\stackMath

Sharp superlevel set estimates for small cap decouplings of the parabola

Yuqiu Fu, Larry Guth, and Dominique Maldague

Abstract.

We prove sharp bounds for the size of superlevel sets $\{x\in\mathbb{R}^{2}:|f(x)|>\alpha\}$ where $\alpha>0$ and $f:\mathbb{R}^{2}\to\mathbb{C}$ is a Schwartz function with Fourier transform supported in an $R^{-1}$ -neighborhood of the truncated parabola $\mathbb{P}^{1}$ . These estimates imply the small cap decoupling theorem for $\mathbb{P}^{1}$ from [DGW20] and the canonical decoupling theorem for $\mathbb{P}^{1}$ from [BD15]. New $(\ell^{q},L^{p})$ small cap decoupling inequalities also follow from our sharp level set estimates.

In this paper, we further develop the high/low frequency proof of decoupling for the parabola [GMW20] to prove sharp level set estimates which recover and refine the small cap decoupling results for the parabola in [DGW20]. We begin by describing the problem and our results in terms of exponential sums. The main results in full generality are in §1.

For $N\geq 1$ , $R\in[N,N^{2}]$ , and $2\leq p$ , let $D(N,R,p)$ denote the smallest constant so that

(1)

|Q_{R}|^{-1}\int_{Q_{R}}|\sum_{\xi\in\Xi}a_{\xi}e((x,t)\cdot(\xi,\xi^{2}))|^{p}dxdt\leq D(N,R,p)N^{p/2}

for any collection $\Xi\subset[-1,1]$ with $|\Xi|\sim N$ consisting of $\sim\frac{1}{N}$ -separated points, $a_{\xi}\in\mathbb{C}$ with $|a_{\xi}|\sim 1$ , and any cube $Q_{R}\subset\mathbb{R}^{2}$ of sidelength $R$ .

A corollary of the small cap decoupling theorem for the parabola in [DGW20] is that if $2\leq p\leq 2+2s$ for $R=N^{s}$ , then

(2)

D(N,R,p)\leq C_{\varepsilon}N^{\varepsilon}.

This estimate is sharp, up to the $C_{\varepsilon}N^{\varepsilon}$ factor, which may be seen by Khintchine’s inequality. The range $2\leq p\leq 2+2s$ is the largest range of $p$ for which $D(N,R,p)$ may be bounded by sub-polynomial factors in $N$ . The case $R=N^{2}$ of (2) follows from the canonical $\ell^{2}$ decoupling theorem of Bourgain and Demeter for the parabola [BD15]. For $R<N^{2}$ and the subset $\Xi=\{k/N\}_{k=1}^{N}$ , the inequality (1) is an estimate for the moments of exponential sums over subsets smaller than the full domain of periodicity (i.e. $N^{2}$ in the $t$ -variable). Bourgain investigated examples of this type of inequality in [Bou17b, Bou17a].

By a pigeonholing argument (see Section 5 of [GMW20]), (2) follows from upper bounds for superlevel sets $U_{\alpha}$ defined by

U_{\alpha}=\{(x,t)\in\mathbb{R}^{2}:|\sum_{\xi\in\Xi}a_{\xi}e((x,t)\cdot(\xi,\xi^{2}))|>\alpha\}.

In particular, (2) is equivalent, up to a $\log N$ factor, to proving that for any $\alpha>0$ and for $R=N^{s}$ ,

(3)

\alpha^{2+2s}|U_{\alpha}\cap Q_{R}|\leq C_{\varepsilon}R^{\varepsilon}N^{1+s}R^{2}

when $\Xi$ , $a_{\xi}$ satisfy the hypotheses following (1). In this paper, we improve the above superlevel set estimate for all $\alpha>0$ strictly between $N^{1/2}$ and $N$ .

Theorem 1.

Let $R\in[N,N^{2}]$ . For any $\varepsilon>0$ , there exists $C_{\varepsilon}<\infty$ such that

|U_{\alpha}\cap Q_{R}|\leq C_{\varepsilon}N^{\varepsilon}\begin{cases}\frac{N^{2}R}{\alpha^{4}}\sum\limits_{\xi\in\Xi}|a_{\xi}|^{2}\quad&\text{if}\quad\alpha^{2}>R\\ \frac{N^{2}R^{2}}{\alpha^{6}}\sum\limits_{\xi\in\Xi}|a_{\xi}|^{2}\quad&\text{if}\quad N\leq\alpha^{2}\leq R\\ R^{2}\quad&\text{if}\quad\alpha^{2}<N.\end{cases}

whenever $\Xi\subset[-1,1]$ is a $\gtrsim\frac{1}{N}$ -separated subset, $|a_{\xi}|\leq 1$ for each $\xi\in\Xi$ , and $Q_{R}\subset\mathbb{R}^{2}$ is a cube of sidelength $R$ .

Our superlevel set estimates are essentially sharp, which follows from analyzing the function $F(x,t)=\sum_{n=1}^{N}e((x,t)\cdot(\frac{n}{N},\frac{n^{2}}{N^{2}}))$ . It is not known whether the implicit constant in the upper bound of (2) goes to infinity with $N$ except in the case that $p=6$ and $s=2$ , when the same example $F(x,t)=\sum_{n=1}^{N}e((x,t)\cdot(\frac{n}{N},\frac{n^{2}}{N^{2}}))$ shows that $D(N,N^{2},6)\gtrsim(\log N)$ [Bou93]. Roughly, the argument is that for each dyadic value $\alpha\in[N^{3/4},N]$ , one can show by counting the “major arcs” that

\alpha^{6}\{(x,t)\in Q_{N^{2}}:|F(x,t)|\sim\alpha\}|\gtrsim N^{4}\cdot N^{3}.

Since there are $\sim\log N$ values of $\alpha$ , the lower bound for $\int_{Q_{N^{2}}}|F|^{6}$ follows. Theorem 1 implies that the corresponding superlevel set estimates (3) are not sharp for $1\leq s<2$ , unless $\alpha\sim N$ or $\alpha^{2}\sim N$ , which leads to the following conjecture.

Conjecture 2.

Let $s\in[1,2)$ and $2\leq p\leq 2+2s$ . There exists $C(s)>0$ so that

D(N,N^{s},p)\leq C(s).

A more refined version of Theorem 1 leads to the following essentially sharp $(\ell^{q},L^{p})$ small cap decoupling theorem, stated here for general exponential sums.

Corollary 1.

Let $\frac{3}{p}+\frac{1}{q}\leq 1,$ and let $R\in[N,N^{2}].$ Then for each $\varepsilon>0$ , there exists $C_{\varepsilon}<\infty$ so that

\|\sum_{\xi\in\Xi}a_{\xi}e((x,t)\cdot(\xi,\xi^{2}))\|_{L^{p}(B_{R})}\leq C_{\varepsilon}N^{\varepsilon}(N^{1-\frac{1}{p}-\frac{1}{q}}R^{\frac{1}{p}}+N^{\frac{1}{2}-\frac{1}{q}}R^{\frac{2}{p}})(\sum_{\xi}|a_{\xi}|^{q})^{1/q}.

In the above corollary, the assumptions are that $\Xi$ is a $\gtrsim\frac{1}{N}$ -separated subset of $[-1,1]$ and that $a_{\xi}\in\mathbb{C}$ .

1. Main results

We state our main results in the more general set-up for decoupling. Let $\mathbb{P}^{1}$ denote the truncated parabola

\{(t,t^{2}):|t|\leq 1\}

and write $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ for the $R^{-1}$ -neighborhood of $\mathbb{P}^{1}$ in $\mathbb{R}^{2}$ , where $R\geq 2$ . For a partition $\{\gamma\}$ of $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ into almost rectangular blocks, an $(\ell^{2},L^{p})$ decoupling inequality is

(4)

\|f\|_{L^{p}(B_{R})}\leq D(R,p)(\sum_{\gamma}\|f_{\gamma}\|_{L^{p}(\mathbb{R}^{2})}^{2})^{1/2}

in which $f:\mathbb{R}^{2}\to\mathbb{C}$ is a Schwartz function with $\mathrm{supp}\widehat{f}\subset\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ and $f_{\gamma}$ means the Fourier projection onto $\gamma$ , defined precisely below. When we refer to canonical caps or to canonical decoupling, we mean that $\gamma$ are approximately $R^{-1/2}\times R^{-1}$ blocks corresponding to the $\ell^{2}$ -decoupling paper of [BD15]. In this paper, we allow $\gamma$ to be approximate $R^{-\beta}\times R^{-1}$ blocks, where $\frac{1}{2}\leq\beta\leq 1$ . This is the “small cap” regime studied in [DGW20]. We also consider $(\ell^{q},L^{p})$ decoupling for small caps, which replaces $(\sum_{\gamma}\|f_{\gamma}\|_{p}^{2})^{1/2}$ by $(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{1/q}$ in the decoupling inequality above (see Corollary 5).

To precisely discuss the collection $\{\gamma\}$ , fix a $\beta\in[\frac{1}{2},1]$ . Let $\mathcal{P}=\mathcal{P}(R,\beta)=\{\gamma\}$ be the partition of $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ given by

(5)

\bigsqcup_{|k|\leq\lceil R^{\beta}\rceil-2}\{(x,t)\in\mathcal{N}_{R^{-1}}(\mathbb{P}^{1}):k\lceil R^{\beta}\rceil^{-1}\leq x<(k+1)\lceil R^{\beta}\rceil^{-1}\}

and the two end pieces

\{(x,t)\in\mathcal{N}_{R^{-1}}(\mathbb{P}^{1}):x<-1+\lceil R^{\beta}\rceil^{-1}\}\sqcup\{(x,t)\in\mathcal{N}_{R^{-1}}(\mathbb{P}^{1}):1-\lceil R^{\beta}\rceil^{-1}\leq x\}.

For a Schwartz function $f:\mathbb{R}^{2}\to\mathbb{C}$ with $\mathrm{supp}\widehat{f}\subset\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ , define for each $\gamma\in\mathcal{P}(R,\beta)$

f_{\gamma}(x):=\int_{\gamma}\widehat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi.

For $a,b>0$ , the notation $a\lesssim b$ means that $a\leq Cb$ where $C>0$ is a universal constant whose definition varies from line to line, but which only depends on fixed parameters of the problem. Also, $a\sim b$ means $C^{-1}b\leq a\leq Cb$ for a universal constant $C$ .

