(\nu^{-O(1)})}$ intervals ${J'}$ for which ${F_{I'} G^2}$ and ${F_{J'} G^2}$ have overlapping Fourier supports, and the claim follows. $\Box$
The final ingredient needed is a simple application of Hölder’s inequality to allow one to (partially) swap ${\rho_1}$ and ${\rho_2}$:

Exercise 20 For any ${0 \leq \delta \leq \rho_1 \leq \rho_2 \leq \nu \leq 1}$, establish the inequality

$\displaystyle M_{2,4}(\delta,\nu,\rho_1,\rho_2) \lesssim M_{2,4}(\delta,\nu,\rho_2,\rho_1)^{1/2} D(\delta/\rho_2)^{1/2}.$

Now we have enough inequalities to establish the claim (9). Let ${\alpha}$ be the least exponent for which we have the bound

$\displaystyle D(\delta) \lesssim_\varepsilon \delta^{-\alpha-\varepsilon}$

for all ${0 < \delta \leq 1}$ and ${\varepsilon>0}$; equivalently, we have

$\displaystyle \alpha = \limsup_{\delta \rightarrow 0} \frac{\log D(\delta)}{\log 1/\delta}.$

Another equivalent formulation is that ${\alpha}$ is the least exponent for which we have the bound

$\displaystyle D(\delta) \leq \delta^{-\alpha-o(1)} \ \ \ \ \ (17)$

as ${\delta \rightarrow 0}$ where ${o(1)}$ denotes a quantity that goes to zero as ${\delta \rightarrow 0}$. Clearly ${\alpha \geq 0}$; our task is to show that ${\alpha=0}$.
Suppose for contradiction that ${\alpha > 0}$. We will establish the bound

$\displaystyle D(\delta) \leq \delta^{-\alpha'-o(1)} \ \ \ \ \ (18)$

as ${\delta \rightarrow 0}$ for some ${\alpha' < \alpha}$, which will give the desired contradiction.
Let ${\nu = \delta^\varepsilon}$ for some small exponent ${\varepsilon>0}$ (independent of ${\delta}$, but depending on ${\alpha}$) to be chosen later. From Proposition 18 and (17) we have

$\displaystyle D(\delta) \lesssim \delta^{-O(\varepsilon)} M_{2,4}(\delta,\nu,\nu,\nu) + \delta^{-(1-\varepsilon)\alpha-o(1)}. \ \ \ \ \ (19)$

Since ${\alpha>0}$, the second term on the right-hand side is already of the desired form; it remains to get a sufficiently good bound on the first term. Note that a direct application of (14), (17) bounds this term by ${\delta^{\alpha + O(\varepsilon) + o(1)}}$; we need to improve this bound by a large multiple of ${\varepsilon}$ to conclude. To obtain this improvement we will repeatedly use Proposition 19 and Exercise 20. Firstly, from Proposition 19 we have

$\displaystyle M_{2,4}(\delta,\nu,\nu,\nu) \lesssim \nu^{-O(1)} M_{2,4}(\delta,\nu,\nu^2,\nu) \ \ \ \ \ (20)$

if ${\varepsilon}$ is small enough. To control the right-hand side, we more generally consider expressions of the form

$\displaystyle M_{2,4}(\delta,\nu,\nu^{2a},\nu^{a})$

for various ${a \geq 1}$; this quantity is well-defined if ${\varepsilon}$ is small enough depending on ${a}$. From Exercise 20 and (17) we have