Let $U_{\alpha}:=\{x\in\mathbb{R}^{2}:|f(x)|\geq\alpha\}$ . In Section 5 of [GMW20], through a wave packet decomposition and series of pigeonholing steps, bounds for $D(R,p)$ in (4) follow (with an additional power of $(\log R)$ ) from bounds on the constant $C(R,p)$ in

\alpha^{p}|U_{\alpha}|\leq C(R,p)(\#\{\gamma:f_{\gamma}\not=0\})^{\frac{p}{2}-1}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}

for any $\alpha>0$ and under the additional assumptions that $\|f_{\gamma}\|_{\infty}\lesssim 1$ , $\|f_{\gamma}\|_{p}^{p}\sim\|f_{\gamma}\|_{2}^{2}$ for each $\gamma$ . Thus decoupling bounds follow from upper bounds on the superlevel set $|U_{\alpha}|$ . In this paper, we consider the question: given $\alpha>0$ and a partition $\{\gamma\}$ , how large can $|U_{\alpha}|$ be, varying over functions $f$ satisfying $\|f_{\gamma}\|_{\infty}\lesssim 1$ for each $\gamma$ ? We answer this question in the following theorem.

Theorem 3.

Let $\beta\in[\frac{1}{2},1]$ , $R\geq 2$ . Let $f:\mathbb{R}^{2}\to\mathbb{C}$ be a Schwartz function with Fourier transform supported in $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ satisfying $\|f_{\gamma}\|_{\infty}\leq 1$ for all $\gamma\in\mathcal{P}(R,\beta)$ . Then for any $\alpha>0$ ,

|U_{\alpha}\cap[-R,R]^{2}|\leq C_{\varepsilon}R^{\varepsilon}\begin{cases}\frac{R^{2\beta-1}}{\alpha^{4}}\sum\limits_{\gamma}\|f_{\gamma}\|_{L^{2}(\mathbb{R}^{2})}^{2}\quad&\text{if}\quad\alpha^{2}>R\\ \frac{R^{2\beta}}{\alpha^{6}}\sum\limits_{\gamma}\|f_{\gamma}\|_{L^{2}(\mathbb{R}^{2})}^{2}\quad&\text{if}\quad R^{\beta}\leq\alpha^{2}\leq R\\ R^{2}\quad&\text{if}\quad\alpha^{2}<R^{\beta}.\end{cases}

Each bound in Theorem 3 is sharp, up to the $C_{\varepsilon}R^{\varepsilon}$ factor, which we show in §2.

Define notation for a distribution function for the Fourier support of a Schwartz function $f$ with Fourier transform supported in $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ as follows. For each $0\leq s\leq 2$ , let

\lambda(s)=\sup_{\omega(s)}\#\{\gamma:\gamma\cap\omega(s)\not=\emptyset,\,\,f_{\gamma}\not=0\}

where $\omega(s)$ is any arc of $\mathbb{P}^{1}$ with projection onto the $\xi_{1}$ -axis equal to an interval of length $s$ . The following theorem implies Theorem 3 and replaces factors of $R^{\beta}$ in the upper bounds from Theorem 3 by expressions involving $\lambda(\cdot)$ , which see the actual Fourier support of the input function $f$ .

Theorem 4.

Let $\beta\in[\frac{1}{2},1]$ , $R\geq 2$ . For any $f$ with Fourier transform supported in $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ satisfying $\|f_{\gamma}\|_{\infty}\lesssim 1$ for each $\gamma\in\mathcal{P}(R,\beta)$ ,

|U_{\alpha}|\leq C_{\varepsilon}R^{\varepsilon}\begin{cases}\frac{1}{\alpha^{4}}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}\quad&\text{if}\quad\alpha^{2}>\frac{\lambda(1)^{2}}{\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)}\\ \frac{\lambda(1)^{2}}{\alpha^{6}}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}\quad&\text{if}\quad\alpha^{2}\leq\frac{\lambda(1)^{2}}{\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)}\end{cases}

in which the maxima are taken over dyadic $s$ , $R^{-\beta}\leq s\leq R^{-1/2}$ .

Corollary 5 ( $(l^{q},L^{p})$ small cap decoupling).

Let $\frac{3}{p}+\frac{1}{q}\leq 1$ . Then

\|f\|_{L^{p}(B_{R})}\leq C_{\varepsilon}R^{\varepsilon}(R^{\beta(1-\frac{1}{q})-\frac{1}{p}(1+\beta)}+R^{\beta(\frac{1}{2}-\frac{1}{q})})(\sum_{\gamma}\|f_{\gamma}\|_{L^{p}(\mathbb{R}^{2})}^{q})^{1/q}

whenever $f$ is a Schwartz function with Fourier transform supported in $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ .

The powers of $R$ in the upper bound come from considering two natural sharp examples for the ratio $\|f\|_{L^{p}(B_{R})}^{p}/(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{p/q}$ . The first is the square root cancellation example, where $|f_{\gamma}|\sim\chi_{B_{R}}$ for all $\gamma$ and $f=\sum_{\gamma}e_{\gamma}f_{\gamma}$ where $e_{\gamma}$ are $\pm 1$ signs chosen (using Khintchine’s inequality) so that $\|f\|_{L^{p}(B_{R})}^{p}\sim R^{\beta p/2}R^{2}$ .

\|f\|_{p}^{p}/(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{p/q}\gtrsim(R^{\beta p/2}R^{2})/(R^{\beta p/q}R^{2})\sim R^{\beta p(\frac{1}{2}-\frac{1}{q})}.

The second example is the constructive interference example. Let $f_{\gamma}=R^{1+\beta}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\gamma}$ where $\eta_{\gamma}$ is a smooth bump function approximating $\chi_{\gamma}$ . Since $|f|=|\sum_{\gamma}f_{\gamma}|$ is approximately constant on unit balls and $|f(0)|\sim R^{\beta}$ , we have

\|f\|_{p}^{p}/(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{p/q}\gtrsim(R^{\beta p})/(R^{\beta p/q}R^{1+\beta})\sim R^{\beta p(1-\frac{1}{q})-1-\beta}.

There is one more example which may dominate the ratio: The block example is $f=R^{1+\beta}\sum_{\gamma\subset\theta}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\gamma}$ where $\theta$ is a canonical $R^{-1/2}\times R^{-1}$ block. Since $f=f_{\theta}$ and $|f_{\theta}|$ is approximately constant on dual $\sim R^{1/2}\times R$ blocks $\theta^{*}$ , we have

\frac{\alpha^{p}|U_{\alpha}|}{(\#\gamma)^{\frac{p}{q}}\|f_{\gamma}\|_{2}^{2}}\gtrsim\frac{R^{(\beta-\frac{1}{2})p}R^{\frac{3}{2}}}{R^{(\beta-\frac{1}{2})\frac{p}{q}}R^{1+\beta}}=R^{(\beta-\frac{1}{2})p(1-\frac{1}{q})+\frac{1}{2}-\beta}.

One may check that the constructive interference examples dominate the block example when $\frac{3}{p}+\frac{1}{q}\leq 1$ . We do not investigate $(l^{q},L^{p})$ small cap decoupling in the range $\frac{3}{p}+\frac{1}{q}>1$ in the present paper.

The paper is organized as follows. In §2, we demonstrate that Theorem 3 is sharp using an exponential sum example. In §3, we show how Theorem 3 follows easily from Theorem 4 and how after some pigeonholing steps, so does Corollary 5. Then in §4, we develop the multi-scale high/low frequency tools we use in the proof of Theorem 4. These tools are very similar to those developed in [GMW20]. It appears that a more careful version of the proof of Theorem 4 could also replace the $C_{\varepsilon}R^{\varepsilon}$ factor by a power of $(\log R)$ , as is done for canonical decoupling in [GMW20]. Finally, in §5, we prove a bilinear version of Theorem 4 and then reduce to the bilinear case to finish the proof.

LG is supported by a Simons Investigator grant. DM is supported by the National Science Foundation under Award No. 2103249.

2. A sharp example

Because we will show that Theorem 4 implies Theorem 3, it suffices to show that Theorem 3 is sharp, which we mean up to a $C_{\varepsilon}R^{\varepsilon}$ factor. Write $N=\lceil R^{\beta}\rceil$ . The function achieving the sharp bounds is

F(x_{1},x_{2})=\sum_{k=1}^{N}e(\frac{k}{N}x_{1}+\frac{k^{2}}{N^{2}}x_{2})\eta(x_{1},x_{2}),

where $\eta$ is a Schwartz function satisfying $\eta\sim 1$ on $[-R,R]^{2}$ and $\text{supp }\widehat{\eta}\subset B_{R^{-1}}$ . We will bound the set

U_{\alpha}=\{(x_{1},x_{2})\in[-R,R]^{2}:|F(x_{1},x_{2})|\geq\alpha\}.

Case 1: $R<\alpha^{2}$ .

Suppose that $\alpha\sim N$ and note that $F(0,0)=N$ and $|F(0,0)|\sim N$ when $|(x_{1},x_{2})|<\frac{1}{10^{3}}$ . Using periodicity in the $x_{1}$ variable, there are $\sim R/N$ many other heavy balls where $|F(x)|\sim N$ in $[-R,R]^{2}$ . For $\alpha$ in the range suppose that $R<\alpha^{2}<N^{2}$ , we will show that $U_{\alpha}$ is dominated by larger neighborhoods of the heavy balls.

Let $r=N^{2}/\alpha^{2}$ and assume without loss of generality that $r$ is in the range $R^{\varepsilon}<r<N^{2}/R\sim R^{2\beta-1}\ll N$ . The upper bound for $|U_{\alpha}|$ in Theorem 3 for this range is

|U_{\alpha}|\leq C_{\varepsilon}R^{\varepsilon}\frac{N^{2}}{\alpha^{4}R}\sum_{\gamma}\|F_{\gamma}\|_{2}^{2}\sim C_{\varepsilon}R^{\varepsilon}\frac{N^{2}}{\alpha^{4}R}NR^{2}.