$\displaystyle M_{2,4}(\delta,\nu,\nu^{2a},\nu^a) \lesssim M_{2,4}(\delta,\nu,\nu^{a},\nu^{2a})^{1/2} \delta^{-\alpha/2-o(1)} \delta^{\alpha a \varepsilon/2}$

and then by Proposition 19

$\displaystyle M_{2,4}(\delta,\nu,\nu^{2a},\nu^a) \lesssim \nu^{-O(1)} M_{2,4}(\delta,\nu,\nu^{4a},\nu^{2a})^{1/2} \delta^{-\alpha/2-o(1)} \delta^{\alpha a \varepsilon/2} \ \ \ \ \ (21)$

if ${\varepsilon}$ is small enough depending on ${a}$. We rearrange this as

$\displaystyle (\delta^{\alpha} M_{2,4}(\delta,\nu,\nu^{2a},\nu^a))^{1/a} \lesssim (\delta^{\alpha} M_{2,4}(\delta,\nu,\nu^{4a},\nu^{2a}))^{1/2a} \delta^{\alpha \varepsilon/2 - O(\varepsilon/a) - o(1)}.$

The crucial fact here is that we gain a small power of ${\delta^\varepsilon}$ on the right-hand side when ${a}$ is large. Iterating this inequality ${k}$ times, we see that

$\displaystyle \delta^{\alpha} M_{2,4}(\delta,\nu,\nu^{2},\nu) \lesssim_k (\delta^{\alpha} M_{2,4}(\delta,\nu,\nu^{2^{k+1}},\nu^{2^k}))^{1/2^k} \delta^{k\alpha \varepsilon/2-O(\varepsilon) - o_k(1)}$

for any given ${k}$, if ${\varepsilon}$ is small enough depending on ${k}$, and ${o_k(1)}$ denotes a quantity that goes to zero in the limit ${\delta \rightarrow 0}$ holding ${k,\varepsilon}$ fixed. Now we can afford to apply (14), (17) and conclude that

$\displaystyle \delta^{\alpha} M_{2,4}(\delta,\nu,\nu^{2},\nu) \lesssim \delta^{k\alpha \varepsilon/2-O(\varepsilon) - o(1)}$

which when inserted back into (20), (19) gives

$\displaystyle D(\delta) \lesssim \delta^{-\alpha + k\alpha \varepsilon/2 - O(\varepsilon) + o(1)} + \delta^{-\alpha + \varepsilon \alpha + o(1)}.$

If we then choose ${k}$ large enough depending on ${\alpha}$, and ${\varepsilon}$ small enough depending on ${k}$, we obtain the desired improved bound (18).

Remark 21 An alternate arrangement of the above argument is as follows. For any exponent ${\theta}$, let ${P(\theta)}$ denote the claim that

$\displaystyle M_{2,4}(\delta,\nu,\nu^{2a},\nu^{a}) \lesssim_\theta \delta^{-\alpha + o(1)} \nu^{a\theta + O_\theta(1)} \ \ \ \ \ (22)$

whenever ${a \geq 1}$, ${\nu = \delta^\varepsilon}$ with ${\varepsilon}$ sufficiently small depending on ${a}$, and ${\delta}$ is sent to zero holding ${a,\varepsilon}$ fixed. The bounds (14), (17) give the claim ${P(0)}$. On the other hand, the bound (21) shows that ${P(\theta)}$ implies ${P(\theta+\alpha/2)}$ for any given ${\theta}$. Thus if ${\alpha>0}$, we can establish ${P(\theta)}$ for arbitrarily large ${\theta}$, and for ${\theta}$ large enough we can insert the bound (22) (with ${a}$ sufficiently large depending on ${\theta}$) into (19), (20) to obtain the required claim (18). See this blog post for a further elaboration of this approach, which allows one to systematically determine the optimal exponents one can conclude from a system of inequalities of the type one sees in Proposition 19 or Exercise 20 (it boils down to computing the Perron-Frobenius eigenvalue of certain matrices).

Exercise 22 By carefully unpacking the above iterative arguments, establish a bound of the form

$\displaystyle D(\delta) \lesssim \exp( O( \frac{\log 1/\delta}{\log\log 1/\delta} ) )$

for all sufficiently small ${\delta>0}$. (This bound was first established by Li.)