To demonstrate that this inequality is sharp, by periodicity in $x_{1}$ , it suffices to show that $|U_{\alpha}\cap B_{r}|\gtrsim r^{2}$ . Let $\phi_{r^{-1}}$ be a nonnegative bump function supported in $B_{r^{-1}/2}$ with $\phi_{r^{-1}}\gtrsim 1$ on $B_{r^{-1}/4}$ . Let $\eta_{r}=r^{4}({\phi_{r^{-1}}*\phi_{r^{-1}}})^{\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\,\,\,\,}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{3.01389pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\,\,\,\,}{\scalebox{-1.0}{\tmpbox}}}$ and analyze the $L^{2}$ norm $\|F\|_{L^{2}(\eta_{r})}$ . By Plancherel’s,

	$\displaystyle\\|F\\|_{L^{2}(\eta_{r})}^{2}=\int\|F\|^{2}\eta_{r}\sim\int\|\sum_{k=1}^{N}e(\frac{k}{N}x_{1}+\frac{k^{2}}{N^{2}}x_{2})\|^{2}\eta_{r}(x_{1},x_{2})$
	$\displaystyle=\sum_{k=1}^{N}\sum_{k^{\prime}=1}^{N}\widehat{\eta}_{r}(\xi(\frac{k-k^{\prime}}{N},\frac{k^{2}-(k^{\prime})^{2}}{N^{2}}))\sim N\cdot N/r\cdot r^{2}=rN^{2}.$

Next we bound $\|F\|_{L^{4}(B_{R^{\varepsilon}r})}$ above. It follows from the local linear restriction statement (see [Dem20] Theorem 1.14, Prop 1.27, and Exercise 1.32)

\|f\|_{L^{4}(B_{R^{\varepsilon}r})}^{4}\lesssim C_{\varepsilon}R^{O(\varepsilon)}r^{-3}\|\widehat{f}\|_{L^{4}(\mathbb{R}^{2})}^{4}

that

	$\displaystyle\\|F\\|_{L^{4}(B_{R^{\varepsilon}r})}^{4}$	$\displaystyle\sim\\|\sum_{k=1}^{N}e(\frac{k}{N}x_{1}+\frac{k^{2}}{N^{2}}x_{2})\eta_{r}(x_{1},x_{2})\\|_{L^{4}(B_{R^{\varepsilon}r})}^{4}$
		$\displaystyle\lesssim C_{\varepsilon}R^{\varepsilon}r^{-3}\\|\sum_{k=1}^{N}\widehat{\eta}_{r}(\xi-(\frac{k}{N},\frac{k^{2}}{N^{2}}))\\|_{L^{4}(\mathbb{R}^{2})}^{4}.$

The $L^{4}$ norm on the right hand side is bounded above by

	$\displaystyle\int_{B_{2}}\|\sum_{k=1}^{N}\widehat{\eta}_{r}(\xi-(\frac{k}{N},\frac{k^{2}}{N^{2}}))\|^{4}d\xi$	$\displaystyle\lesssim(Nr^{-1})^{3}\int_{B_{2}}\sum_{k=1}^{N}\|\widehat{\eta}_{r}(\xi-(\frac{k}{N},\frac{k^{2}}{N^{2}}))\|^{4}d\xi$
		$\displaystyle\lesssim(Nr^{-1})^{3}(r^{2})^{3}\int_{B_{2}}\sum_{k=1}^{N}\|\widehat{\eta}_{r}(\xi-(\frac{k}{N},\frac{k^{2}}{N^{2}}))\|d\xi\sim N^{4}r^{3}.$

This leads to the upper bound $\|F\|_{L^{4}(B_{R^{\varepsilon}r})}^{4}\lesssim(\log R)N^{4}$ .

Finally, by dyadic pigeonholing, there is some $\lambda\in[R^{-1000},N]$ so that $\|F\|_{L^{2}(\eta_{r})}^{2}\lesssim(\log R)\lambda^{2}|\{x\in B_{R^{\varepsilon}r}:|F(x)|\sim\lambda\}|+C_{\varepsilon}R^{-2000}$ . The lower bound for $\|F\|_{L^{2}(\eta_{r})}^{2}$ and the upper bound for $\|F\|_{L^{4}(B_{R^{\varepsilon}r})}^{4}$ tell us that

	$\displaystyle\lambda^{2}rN^{2}\sim\lambda^{2}\\|F\\|_{L^{2}(\eta_{r})}^{2}$	$\displaystyle\lesssim(\log R)\lambda^{4}\|\{x\in B_{R^{\varepsilon}r}:\|F(x)\|\sim\lambda\}\|+C_{\varepsilon}\lambda^{4}R^{-2000}$
		$\displaystyle\lesssim(\log R)\\|F\\|_{L^{4}(B_{R^{\varepsilon}r})}^{4}+C_{\varepsilon}\lambda^{4}R^{-2000}\lesssim C_{\varepsilon}R^{\varepsilon}N^{4}+C_{\varepsilon}\lambda^{4}R^{-2000}.$

Conclude that $\lambda^{2}\lesssim C_{\varepsilon}R^{\varepsilon}N^{2}/r\sim C_{\varepsilon}R^{\varepsilon}\alpha^{2}$ . Assuming $R$ is sufficiently large depending on $\varepsilon$ ,

rN^{2}\sim(\log R)\lambda^{2}|\{x\in B_{R^{\varepsilon}r}:|F(x)|\sim\lambda\}|\lesssim C_{\varepsilon}R^{\varepsilon}(N^{2}/r)|\{x\in B_{R^{\varepsilon}r}:|F(x)|\sim\lambda\}|,

so $|\{x\in B_{R^{\varepsilon}r}:|F(x)|\sim\lambda\}|\gtrsim C_{\varepsilon}^{-1}R^{-\varepsilon}r^{2}$ and $\lambda^{2}\gtrsim C_{\varepsilon}^{-1}R^{-\varepsilon}N^{2}/r\sim C_{\varepsilon}^{-1}R^{-\varepsilon}\alpha^{2}$ .

Case 2: $R^{\beta}<\alpha^{2}\leq R$ . Let $q$ , $a$ , and $b$ be integers satisfying

(6)

q\text{ odd},\quad 1\leq b\leq q\leq N^{2/3},\quad(b,q)=1,\quad\text{and}\quad 0\leq a\leq q.

Define the set $M(q,a,b)$ to be

M(q,a,b):=\{(x_{1},x_{2})\in[0,N]\times[0,N^{2}]:|x_{1}-\frac{a}{q}N|\leq\frac{1}{10^{10}},\quad|x_{2}-\frac{b}{q}N^{2}|\leq\frac{1}{10^{10}}\}.

Lemma 6.

For each $(q,a,b)\not=(q^{\prime},a^{\prime},b^{\prime})$ , both tuples satisfying (6), $M(q,a,b)\cap M(q^{\prime},a^{\prime},b^{\prime})=\emptyset$ .

Proof.

If $\frac{b}{q}=\frac{b^{\prime}}{q^{\prime}}$ , then using the relatively prime part of (6), $b=b^{\prime}$ and $q=q^{\prime}$ . Then we must have $a\not=a^{\prime}$ , meaning that if $x_{1}$ is the first coordinate of a point in $M(q,a,b)\cap M(q,a^{\prime},b^{\prime})$ , then

\frac{2}{10^{10}}\geq|x_{1}-\frac{a}{q}N|+|x_{1}-\frac{a^{\prime}}{q}N|\geq\frac{|a-a^{\prime}|N}{q}\geq N^{1/3}

which is clearly a contradiction. The alternative is that $\frac{b}{q}\not=\frac{b^{\prime}}{q^{\prime}}$ in which case for $x_{2}$ the second coordinate of a point in $M(q,a,b)\cap M(q^{\prime},a^{\prime},b^{\prime})$ ,

\frac{2}{10^{10}}\geq|x_{2}-\frac{b}{q}N^{2}|+|x_{2}-\frac{b^{\prime}}{q^{\prime}}N^{2}|\geq\frac{|b^{\prime}q-bq^{\prime}|N^{2}}{qq^{\prime}}\geq\frac{N^{2}}{qq^{\prime}}\geq N^{2/3},

which is another contradiction. ∎

Lemma 7.

For each $(x_{1},x_{2})\in M(q,a,b)$ , $|F(x_{1},x_{2})|\sim\frac{N}{q^{1/2}}$ , here meaning within a factor of $4$ .

Proof.

This follows from Proposition 13.4 in [Dem20]. ∎

Proposition 8.

Let $R^{\beta}<\alpha^{2}\leq R$ be given. There exists $v\in[0,N^{2}]$ satisfying

|\{(x_{1},x_{2})\in[0,R]^{2}:|F(x_{1},x_{2}+v))|\geq\alpha\}|\gtrsim\frac{R^{2}N^{3}}{\alpha^{6}}.

Proof.

First note that by $N$ -periodicity in $x_{1}$ ,

|\{(x_{1},x_{2})\in[0,R]^{2}:|F(x_{1},x_{2}+v))|\geq\alpha\}|\gtrsim\frac{R}{N}|\{(x_{1},x_{2})\in([0,N]\times[0,R]):|F(x_{1},x_{2}+v))|\geq\alpha\}|.

The function $F$ is $N^{2}$ periodic in $x_{2}$ , but $R<N^{2}$ so we need to find $v\in[0,N^{2}]$ making the set in the lower bound above largest.

By Lemma 7, it suffices to count the tuples $(q,a,b)$ satisfying (6), $q\leq N^{2}/(16\alpha^{2})$ , and $|\frac{b}{q}N^{2}-v|\leq R$ , where $v$ is to be determined. Begin by considering the distribution of points $\frac{b}{q}$ in $[0,1]$ , where $1\leq b\leq q\sim\frac{N^{2}}{\alpha^{2}}$ , $(b,q)=1$ . As in the proof of Lemma 6, if $\frac{b}{q}\not=\frac{b^{\prime}}{q^{\prime}}$ , then $|\frac{b}{q}-\frac{b^{\prime}}{q^{\prime}}|\gtrsim\frac{\alpha^{2}}{N^{4}}$ . Fix $b_{0},q_{0}$ and consider the set $\{\frac{b}{q}:\frac{b}{q}=\frac{b_{0}}{q_{0}},\quad 1\leq b\leq q\sim N^{2}/\alpha^{2}\}$ . Let $q_{m}$ be maximal such that for some $1\leq b_{m}\leq q_{m}\sim N^{2}/\alpha^{2}$ and $(b_{m},q_{m})=1$ , $\frac{b_{m}}{q_{m}}=\frac{b_{0}}{q_{0}}$ . Then $q_{0}=q_{m}-k$ for some integer $k$ and $b_{m}(q_{m}-k)=b_{0}q_{m}$ . Rearrange to get $q_{m}(1-\frac{b_{0}}{b_{m}})=k$ . Thus $q_{0}=q_{m}\frac{b_{0}}{b_{m}}\sim N^{2}/\alpha^{2}$ , which implies that $\frac{b_{0}}{b_{m}}\sim 1$ . Conclude that there are $\gtrsim\sum_{q\sim N^{2}/\alpha^{2}}\varphi(q)$ many unique points $\frac{b}{q}$ in $[0,1]$ satisfying our prescribed conditions for $\varphi$ denoting the Euler totient function. Use Theorem 3.7 in [Apo76] to estimate $\sum_{q\sim N^{2}/\alpha^{2}}\varphi(q)\sim N^{4}/\alpha^{4}$ , as long as $N/\alpha$ is larger than some absolute constant. By the pigeonhole principle, there exists some $R/N^{2}$ interval $I\subset[0,1]$ containing $\sim\lceil\frac{N^{4}}{\alpha^{4}}\frac{R}{N^{2}}\rceil$ many points $\frac{b}{q}$ with $1\leq b\leq q\sim N^{2}/\alpha^{2}$ , $(b,q)=1$ . There are also $\sim N^{2}/\alpha^{2}$ many choices for $a$ to complete the tuple $(q,a,b)$ satisfying (6). Let $c$ denote the center of $I$ and take $v=cN^{2}$ in the proposition statement and conclude that

|\{(x_{1},x_{2})\in([0,N]\times[0,R]):|F(x_{1},x_{2}+v))|\geq\alpha\}|\gtrsim\frac{RN^{4}}{\alpha^{6}}

to finish the proof. ∎

Note that Proposition 8 shows the sharpness of Theorem 3 in the range $R^{\beta}<\alpha\leq R$ since