Exercise 23 (Localised decoupling) Let ${0 < \delta < 1}$, and let ${{\mathcal I}}$ be a family of boundedly overlapping intervals ${I}$ in ${[-1,1]}$ of length ${\delta}$. For each ${I \in {\mathcal I}}$, let ${g_I \in L^1({\bf R})}$ be an integrable function supported on ${I}$, and let ${{\mathcal E} g_I}$ denote the extension operator

$\displaystyle {\mathcal E} g_I(x_1,x_2) := \int_{\bf R} g_I(\xi) e^{2\pi i (\xi x_1 + \xi^2 x_2)}\ d\xi.$

For any ball ${B = B(x_0, \delta^{-2})}$ of radius ${\delta^{-2}}$, use Theorem 13 to establish the local decoupling inequality

$\displaystyle \| \sum_{I \in {\mathcal I}} {\mathcal E} g_I \|_{L^6(B)} \lesssim_\varepsilon \delta^{-\varepsilon} (\sum_{I \in {\mathcal I}} \|{\mathcal E} g_I\|_{L^6({\bf R}^2, w_B\ dx)}^2)^{1/2}$

for any ${\varepsilon>0}$, where ${w_B}$ is the weight function

$\displaystyle w_B(x) := \langle \frac{x-x_0}{\delta^{-2}} \rangle^{-100}.$

The decoupling theorem for the parabola has been extended in a number of directions. Bourgain and Demeter obtained the analogous decoupling theorem for the paraboloid:

Theorem 24 (Decoupling for the paraboloid) Let ${d \geq 2}$, let ${0 < \delta < 1}$, let ${\Omega = \phi(\Sigma)}$ for some ${\delta}$-separated subset ${\Sigma}$ of ${[-1,1]^{d-1}}$, where ${\phi: {\bf R}^{d-1} \rightarrow {\bf R}^d}$ is the map ${\phi(\xi) := (\xi, |\xi|^2)}$, and to each ${\omega = \phi(\xi) \in \Omega}$ let ${D_\omega}$ be the disk

$\displaystyle D_\omega := \{ (\xi', \xi_d): |\xi' - \xi| \leq \delta; |\xi_d - |\xi|^2 - 2 \xi \cdot (\xi'-\xi)| \leq \delta^2 \}.$

Then one has

$\displaystyle \mathrm{Dec}_{\frac{2(d+1)}{d-1}}( \{ D_{\omega}: \omega \in \Omega \} ) \lesssim_{d,\varepsilon} \delta^{-\varepsilon}$

for any ${\varepsilon>0}$.

Clearly Theorem 13 is the ${d=2}$ case of Theorem 24.

Exercise 25 Show that the exponent ${\frac{2(d+1)}{d-1}}$ in Theorem 24 cannot be replaced by any larger exponent.

We will not prove Theorem 24 here; the proof in Bourgain-Demeter shares some features in common with the one given above (for instance, it focuses on a ${d}$-linear formulation of the decoupling problem, though not one that corresponds precisely to the bilinear formulation given above), but also involves some additional ingredients, such as the wave packet decomposition and the multilinear restriction theorem from Notes 1.
Somewhat analogously to how the multilinear Kakeya conjecture could be used in Notes 1 to establish the multilinear restriction conjecture (up to some epsilon losses) by an induction on scales argument, the decoupling theorem for the paraboloid can be used to establish decoupling theorems for other surfaces, such as the sphere:

Exercise 26 (Decoupling for the sphere) Let ${d \geq 2}$, let ${0 < \delta < 1}$, and let ${\Omega}$ be a ${\delta}$-separated subset of the sphere ${S^{d-1}}$. To each ${\omega \in \Omega}$, let ${\tilde D_\omega}$ be the disk

$\displaystyle \tilde D_\omega := \{ \xi: |\xi \cdot \omega - 1| \leq \delta; |\xi - (\xi \cdot \omega) \omega| \leq \delta^2\}.$

Assuming Theorem 24, establish the bound

$\displaystyle \mathrm{Dec}_{\frac{2(d+1)}{d-1}}( \{ \tilde D_{\omega}: \omega \in \Omega \} ) \lesssim_{d,\varepsilon} \delta^{-\varepsilon} \ \ \ \ \ (23)$

for any ${\varepsilon>0}$. (Hint: if one lets ${\tilde D(\delta)}$ denote the supremum over all expressions of the form of the left-hand side of (23), use Exercise 10 and Theorem 24 to establish a bound of the form ${\tilde D(\delta) \lesssim_{d,\varepsilon} \delta^{-\varepsilon} \tilde D(\sqrt{\delta})}$, taking advantage of the fact that a sphere resembles a paraboloid at small scales. This argument can also be found in the above-mentioned paper of Bourgain and Demeter.)

An induction on scales argument (somewhat similar to the one used to establish the multilinear Kakeya estimate in Notes 1) can similarly be used to establish decoupling theorems for the cone

$\displaystyle \{ (\xi_1,\xi_2,\xi_3): \xi_1^2 + \xi_2^2 = \xi_3^2\}$

from the decoupling theorem for the parabola (Theorem 13). It will be convenient to rewrite the equation for the cone as ${\xi_1^2 = (\xi_2+\xi_3) (\xi_3-\xi_2)}$, then perform a linear change of variables to work with the tilted cone

$\displaystyle \{ (\xi_1,\xi_2,\xi_3): \xi_1^2 = \xi_2 \xi_3 \}$

which can be viewed as a projective version of the parabola ${\{ (\xi_1,\xi_2): \xi_1^2 = \xi_2 \}}$.

Exercise 27 (Decoupling for the cone) For ${0 < \delta \leq 1}$, let ${D_C(\delta)}$ denote the supremum of the decoupling constants

$\displaystyle \mathrm{Dec}_6( \{ S_{\theta,\delta}: \theta \in \Sigma \} )$

where ${\Sigma}$ ranges over ${\delta}$-separated subsets of ${[-1,1]}$, and ${S_{\theta,\delta}}$ denotes the sector

$\displaystyle \{ (\xi_1,\xi_2,\xi_3): |\frac{\xi_1}{\xi_3} - \theta| \leq \delta; |\frac{\xi_2}{\xi_3} - \frac{\xi_1^2}{\xi^2_3}| \leq \delta^2; 1 \leq \xi_3 \leq 2 \}.$

More generally, if ${0 < \nu \leq 1}$, let ${D_C(\delta,\nu)}$ denote the supremum of the decoupling constants

$\displaystyle \mathrm{Dec}_6( \{ S_{\theta,\delta,\nu}: \theta \in \Sigma \} )$

where ${\Sigma}$ ranges over ${\delta}$-separated subsets of ${[-1,1]}$, and ${S_{\theta,\delta,\nu}}$ denotes the shortened sector

$\displaystyle \{ (\xi_1,\xi_2,\xi_3): |\frac{\xi_1}{\xi_3} - \theta| \leq \delta; |\frac{\xi_2}{\xi_3} - \frac{\xi_1^2}{\xi^2_3}| \leq \delta^2; 1 \leq \xi_3 \leq 1+\nu \}.$