\frac{R^{2\beta}}{\alpha^{6}}\sum_{\gamma}\|F_{\gamma}\|_{2}^{2}\sim\frac{R^{2\beta}}{\alpha^{6}}R^{\beta}R^{2}=\frac{N^{3}R^{2}}{\alpha^{6}}.

The sharpness of the trivial estimate $|U_{\alpha}\cap[-R,R]^{2}|\lesssim R^{2}$ in the range $\alpha^{2}<R^{\beta}$ follows from Case 2 since for $\alpha^{2}<R^{\beta}$ ,

|U_{\alpha}\cap[-R,R]^{2}|\geq|U_{R^{\beta/2}}\cap[-R,R]^{2}|\gtrsim\frac{R^{2\beta}}{(R^{\beta/2})^{6}}\sum_{\gamma}\|F_{\gamma}\|_{2}^{2}\sim R^{2}.

3. Implications of Theorem 4

Proof of Theorem 3 from Theorem 4.

First suppose that $\alpha^{2}>\frac{\lambda(1)^{2}}{\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)}$ . Then

	$\displaystyle\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)$	$\displaystyle\lesssim\max_{s}(s^{-1}R^{-1}R^{\beta})(sR^{\beta})=R^{2\beta-1}$
		$\displaystyle\leq\begin{cases}R^{2\beta-1}\quad&\text{if}\quad\alpha^{2}>R\\ \frac{R^{2\beta}}{\alpha^{2}}\quad&\text{if}\quad R^{\beta}\leq\alpha^{2}\leq R\end{cases}.$

Now suppose that $\alpha^{2}\leq\frac{\lambda(1)^{2}}{\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)}$ . Then

\displaystyle\frac{\lambda(1)^{2}}{\alpha^{2}}

\displaystyle\lesssim\begin{cases}R^{2\beta-1}\quad&\text{if}\quad\alpha^{2}>R\\ \frac{R^{2\beta}}{\alpha^{2}}\quad&\text{if}\quad R^{\beta}\leq\alpha^{2}\leq R\end{cases}.

∎

Proof of Corollary 5 from Theorem 4.

To see how this corollary follows from Theorem 4, first use an analogous series of pigeonholing steps as in Section 5 of [GMW20] to reduce to the case where $\|f_{\gamma}\|_{\infty}\lesssim 1$ for all $\gamma$ and there exists $C>0$ so that $\|f_{\gamma}\|_{p}^{p}$ is either $0$ or comparable to $C$ for all $\gamma$ . Split the integral

\int|f|^{p}=\sum_{R^{-1000}\leq\alpha\lesssim R^{\beta}}\int_{U_{\alpha}}|f|^{p}+\int_{|f|<R^{-1000}}|f|^{p}

where $U_{\alpha}=\{x:|f(x)|\sim\alpha\}$ and assume via dyadic pigeonholing that

\int|f|^{p}\lesssim\alpha^{p}|U_{\alpha}|

(ignoring the case that the set where $|f|\leq R^{-1000}$ dominates the integral which may be handled trivially). The result of all of the pigeonholing steps is that the statement of Corollary 5 follows from showing that

\alpha^{p}|U_{\alpha}|\leq C_{\varepsilon}R^{\varepsilon}(R^{\beta p(1-\frac{1}{q})-(1+\beta)}+R^{\beta p(\frac{1}{2}-\frac{1}{q})})\lambda(1)^{\frac{p}{q}-1}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}

where $f$ satisfies the hypotheses of Theorem 4. The full range $\frac{3}{p}+\frac{1}{q}\leq 1$ follows from $p$ in the critical range $4\leq p\leq 6$ , which we treat first.
$4\leq p\leq 6$ : There are two cases depending on which upper bound is larger in Theorem 4. First we assume the $L^{4}$ bound holds, in which case

	$\displaystyle\alpha^{p}\|U_{\alpha}\|$	$\displaystyle\leq C_{\varepsilon}R^{\varepsilon}\alpha^{p-4}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle\sim C_{\varepsilon}R^{\varepsilon}\frac{\alpha^{p-4}}{\lambda(1)^{\frac{p}{q}-1}}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}$
		$\displaystyle\lesssim C_{\varepsilon}R^{\varepsilon}\frac{\lambda(1)^{p-4}}{\lambda(1)^{\frac{p}{q}-1}}\underset{s}{\max}(R^{\beta}s^{-1}R^{-1})(R^{\beta}s)(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}$
		$\displaystyle\lesssim C_{\varepsilon}R^{\varepsilon}\lambda(1)^{p(1-\frac{1}{q})-3}R^{2\beta-1}(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}.$

Since $p(1-\frac{1}{q})-3\geq 0$ , we may use the bound $\lambda(1)\lesssim R^{\beta}$ to conclude that

\lambda(1)^{p(1-\frac{1}{q})-3}R^{2\beta-1}\leq R^{\beta p(1-\frac{1}{q})-3\beta+2\beta-1}=R^{\beta p(1-\frac{1}{q})-(1+\beta)}.

The other case is that the $L^{6}$ bound holds in Theorem 4. We may also assume that $\alpha^{2}>\lambda(1)$ since otherwise we trivially have

\alpha^{p}|U_{\alpha}|\leq\lambda(1)^{\frac{p}{2}-1}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}\sim\lambda(1)^{\frac{p}{2}-1+1-\frac{p}{q}}(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{\frac{p}{q}}\lesssim R^{\beta p(\frac{1}{2}-\frac{1}{q})}(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{\frac{p}{q}}

where we used that $q\geq 2$ since $4\leq p\leq 6$ and $\frac{3}{p}+\frac{1}{q}\leq 1$ . Now using the assumptions $\alpha^{2}>\lambda(1)$ and $p\leq 6$ , we have

	$\displaystyle\alpha^{p}\|U_{\alpha}\|$	$\displaystyle\leq C_{\varepsilon}R^{\varepsilon}\alpha^{p-6}\lambda(1)^{2}\lambda(1)^{1-\frac{p}{q}}(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}$
		$\displaystyle\sim C_{\varepsilon}R^{\varepsilon}\lambda(1)^{p(\frac{1}{2}-\frac{1}{q})}(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}\lesssim C_{\varepsilon}R^{\varepsilon}R^{\beta p(\frac{1}{2}-\frac{1}{q})}(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}.$

$3\leq p<4$ : Suppose that $\alpha<R^{\beta/2}$ . Then using $L^{2}$ -orthogonality,

\alpha^{p}|U_{\alpha}|\leq R^{\frac{\beta}{2}(p-2)}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}\sim R^{\frac{\beta}{2}(p-2)}\lambda(1)^{1-\frac{p}{q}}(\sum_{\gamma}\|f_{\gamma}\|_{p}^{q})^{\frac{p}{q}}.

Since in this subcase, $1-\frac{p}{q}\geq 1-(p-3)>0$ , we are done after noting that $R^{\frac{\beta}{2}(p-2)}\lambda(1)^{1-\frac{p}{q}}\leq R^{\beta p(\frac{1}{2}-\frac{1}{q})}$ . Now assume that $\alpha\geq R^{\beta/2}$ and use the $p=4$ case above (noting that $R^{4\beta(1-\frac{1}{q})-(1+\beta)}\leq R^{4\beta(\frac{1}{2}-\frac{1}{q})}$ ) to get

	$\displaystyle\alpha^{p}\|U_{\alpha}\|$	$\displaystyle\leq\frac{\alpha^{4}}{(R^{\beta/2})^{4-p}}\|U_{\alpha}\|\leq R^{-\frac{\beta}{2}(4-p)}C_{\varepsilon}R^{\varepsilon}R^{4\beta(\frac{1}{2}-\frac{1}{q})}\lambda(1)^{\frac{4}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle\leq C_{\varepsilon}R^{\varepsilon}R^{\beta p(\frac{1}{2}-\frac{1}{q})}\lambda(1)^{\frac{p}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}.$

$6<p$ : In this range, we use the trivial bound $\alpha\leq\lambda(1)$ and the $p=6$ case above (noting that $R^{6\beta(\frac{1}{2}-\frac{1}{q})}\leq R^{6\beta(1-\frac{1}{q})-(1+\beta)}$ ) to get

	$\displaystyle\alpha^{p}\|U_{\alpha}\|$	$\displaystyle\leq\lambda(1)^{p-6}\alpha^{6}\|U_{\alpha}\|\leq\lambda(1)^{p-6}C_{\varepsilon}R^{\varepsilon}R^{6\beta(1-\frac{1}{q})-(1+\beta)}\lambda(1)^{\frac{6}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle=\Big{(}\frac{\lambda(1)}{R^{\beta}}\Big{)}^{(p-6)(1-\frac{1}{q})}C_{\varepsilon}R^{\varepsilon}R^{p\beta(1-\frac{1}{q})-(1+\beta)}\lambda(1)^{\frac{p}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle\leq C_{\varepsilon}R^{\varepsilon}R^{p\beta(1-\frac{1}{q})-(1+\beta)}\lambda(1)^{\frac{p}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}.$

∎

4. Tools to prove Theorem 4

The proof of Theorem 4 follows the high/low frequency decomposition and pruning approach from [GMW20]. In this section, we introduce notation for different scale neighborhoods of $\mathbb{P}^{1}$ , a pruning process for wave packets at various scales, some high/low lemmas which are used to analyze the high/low frequency parts of square functions, and a version of a bilinear restriction theorem for $\mathbb{P}^{1}$ .

Begin by fixing some notation, as above. Let $\beta\in[\frac{1}{2},1]$ and $R\geq 2$ . The parameter $\alpha>0$ describes the superlevel set

U_{\alpha}=\{x\in\mathbb{R}^{2}:|f(x)|\geq\alpha\}.

For $\varepsilon>0$ , we analyze scales $R_{k}=R^{k\varepsilon}$ , noting that $R^{-1/2}\leq R_{k}^{-1/2}\leq 1$ . Let $N$ distinguish the index so that $R_{N}$ is closest to $R$ . Since $R$ and $R_{N}$ differ at most by a factor of $R^{\varepsilon}$ , we will ignore the distinction between $R_{N}$ and $R$ in the rest of the argument.

Define the following collections, each of which partitions a neighborhood of $\mathbb{P}$ into approximate rectangles.

(1)

$\{\gamma\}$ is a partition of $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ by approximate $R^{-\beta}\times R^{-1}$ rectangles, described explicitly in (5).
(2)

$\{\theta\}$ is a partition of $\mathcal{N}_{R^{-1}}(\mathbb{P}^{1})$ by approximate $R^{-1/2}\times R^{-1}$ rectangles. In particular, let each $\theta$ be a union of adjacent $\gamma$ .
(3)

$\{\tau_{k}\}$ is a partition of $\mathcal{N}_{R_{k}^{-1}}(\mathbb{P}^{1})$ by approximate $R_{k}^{-1/2}\times R_{k}^{-1}$ rectangles. Assume the additional property that $\gamma\cap\tau_{k}=\emptyset$ or $\gamma\subset\tau_{k}$ .

4.1. A pruning step

We will define wave packets at each scale $\tau_{k}$ , and prune the wave packets associated to $f_{\tau_{k}}$ according to their amplitudes.

For each $\tau_{k}$ , fix a dual rectangle $\tau_{k}^{*}$ which is a $2R_{k}^{1/2}\times 2R_{k}$ rectangle centered at the origin and comparable to the convex set

\{x\in\mathbb{R}^{2}:|x\cdot\xi|\leq 1\quad\forall\xi\in\tau_{k}\}.

Let $\mathbb{T}_{\tau_{k}}$ be the collection of tubes $T_{\tau_{k}}$ which are dual to $\tau_{k}$ , contain $\tau_{k}^{*}$ , and which tile $\mathbb{R}^{2}$ . Next, we will define an associated partition of unity $\psi_{T_{\tau_{k}}}$ . First let $\varphi(\xi)$ be a bump function supported in $[-\frac{1}{4},\frac{1}{4}]^{2}$ . For each $m\in\mathbb{Z}^{2}$ , let

\psi_{m}(x)=c\int_{[-\frac{1}{2},\frac{1}{2}]^{2}}|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\varphi}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\varphi}{\scalebox{-1.0}{\tmpbox}}|^{2}(x-y-m)dy,

where $c$ is chosen so that $\sum_{m\in\mathbb{Z}^{2}}\psi_{m}(x)=c\int_{\mathbb{R}^{2}}|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\varphi}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\varphi}{\scalebox{-1.0}{\tmpbox}}|^{2}=1$ . Since $|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\varphi}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\varphi}{\scalebox{-1.0}{\tmpbox}}|$ is a rapidly decaying function, for any $n\in\mathbb{N}$ , there exists $C_{n}>0$ such that

\psi_{m}(x)\leq c\int_{[0,1]^{2}}\frac{C_{n}}{(1+|x-y-m|^{2})^{n}}dy\leq\frac{\tilde{C}_{n}}{(1+|x-m|^{2})^{n}}.

Define the partition of unity $\psi_{T_{\tau_{k}}}$ associated to ${\tau_{k}}$ to be $\psi_{T_{\tau_{k}}}(x)=\psi_{m}\circ A_{\tau_{k}}$ , where $A_{\tau_{k}}$ is a linear transformation taking $\tau_{k}^{*}$ to $[-\frac{1}{2},\frac{1}{2}]^{2}$ and $A_{\tau_{k}}(T_{\tau_{k}})=m+[-\frac{1}{2},\frac{1}{2}]^{2}$ . The important properties of $\psi_{T_{\tau_{k}}}$ are (1) rapid decay off of $T_{\tau_{k}}$ and (2) Fourier support contained in $\tau_{k}$ .

To prove upper bounds for the size of $U_{\alpha}$ , we will actually bound the sizes of $\sim\varepsilon^{-1}$ many subsets which will be denoted $U_{\alpha}\cap\Omega_{k}$ , $U_{\alpha}\cap H$ , and $U_{\alpha}\cap L$ . The pruning process sorts between important and unimportant wave packets on each of these subsets, as described in Lemma 12 below.

Partition $\mathbb{T}_{\theta}=\mathbb{T}_{\theta}^{g}\sqcup\mathbb{T}_{\theta}^{b}$ into a “good” and a “bad” set as follows. Let $\delta>0$ be a parameter to be chosen in §5.2 and set

T_{\theta}\in\mathbb{T}_{\theta}^{g}\quad\text{if}\quad\|\psi_{T_{\theta}}f_{\theta}\|_{L^{\infty}(R^{2})}\leq R^{M\delta}\frac{\lambda(1)}{\alpha}

where $M>0$ is a universal constant we will choose in the proof of Proposition 1.

Definition 1 (Pruning with respect to $\tau_{k}$ ).

For each $\theta$ and $\tau_{N-1}$ , define the notation $f_{\theta}^{N}=\sum_{T_{\theta}\in\mathbb{T}_{\theta}^{g}}\psi_{T_{\theta}}f_{\theta}$ and $f_{\tau_{N-1}}^{N}=\sum_{\theta\subset\tau_{N-1}}f_{\theta}^{N}$ . For each $k<N$ , let

	$\displaystyle\mathbb{T}_{\tau_{k}}^{g}$	$\displaystyle=\{T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}:\\|\psi_{T_{\tau_{k}}}f_{\tau_{k}}^{k+1}\\|_{L^{\infty}(R^{2})}\leq R^{M\delta}\frac{\lambda(1)}{\alpha}\},$
	$\displaystyle f_{\tau_{k}}^{k}=\sum_{T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}^{g}}$	$\displaystyle\psi_{T_{\tau_{k}}}f^{k+1}_{\tau_{k}}\qquad\text{and}\qquad f_{\tau_{k-1}}^{k}=\sum_{\tau_{k}\subset\tau_{k-1}}f_{\tau_{k}}^{k}.$

For each $k$ , define the $k$ th version of $f$ to be $f^{k}=\underset{\tau_{k}}{\sum}f_{\tau_{k}}^{k}$ .

Lemma 9 (Properties of $f^{k}$ ).

(1)

$|f_{\tau_{k}}^{k}(x)|\leq|f_{\tau_{k}}^{k+1}(x)|\leq\#\gamma\subset\tau_{k}.$
(2)

$\|f_{\tau_{k}}^{k}\|_{L^{\infty}}\leq C_{\varepsilon}R^{O(\varepsilon)}R^{M\delta}\frac{\lambda(1)}{\alpha}$ .
(3)

$\text{supp}\widehat{f_{\tau_{k}}^{k}}\subset 2\tau_{k}.$
(4)

$\text{supp}\widehat{f_{\tau_{k-1}}^{k}}\subset(1+(\log R)^{-1})\tau_{k-1}.$

Proof.

The first property follows because $\sum_{T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}}\psi_{T_{\tau_{k}}}$ is a partition of unity, and

f_{\tau_{k}}^{k}=\sum_{T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}^{h}}}\psi_{T_{\tau_{k}}}f_{\tau_{k}}^{k+1}.

Furthermore, by definition of $f_{\tau_{k}}^{k+1}$ and iterating, we have

	$\displaystyle\|f_{\tau_{k}}^{k}\|\leq\|f_{\tau_{k}}^{k+1}\|$	$\displaystyle\leq\sum_{\tau_{k+1}\subset\tau_{k}}\|f_{\tau_{k+1}}^{k+1}\|\leq\cdots\leq\sum_{\tau_{N}\subset\tau_{k}}\|f_{\tau_{N}}^{N}\|$
		$\displaystyle\leq\sum_{\theta\subset\tau_{k}}\|f_{\theta}\|\leq\sum_{\gamma\subset\tau_{k}}\|f_{\gamma}\|\lesssim\#\gamma\subset\tau_{k}$

where we used the assumption $\|f_{\gamma}\|_{\infty}\lesssim 1$ for all $\gamma$ . Now consider the $L^{\infty}$ bound in the second property. We write

f_{\tau_{k}}^{k}(x)=\sum_{\begin{subarray}{c}T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}^{h}},\\ x\in R^{\varepsilon}T_{\tau_{k}}\end{subarray}}\psi_{T_{\tau_{k}}}f_{\tau_{k}}^{k+1}+\sum_{\begin{subarray}{c}T_{\tau_{k}}\in\mathbb{T}_{\tau_{k},\lambda},\\ x\notin R^{\varepsilon}T_{\tau_{k}}\end{subarray}}\psi_{T_{\tau_{k}}}f_{k+1,\tau_{k}}.

The first sum has at most $CR^{2\varepsilon}$ terms, and each term has norm bounded by $R^{M\delta}\frac{\lambda(1)}{\alpha}$ by the definition of $\mathbb{T}_{\tau_{k}}^{h}$ . By the first property, we may trivially bound $f_{\tau_{k}}^{k+1}$ by $R\max_{\gamma}\|f_{\gamma}\|_{\infty}$ . But if $x\notin R^{\varepsilon}T_{\tau_{k}}$ , then $\psi_{T_{\tau_{k}}}(x)\leq R^{-1000}$ . Thus

\displaystyle|\sum_{\begin{subarray}{c}T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}^{h},\\ x\notin R^{\varepsilon}T_{\tau_{k}}\end{subarray}}\psi_{T_{\tau_{k}}}f^{k+1}_{\tau_{k}}|

\displaystyle\leq\sum_{\begin{subarray}{c}T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}^{h},\\ x\notin R^{\varepsilon}T_{\tau_{k}}\end{subarray}}R^{-500}\psi_{T_{\tau_{k}}}^{1/2}(x)\|f^{k+1}_{\tau_{k}}\|_{\infty}\leq R^{-250}\max_{\gamma}\|f_{\gamma}\|_{\infty}.