• (i) For any ${0 < \delta,\nu \leq 1}$, show that ${D_C(\delta) \lesssim \nu^{-O(1)} D_C(\delta,\nu)}$.
• (ii) For any ${0 < \nu \leq 1}$, show that ${D_C(\nu^2, \nu) \lesssim_\varepsilon \nu^{-\varepsilon}}$ for any ${\varepsilon>0}$. (Hint: use Theorem 13 and various parts of Exercise 10, exploiting the geometric fact that thin slices of the tilted cone resemble the Cartesian product of a parabola and a short interval.)
• (iii) For any ${0 < \delta_1, \delta_2, \nu \leq 1}$, show that ${D_C(\delta_1 \delta_2, \nu) \lesssim D_C(\delta_1, \nu) D_C(\delta_2,\nu)}$. (Hint: adapt the argument used to establish Exercise 16, taking advantage of the invariance of the tilted light cone under projective parabolic rescaling ${(\xi_1,\xi_2,\xi_3) \mapsto (\lambda \xi_1, \lambda^2 \xi_2, \xi_3)}$ and projective Galilean transformations ${(\xi_1,\xi_2,\xi_3) \mapsto (\xi_1 + \eta \xi_3, \xi_2 + 2 \eta \xi_1 + \eta^2 \xi_3, \xi_3)}$; these maps can also be viewed as tilted (conformal) Lorentz transformations ).
• (iv) Show that ${D_C(\delta) \lesssim_\varepsilon \delta^{-\varepsilon}}$ for any ${\varepsilon>0}$ and ${0 < \delta \leq 1}$.
• (v) State and prove a generalisation of (iv) to higher dimensions, using Theorem 24 in place of Theorem 13.

This argument can also be found in the above-mentioned paper of Bourgain and Demeter.

A separate generalization of Theorem 13, to the moment curve

$\displaystyle \{ (t, t^2, \dots, t^d): t \in {\bf R}\}$

was obtained by Bourgain, Demeter, and Guth:

Theorem 28 (Decoupling for the moment curve) Let ${d \geq 2}$, let ${0 < \delta \leq 1}$, and let ${\Sigma}$ be a ${\delta}$-separated subset of ${[-1,1]}$. For each ${t \in \Sigma}$, let ${R_{t,\delta} \subset {\bf R}^d}$ denote the region

$\displaystyle R_{t,\delta} := \{ \xi \in {\bf R}^d: |\xi - \phi(t')| \leq \delta^d \hbox{ for some } t' \in [t-\delta,t+\delta] \}$

where ${\phi: {\bf R} \rightarrow {\bf R}^d}$ is the map

$\displaystyle \phi(t) := (t,t^2,\dots,t^d).$

Then

$\displaystyle \mathrm{Dec}_{d(d+1)}( \{ R_{t,\delta}: t \in \Sigma \} ) \lesssim_{d,\varepsilon} \delta^{-\varepsilon}$

for any ${\varepsilon>0}$.

Exercise 29 Show that the exponent ${d(d+1)}$ in Theorem 28 cannot be replaced by any higher exponent.

It is not difficult to use Exercise 10 to deduce Theorem 13 from the ${d=2}$ case of Theorem 28 (the only issue being that the regions ${R_{t,\delta}}$ are not quite the same the parallelograms ${\tilde R_{(t,t^2),\delta}}$ appearing in Theorem 13).
The original proof of Theorem 28 by Bourgain-Demeter-Guth was rather intricate, using for instance a version of the multilinear Kakeya estimate from Notes 1. A shorter proof, similar to the one used to prove Theorem 13 in these notes, was recently given by Guo, Li, Yung, and Zorin-Kranich, adapting the “nested efficient congruencing” method of Wooley, which we will not discuss here, save to say that this method can be viewed as a ${p}$-adic counterpart to decoupling techniques. See also this paper of Wooley for an alternate approach to (a slightly specialised version of) Theorem 28.
Perhaps the most striking application of Theorem 28 is the following conjecture of Vinogradov:

Exercise 30 (Main conjecture for the Vinogradov mean value theorem) Let ${d \geq 2}$. For any ${s \geq 1}$ and any ${N \geq 1}$, let ${J_{s,d}(N)}$ denote the quantity

$\displaystyle J_{s,d}(N) := \int_{[0,1]^d} |\sum_{n=1}^N e( x_1 n + x_2 n^2 + \dots + x_d n^d )|^{2s}\ dx_1 \dots dx_n$

where ${e(\theta) := e^{2\pi i \theta}}$.