Since $\alpha\lesssim|f(x)|\lesssim\sum_{\gamma}\|f_{\gamma}\|_{\infty}\lesssim\lambda(1)$ , (recalling the assumption that each $\|f_{\gamma}\|_{\infty}\lesssim 1$ ), we note $R^{-250}\leq CR^{2\varepsilon}\frac{\lambda(1)}{\alpha}$ .

The fourth and fifth properties depend on the Fourier support of $\psi_{T_{\tau_{k}}}$ , which is contained in $\frac{1}{2}\tau_{k}$ . Initiate a 2-step induction with base case $k=N$ : $f_{\theta}^{N}$ has Fourier support in $2\theta$ because of the above definition. Then

f_{\tau_{N-1}}^{N}=\sum_{\theta\subset\tau_{N-1}}f_{\theta}^{N}

has Fourier support in $\underset{\theta\subset\tau_{N-1}}{\cup}2\theta$ , which is contained in $(1+(\log R)^{-1})\tau_{N-1}$ . Since each $\psi_{T_{\tau_{N-1}}}$ has Fourier support in $\frac{1}{2}\tau_{N-1}$ ,

f_{\tau_{N-1}}^{N-1}=\sum_{T_{\tau_{N-1}}\in\mathbb{T}_{\tau_{N-1},\lambda}}\psi_{\tau_{N-1}}f_{\tau_{N-1}}^{N}

has Fourier support in $\frac{1}{2}\tau_{N-1}+(1+(\log R)^{-1})\tau_{N-1}\subset 2\tau_{N-1}$ . Iterating this reasoning until $k=1$ gives (3) and (4).

∎

Definition 2.

For each $\tau_{k}$ , let $w_{\tau_{k}}$ be the weight function adapted to $\tau_{k}^{*}$ defined by

w_{\tau_{k}}(x)=w_{k}\circ R_{\tau_{k}}(x)

where

w_{k}(x,y)=\frac{c}{(1+\frac{|x|^{2}}{R_{k}})^{10}(1+\frac{|y|^{2}}{R_{k}^{2}})^{10}},\qquad\|w\|_{1}=1,

and $R_{\tau_{k}}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is the rotation taking $\tau_{k}^{*}$ to $[-R_{k}^{1/2},R_{k}^{1/2}]\times[-R_{k},R_{k}]$ . For each $T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}$ , let $w_{T_{\tau_{k}}}=w_{\tau_{k}}(x-c_{T_{\tau_{k}}})$ where $c_{T_{\tau_{k}}}$ is the center of $T_{\tau_{k}}$ . For $s>0$ , we also use the notation $w_{s}$ to mean

(7)

w_{s}(x)=\frac{c^{\prime}}{(1+|x|^{2}/s^{2})^{10}},\qquad\|w_{s}\|_{1}=1.

The weights $w_{\tau_{k}}$ , $w_{\theta}=w_{\tau_{N}}$ , and $w_{s}$ are useful when we invoke the locally constant property. By locally constant property, we mean generally that if a function $f$ has Fourier transform supported in a convex set $A$ , then for a bump function $\varphi_{A}\equiv 1$ on $A$ , $f=f*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\varphi_{A}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\varphi_{A}}{\scalebox{-1.0}{\tmpbox}}$ . Since $|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\varphi_{A}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\varphi_{A}}{\scalebox{-1.0}{\tmpbox}}|$ is an $L^{1}$ -normalized function which is positive on a set dual to $A$ , $|f|*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\varphi_{A}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\varphi_{A}}{\scalebox{-1.0}{\tmpbox}}|$ is an averaged version of $|f|$ over a dual set $A^{*}$ . We record some of the specific locally constant properties we need in the following lemma.

Lemma 10 (Locally constant property).

For each $\tau_{k}$ and $T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}$ ,

\|f_{\tau_{k}}\|_{L^{\infty}(T_{\tau_{k}})}^{2}\lesssim|f_{\tau_{k}}|^{2}*w_{\tau_{k}}(x)\qquad\text{for any}\quad x\in T_{\tau_{k}}.

For any collection of $\sim s^{-1}\times s^{-2}$ blocks $\theta_{s}$ partitioning $\mathcal{N}_{s^{-2}}(\mathbb{P}^{1})$ and any $s$ -ball $B$ ,

\|\sum_{\theta_{s}}|f_{\theta_{s}}|^{2}\|_{L^{\infty}(B)}\lesssim\sum_{\theta_{s}}|f_{\theta_{s}}|^{2}*w_{s}(x)\qquad\text{for any}\quad x\in B.

Because the pruned versions of $f$ and $f_{\tau_{k}}$ have essentially the same Fourier supports as the unpruned versions, the locally constant lemma applies to the pruned versions as well.

Proof of Lemma 10.

Let $\rho_{\tau_{k}}$ be a bump function equal to $1$ on $\tau_{k}$ and supported in $2\tau_{k}$ . Then using Fourier inversion and Hölder’s inequality,

|f_{\tau_{k}}(y)|^{2}=|f_{\tau_{k}}*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\tau_{k}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\tau_{k}}}{\scalebox{-1.0}{\tmpbox}}(y)|^{2}\leq\|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\tau_{k}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\tau_{k}}}{\scalebox{-1.0}{\tmpbox}}\|_{1}|f_{\tau_{k}}|^{2}*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\tau_{k}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\tau_{k}}}{\scalebox{-1.0}{\tmpbox}}|(y).

Since $\rho_{\tau_{k}}$ may be taken to be an affine transformation of a standard bump function adapted to the unit ball, $\|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\tau_{k}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\tau_{k}}}{\scalebox{-1.0}{\tmpbox}}\|_{1}$ is a constant. The function $\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\tau_{k}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\tau_{k}}}{\scalebox{-1.0}{\tmpbox}}$ decays rapidly off of $\tau_{k}^{*}$ , so $|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\tau_{k}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\tau_{k}}}{\scalebox{-1.0}{\tmpbox}}|\lesssim w_{{\tau_{k}}}$ . Since for any $T_{\tau_{k}}\in\mathbb{T}_{\tau_{k}}$ , $w_{\tau_{k}}(y)$ is comparable for all $y\in T_{\tau_{k}}$ , we have

	$\displaystyle\sup_{x\in T_{\tau_{k}}}\|f_{\tau_{k}}\|^{2}*w_{\tau_{k}}(x)$	$\displaystyle\leq\int\|f_{\tau_{k}}\|^{2}(y)\sup_{x\in T_{\tau_{k}}}w_{\tau_{k}}(x-y)dy$
		$\displaystyle\sim\int\|f_{\tau_{k}}\|^{2}(y)w_{\tau_{k}}(x-y)dy\qquad\text{for all}\quad x\in T_{\tau_{k}}.$

For the second part of the lemma, repeat analogous steps as above, except begin with $\rho_{\theta_{s}}$ which is identically $1$ on a ball of radius $2s^{-1}$ containing $\theta_{s}$ . Then

\sum_{\theta_{s}}|f_{\theta_{s}}(y)|^{2}=\sum_{\theta_{s}}|f_{\theta_{s}}*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{\theta_{s}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{\theta_{s}}}{\scalebox{-1.0}{\tmpbox}}(y)|^{2}\lesssim\sum_{\theta_{s}}|f_{\theta_{s}}|^{2}*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\rho_{s^{-1}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\rho_{s^{-1}}}{\scalebox{-1.0}{\tmpbox}}|(y),

where we used that each $\rho_{\theta_{s}}$ is a translate of a single function $\rho_{s^{-1}}$ . The rest of the argument is analogous to the first part. ∎

Definition 3 (Auxiliary functions).

Let $\varphi(x):\mathbb{R}^{2}\to[0,\infty)$ be a radial, smooth bump function satisfying $\varphi(x)=1$ on $B_{1}$ and $\mathrm{supp}\varphi\subset B_{2}$ .

\displaystyle\varphi(2^{J+1}\xi)+\sum_{j=-2}^{J}[\varphi(2^{j}\xi)-\varphi(2^{j+1}\xi)]

where $J$ is defined by $2^{J}\leq\lceil R^{\beta}\rceil<2^{J+1}$ . Then for each dyadic $s=2^{j}$ , let

\eta_{\sim s}(\xi)=\varphi(2^{j}\xi)-\varphi(2^{j+1}\xi)

and let

\eta_{<\lceil R^{\beta}\rceil^{-1}}(\xi)=\varphi(2^{J+1}\xi).

Finally, for $k=1,\ldots,N-1$ , define

\eta_{k}(\xi)=\varphi(R_{k+1}^{1/2}x).

Definition 4.

Let $G(x)=\sum_{\theta}|f_{\theta}|^{2}*w_{\theta}$ , $G^{\ell}(x)=G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{<\lceil R^{\beta}\rceil^{-1}}$ , $G^{h}(x)=G(x)-G^{\ell}(x)$ . For $k=1,\ldots,N-1$ , let

g_{k}(x)=\sum_{\tau_{k}}|f_{\tau_{k}}^{k+1}|^{2}*w_{\tau_{k}},\qquad g_{k}^{\ell}(x)=g_{k}*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{k},\qquad\text{and}\qquad g_{k}^{h}(x)=g_{k}-g_{k}^{\ell}.

Definition 5.

Define the high set

H=\{x\in B_{R}:G(x)\leq 2|G^{h}(x)|\}.

For each $k=1,\ldots,N-1$ , let

\Omega_{k}=\{x\in B_{R}\setminus H:g_{k}\leq 2|g_{k}^{h}|,\,g_{k+1}\leq 2|g_{k+1}^{\ell}|,\,\ldots,\,g_{N}\leq 2|g_{N}^{\ell}|\}

and for each $k=1,\ldots,N$ . Define the low set

L=\{x\in B_{R}\setminus H:g_{1}\leq 2|g_{1}^{\ell}|,\,\ldots,\,g_{N}\leq 2|g_{N}^{\ell}|,G(x)\leq 2|G^{\ell}(x)|\}.

4.2. High/low frequency lemmas

Lemma 11 (Low lemma).

For each $x$ , $|G^{\ell}(x)|\lesssim\lambda(1)$ and $|g_{k}^{\ell}(x)|\lesssim g_{k+1}(x)$ .