• (i) If ${s}$ is a natural number, show that ${J_{s,d}(N)}$ is equal to the number of tuples ${(X_1,\dots,X_{2s}) \in \{1,\dots,N\}^{2s}}$ of natural numbers between ${1}$ and ${N}$ obeying the system of equations

$\displaystyle X_1^i + \dots + X_s^i = X_{s+1}^i + \dots + X_{2s}^i$

for ${i=1,\dots,d}$.

• (ii) Using Theorem 28, establish the bound

$\displaystyle J_{d(d+1)/2,d}(N) \lesssim_{d,\varepsilon} N^{d(d+1)+\varepsilon}$

for all ${\varepsilon >0}$. (Hint: set ${\delta = 1/N}$ and ${\Sigma = \{i/N: i=1,\dots,N\}}$, and apply the decoupling inequality to functions ${f_t}$ that are adapted to a small ball around ${\phi(t)}$.)

• (iii) More generally, establish the bound

$\displaystyle J_{s,d}(N) \lesssim_{d,s,\varepsilon} N^{s+\varepsilon} + N^{2s - \frac{d(d+1)}{2} + \varepsilon} \ \ \ \ \ (24)$

for any ${s \geq 1}$ and ${\varepsilon>0}$. Show that this bound is best possible up to the implied constant and the loss of ${N^\varepsilon}$ factors.

Remark 31 Estimates of the form (24) are known as mean value theorems, and were first established by Vinogradov in 1937 in the case when ${s}$ was sufficiently large (and by Hua when ${s}$ was sufficiently small). These estimates in turn had several applications in analytic number theory, most notably the Waring problem and in establishing zero-free regions for the Riemann zeta function; see these previous lecture notes for more discussion. The ranges of ${d,s}$ for which (24) was established was improved over the years, with much recent progress by Wooley using his method of efficient congruencing; see this survey of Pierce for a detailed history. In particular, these methods can supply an alternate proof of (24); see this paper of Wooley.

Exercise 32 (Discrete restriction) Let ${0 < \delta < 1}$ and ${d \geq 2}$, and let ${\Omega}$ be a ${\delta}$-separated subset of either the unit sphere ${S^{d-1}}$ or the paraboloid ${\{ (\xi', |\xi'|^2): \xi' \in [-1,1]^{d-1} \}}$. Using Theorem 24 and Exercise 26, show that for any radius ${R \geq \delta^{-2}}$ and any complex numbers ${a_\omega, \omega \in \Omega}$, one has the discrete restriction estimate

$\displaystyle (\frac{1}{|B(0,R)|} \int_{B(0,R)} |\sum_{\omega \in \Omega} a_\omega e^{2\pi i x \cdot \omega}|^{\frac{2(d+1)}{d-1}}\ dx)^{\frac{d-1}{2(d+1)}}$

$\displaystyle \lesssim_\varepsilon \delta^{-\varepsilon} (\sum_{\omega \in \Omega} |a_\omega|^2)^{1/2}.$

Explain why the exponent ${\frac{2(d+1)}{d-1}}$ here cannot be replaced by any larger exponent, and also explain why the exponent ${2}$ in the condition ${R \geq \delta^{-2}}$ cannot be lowered.

For further applications of decoupling estimates, such as restriction and Strichartz estimates on tori, and application to combinatorial incidence geometry, see the text of Demeter.
[These exercises will be moved to a more appropriate location at the end of the course, but are placed here for now so as not to affect numbering of existing exercises.]

Exercise 33 Show that the inequality in (8) is actually an equality, if ${A_2 = \| \sum_{j=1}^n 1_{U_j} \|_{L^\infty({\bf R}^d)}}$ is the maximal overlap of the ${U_j}$.

Exercise 34 Show that ${D(\delta') \lesssim D(\delta)}$ whenever ${0 < \delta \leq \delta' \leq 1}$. (Hint: despite superficial similarity, this is not related to Lemma 14. Instead, adapt the parabolic rescaling argument used to establish Proposition 15.)