Proof.

For each $\theta$ , by Plancherel’s theorem,

The integrand is supported in $(\gamma\setminus\gamma^{\prime})\cap B_{2\lceil R^{\beta}\rceil^{-1}}$ . This means that the integral vanishes unless $\gamma$ is within $CR^{-\beta}$ of $\gamma^{\prime}$ for some constant $C>0$ , in which case we write $\gamma\sim\gamma^{\prime}$ . Then

\sum_{\gamma,\gamma^{\prime}\subset\theta}\int_{\mathbb{R}^{2}}e^{-2\pi ix\cdot\xi}\widehat{f}_{\gamma}*\widehat{\overline{f}}_{\gamma^{\prime}}(\xi)\eta_{<\lceil R^{\beta}\rceil^{-1}}(\xi)d\xi=\sum_{\begin{subarray}{c}\gamma,\gamma^{\prime}\subset\theta\\ \gamma\sim\gamma^{\prime}\end{subarray}}\int_{\mathbb{R}^{2}}e^{-2\pi ix\cdot\xi}\widehat{f}_{\gamma}*\widehat{\overline{f}}_{\gamma^{\prime}}(\xi)\eta_{<\lceil R^{\beta}\rceil^{-1}}(\xi)d\xi

Use Plancherel’s theorem again to get back to a convolution in $x$ and conclude that

By an analogous argument as above, we have that

|g_{k}^{\ell}(x)|\lesssim\sum_{\tau_{k+1}}|f_{\tau_{k+1}}^{k+1}|^{2}*w_{\tau_{k}}*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{k}|(x)

where for each summand, $w_{\tau_{k}}$ corresponds to the $\tau_{k}$ containing $\tau_{k+1}$ . By definition, $|f_{\tau_{k+1}}^{k+1}|\leq|f_{\tau_{k+1}}^{k}|$ . By the locally constant property, $|f_{\tau_{k+1}}^{k}|^{2}\lesssim|f_{\tau_{k+1}}|^{2}*w_{\tau_{k+1}}$ . It remains to note that

w_{\tau_{k+1}}*w_{\tau_{k}}*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{k}|(x)\lesssim w_{\tau_{k+1}}(x)

since $\tau_{k}^{*}\subset\tau_{k+1}^{*}$ and $\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{k}$ is an $L^{1}$ -normalized function that is rapidly decaying away from $B_{R_{k+1}^{1/2}}(0)$ .

∎

Lemma 12 (Pruning lemma).

For any $\tau$ ,

	$\displaystyle\|\sum_{\tau_{k}\subset\tau}f_{\tau_{k}}-\sum_{\tau_{k}\subset\tau}f_{\tau_{k}}^{k+1}(x)\|$	$\displaystyle\leq C_{\varepsilon}R^{-M\delta}\alpha\qquad\text{for all $x\in\Omega_{k}$}$
	$\displaystyle\text{and}\qquad\|\sum_{\tau_{1}\subset\tau}f_{\tau_{1}}-\sum_{\tau_{1}\subset\tau}f_{\tau_{1}}^{1}(x)\|$	$\displaystyle\leq C_{\varepsilon}R^{-M\delta}\alpha\qquad\text{ for all $x\in L$}.$

Proof.

By the definition of the pruning process, we have

f_{\tau}=f^{N}_{\tau}+(f_{\tau}-f^{N}_{\tau})=\cdots=f^{k+1}_{\tau}(x)+\sum_{m=k+1}^{N}(f^{m+1}_{\tau}-f^{m}_{\tau})

with the understanding that $f^{N+1}=f$ and formally, the subscript $\tau$ means $f_{\tau}=\sum_{\gamma\subset\tau}f_{\gamma}$ and $f_{\tau}^{m}=\sum_{\tau_{m}\subset\tau}f_{\tau_{m}}^{m}$ . We will show that each difference in the sum is much smaller than $\alpha$ . For each $m\geq k+1$ and $\tau_{m}$ ,

	$\displaystyle\|f_{\tau_{m}}^{m}(x)-f_{\tau_{m}}^{m+1}(x)\|$	$\displaystyle=\|\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}\psi_{T_{\tau_{m}}}(x)f_{\tau_{m}}^{m+1}(x)\|=\sum_{T_{\tau_{m}}\in T_{\tau_{m}}^{b}}\|\psi_{T_{\tau_{m}}}^{1/2}(x)f_{\tau_{m}}^{m+1}(x)\|\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}R^{-M\delta}\frac{\alpha}{\lambda(1)}\lambda^{-1}\\|\psi_{T_{\tau_{m}}}f_{{\tau_{m}}}^{m+1}\\|_{L^{\infty}(\mathbb{R}^{2})}\\|\psi_{T_{\tau_{m}}}^{1/2}f_{{\tau_{m}}}^{m+1}\\|_{L^{\infty}(\mathbb{R}^{2})}\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}\\|\psi_{T_{\tau_{m}}}^{1/2}f_{{\tau_{m}}}^{m+1}\\|_{L^{\infty}(\mathbb{R}^{2})}^{2}\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}\sum_{\tilde{T}_{{\tau_{m}}}}\\|\psi_{T_{\tau_{m}}}\|f_{{\tau_{m}}}^{m+1}\|^{2}\\|_{L^{\infty}(\tilde{T}_{{\tau_{m}}})}\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{T_{\tau_{m}},\tilde{T}_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}}\\|\psi_{T_{\tau_{m}}}\\|_{L^{\infty}(\tilde{T}_{\tau_{m}})}\\|\|f_{{\tau_{m}}}^{m+1}\|^{2}\\|_{{L}^{\infty}(\tilde{T}_{{\tau_{m}}})}\psi_{T_{\tau_{m}}}^{1/2}(x).$

Let $c_{\tilde{T}_{\tau_{m}}}$ denote the center of $\tilde{T}_{\tau_{m}}$ and note the pointwise inequality

\sum_{{T}_{\tau_{m}}}\|\psi_{T_{\tau_{m}}}\|_{L^{\infty}(\tilde{T}_{\tau_{m}})}\psi_{T_{\tau_{m}}}^{1/2}(x)\lesssim R_{m}^{3/2}w_{\tau_{m}}(x-c_{\tilde{T}_{\tau_{m}}}),

which means that

	$\displaystyle\|f_{\tau_{m}}^{m}(x)-f_{\tau_{m}}^{m+1}(x)\|$	$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}R_{m}^{3/2}\sum_{\tilde{T}_{\tau_{m}}\in T_{\tau_{m}}}w_{\tau_{m}}(x-c_{\tilde{T}_{\tau_{m}}})\\|\|f_{{\tau_{m}}}^{m+1}\|^{2}\\|_{{L}^{\infty}(\tilde{T}_{{\tau_{m}}})}$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}R_{m}^{3/2}\sum_{\tilde{T}_{\tau_{m}}\in T_{\tau_{m}}}w_{\tau_{m}}(x-c_{\tilde{T}_{\tau_{m}}})\|f_{{\tau_{m}}}^{m+1}\|^{2}*w_{\tau_{m}}(c_{\tilde{T}_{\tau_{m}}})$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\|f_{{\tau_{m}}}^{m+1}\|^{2}*w_{\tau_{m}}(x).$

where we used the locally constant property in the second to last inequality and the pointwise relation $w_{\tau_{m}}*w_{\tau_{m}}\lesssim w_{\tau_{m}}$ for the final inequality. Then

|\sum_{\tau_{m}\subset\tau}f_{\tau_{m}}^{m}(x)-f_{\tau_{m}}^{m+1}(x)|\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{\tau_{m}\subset\tau}|f_{\tau_{m}}^{m+1}|^{2}*w_{\tau_{m}}(x)\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}g_{m}(x).

By the definition of $\Omega_{k}$ and Lemma 11, $g_{m}(x)\leq 2|g_{m}^{\ell}(x)|\leq 2Cg_{m+1}(x)\leq\cdots\leq(2C)^{\varepsilon^{-1}}G(x)\lesssim(2C)^{\varepsilon^{-1}}r$ . Conclude that

|\sum_{\tau_{m}\subset\tau}f_{\tau_{m}}^{m}(x)-f_{\tau_{m}}^{m+1}(x)|\lesssim(2C)^{\varepsilon^{-1}}R^{-M\delta}\alpha.

The claim for $L$ follows immediately from the above argument, using the low-dominance of $g_{k}$ for all $k$ . ∎

Definition 6.

Call the distribution function $\lambda$ associated to a function $f$ $(R,\varepsilon)$ -normalized if for any $\tau_{k},\tau_{m}$ ,

\displaystyle\#\{\tau_{k}\subset\tau_{m}:f_{\tau_{k}}\not=0\}

\displaystyle\leq 100\frac{\lambda(R_{m}^{-1/2})}{\lambda(R_{k}^{-1/2})}.

Lemma 13 (High lemma I).

Assume that $f$ has an $(R,\varepsilon)$ -normalized distribution function $\lambda(\cdot)$ . For each dyadic $s$ , $R^{-\beta}\leq s\leq R^{-1/2}$ ,

\int_{\mathbb{R}^{2}}|G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}|^{2}\lesssim C_{\varepsilon}R^{2\varepsilon}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}.

Proof.

Organize the $\{\gamma\}$ into subcollections $\{\theta_{s}\}$ in which each $\theta_{s}$ is a union of $\gamma$ which intersect the same $\sim s$ -arc of $\mathbb{P}^{1}$ , where here for concreteness, $\sim s$ means within a factor of $2$ . Then by Plancherel’s theorem, since $\overline{\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}}_{\sim s}={\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}}_{\sim s}$ , we have for each $\theta$



(8)

The support of $\widehat{\overline{f}}_{\theta_{s}^{\prime}}(\xi)=\int e^{-2\pi ix\cdot\xi}\overline{f}_{\theta_{s}^{\prime}}(x)dx=\overline{\widehat{f}}_{\theta_{s}^{\prime}}(-\xi)$ is contained in $-\theta_{s}^{\prime}$ . This means that the support of $\widehat{f}_{\theta_{s}}*\widehat{\overline{f}}_{\theta_{s}^{\prime}}(\xi)$ is contained in $\theta_{s}-\theta_{s}^{\prime}$ . Since the support of $\eta_{\sim s}(\xi)$ is contained in the ball of radius $2s$ , for each $\theta_{s}\subset\theta$ , there are only finitely many $\theta_{s}^{\prime}\subset\theta$ so that the integral in (8) is nonzero. Thus we may write

G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}(x)=\sum_{\theta}|f_{\theta}|^{2}*w_{\theta}*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}(x)=\sum_{\theta}\sum_{\begin{subarray}{c}\theta_{s},\theta_{s}^{\prime}\subset\theta\\ \theta_{s}\sim\theta_{s}^{\prime}\end{subarray}}(f_{\theta_{s}}\overline{f}_{\theta_{s}^{\prime}})*w_{\theta}*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}(x).

where the second sum is over $\theta_{s},\theta_{s}^{\prime}\subset\theta$ with $\text{dist}(\theta_{s},\theta_{s}^{\prime})<2s$ . Using the above pointwise expression and then Plancherel’s theorem, we have

For each $\theta$ , $\sum_{\begin{subarray}{c}\theta_{s},\theta_{s}^{\prime}\subset\theta\\ \theta_{s}\sim\theta_{s}^{\prime}\end{subarray}}(\widehat{f_{\theta_{s}}}*\widehat{\overline{f}}_{\theta_{s}^{\prime}})$ is supported in $\theta-\theta$ , since each summand is supported in $\theta_{s}-\theta_{s}^{\prime}$ and $\theta_{s},\theta_{s}^{\prime}\subset\theta$ . For each $\xi\in\mathbb{R}^{2}$ , $|\xi|>\frac{1}{2}r$ , the maximum number of $\theta-\theta$ containing $\xi$ is bounded by the maximum number of $\theta$ intersecting an $R^{-1/2}\cdot s^{-1}R^{-1/2}$ -arc of the parabola. Using that $\lambda(\cdot)$ is $(R,\varepsilon)$ -normalized, this number is bounded above by $C_{\varepsilon}R^{\varepsilon}\frac{\lambda(s^{-1}R^{-1})}{\lambda(R^{-1/2})}$ . Since $\eta_{\sim s}$ is supported in the region $|\xi|>\frac{1}{2}r$ , by Cauchy-Schwarz

It remains to analyze each of the integrals above:

Bound the $L^{\infty}$ norms using the assumption that $\|f_{\gamma}\|_{\infty}\lesssim 1$ for all $\gamma$ :

\displaystyle\|\sum_{\theta_{s}\subset\theta}|f_{\theta_{s}}|^{2}*w_{\theta}*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}|\|_{\infty}\lesssim\sum_{\theta_{s}\subset\theta}\|f_{\theta_{s}}\|_{\infty}^{2}\lesssim\sum_{\theta_{s}\subset\theta}\|\sum_{\gamma\subset\theta_{s}}|f_{\gamma}|\|_{\infty}

\displaystyle\lesssim\lambda(R^{-1/2})\lambda(s).

Finally, using Young’s convolution inequality and the $L^{2}$ -orthogonality of the $f_{\gamma}$ , we have

\int_{\mathbb{R}^{2}}\sum_{\theta_{s}\subset\theta}|f_{\theta_{s}}|^{2}*w_{\theta}*|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}|\lesssim\int_{\mathbb{R}^{2}}\sum_{\theta_{s}\subset\theta}|f_{\theta_{s}}|^{2}=\sum_{\gamma\subset\theta}\|f_{\gamma}\|_{2}^{2}.

∎

Lemma 14 (High lemma II).

For each $k$ ,

\int_{\mathbb{R}^{2}}|g_{k}^{h}|^{2}\lesssim R^{3\varepsilon}\sum_{\tau_{k}}\int_{\mathbb{R}^{2}}|f_{\tau_{k+1}}^{k+1}|^{4}.

Proof.

By Plancherel’s theorem, we have

	$\displaystyle\int_{\mathbb{R}^{2}}\|g_{k}^{h}\|^{2}$	$\displaystyle=\int_{\mathbb{R}^{2}}\|g_{k}-g_{k}^{\ell}\|^{2}$
		$\displaystyle=\int_{\mathbb{R}^{2}}\|\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}-\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}\widehat{\overline{f^{k+1}_{\tau_{k}}}})\widehat{w}_{\tau_{k}}\eta_{k}\|^{2}$
		$\displaystyle\leq\int_{\|\xi\|>cR_{k+1}^{-1/2}}\|\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}*\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}\|^{2}$

since $(1-\eta_{k})$ is supported in the region $|\xi|>cR_{k+1}^{-1/2}$ for some constant $c>0$ . For each $\tau_{k}$ , $\widehat{f_{\tau_{k}}^{k+1}}*\widehat{\overline{f_{\tau_{k}}^{k+1}}}$ is supported in $2\tau_{k}-2\tau_{k}$ , using property (4) of Lemma 9. The maximum overlap of the sets $\{2\tau_{k}-2\tau_{k}\}$ in the region $|\xi|\geq cR_{k+1}^{-1/2}$ is bounded by $\sim\frac{R_{k}^{-1/2}}{R_{k+1}^{-1/2}}\lesssim R^{\varepsilon}$ . Thus using Cauchy-Schwarz,

	$\displaystyle\int_{\|\xi\|>cR_{k+1}^{-1/2}}\|\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}*\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}\|^{2}$	$\displaystyle\lesssim R^{\varepsilon}\sum_{\tau_{k}}\int_{\|\xi\|>cR_{k+1}^{-1/2}}\|(\widehat{f_{\tau_{k}}^{k+1}}*\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}\|^{2}$
		$\displaystyle\leq R^{\varepsilon}\sum_{\tau_{k}}\int_{\mathbb{R}^{2}}\|(\widehat{f_{\tau_{k}}^{k+1}}*\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}\|^{2}$
		$\displaystyle=R^{\varepsilon}\sum_{\tau_{k}}\int_{\mathbb{R}^{2}}\|\|f_{\tau_{k}}^{k+1}\|^{2}*{w}_{\tau_{k}}\|^{2}\leq R^{3\varepsilon}\sum_{\tau_{k+1}}\int_{\mathbb{R}^{2}}\|f_{\tau_{k+1}}^{k+1}\|^{4}$

where we used Young’s inequality with $\|w_{\tau_{k}}\|_{1}\lesssim 1$ and $f_{\tau_{k}}^{k+1}=\sum_{\tau_{k+1}\subset\tau_{k}}f_{\tau_{k+1}}^{k+1}$ with Cauchy-Schwarz again in the last line. ∎

4.3. Bilinear restriction

We will use the following version of a local bilinear restriction theorem, which follows from a standard Córdoba argument [Cor77] included here for completeness.

Theorem 15.

Let $S\geq 4$ , $\frac{1}{2}\geq D\geq S^{-1/2}$ , and $X\subset\mathbb{R}^{2}$ be any Lebesgue measurable set. Suppose that $\tau$ and $\tau^{\prime}$ are $D$ -separated subsets of $\mathcal{N}_{S^{-1}}(\mathbb{P}^{1})$ . Then for a partition $\{\theta_{S}\}$ of $\mathcal{N}_{S^{-1}}(\mathbb{P}^{1})$ into $\sim S^{-1/2}\times S^{-1}$ -blocks, we have

\int_{X}|f_{\tau}|^{2}(x)|f_{\tau^{\prime}}|^{2}(x)dx\lesssim D^{-2}\int_{\mathcal{N}_{S^{1/2}}(X)}|\sum_{\theta_{S}}|f_{\theta_{S}}|^{2}*w_{S^{1/2}}(x)|^{2}dx.

In the following proof, the exact definition of the $\sim S^{-1}\times S^{-1}$ blocks $\theta_{S}$ is not important. However, by $f_{\tau}$ and $f_{\tau^{\prime}}$ , we mean more formally $f_{\tau}=\sum_{\theta_{S}\cap\tau\not=\emptyset}f_{\theta_{S}}$ and $f_{\tau^{\prime}}=\sum_{\theta_{S}\cap\tau^{\prime}\not=\emptyset}f_{\theta_{S}}$ .

Proof.

Let $B$ be a ball of radius $S^{1/2}$ centered at a point in $X$ . Let $\varphi_{B}$ be a smooth function satisfying $\varphi_{B}\gtrsim 1$ in $B$ , $\varphi_{B}$ decays rapidly away from $B$ , and $\widehat{\varphi_{B}}$ is supported in the $S^{-1/2}$ neighborhood of the origin. Then

\displaystyle\int_{X\cap B}|f_{\tau}|^{2}|f_{\tau^{\prime}}|^{2}

\displaystyle\lesssim\int_{\mathbb{R}^{2}}|f_{\tau}|^{2}|f_{\tau^{\prime}}|^{2}\varphi_{B}.

Since $S$ is a fixed parameter and $\theta_{S}$ are fixed $\sim S^{-1/2}\times S^{-1}$ blocks, simplify notation by dropping the $S$ . Expand the squared terms in the integral above to obtain

\int_{\mathbb{R}^{2}}|f_{\tau}|^{2}|f_{\tau^{\prime}}|^{2}\varphi_{B}=\sum_{\begin{subarray}{c}\theta_{i}\cap\tau\not=\emptyset\\ \theta^{\prime}_{i}\cap\tau^{\prime}\not=\emptyset\end{subarray}}\int_{\mathbb{R}^{2}}f_{\theta_{1}}\overline{f}_{\theta_{2}}f_{\theta_{2}^{\prime}}\overline{f}_{\theta_{1}^{\prime}}\varphi_{B}.

By Placherel’s theorem, each integral vanishes unless

(9)

(\theta_{1}-\theta_{2})\cap\mathcal{N}_{S^{-1/2}}(\theta_{1}^{\prime}-\theta_{2}^{\prime})\not=\emptyset.

Next we check that the number of tuples $(\theta_{1},\theta_{2},\theta_{1}^{\prime},\theta_{2}^{\prime})$ (with $\theta_{1},\theta_{2}$ having nonempty intersection with $\tau$ and $\theta_{1}^{\prime},\theta_{2}^{\prime}$ having nonempty intersection with $\tau^{\prime}$ ) satisfying (9) is $O(D^{-1})$ . Indeed, suppose that $\xi<\xi^{\prime}<\xi^{\prime\prime}<\xi^{\prime\prime\prime}$ satisfy

(\xi,\xi^{2})\in\theta_{1},\quad(\xi^{\prime},(\xi^{\prime})^{2})\in\theta_{2},\quad(\xi^{\prime\prime},(\xi^{\prime\prime})^{2})\in\theta_{1}^{\prime},\quad(\xi^{\prime\prime\prime},(\xi^{\prime\prime\prime})^{2})\in\theta_{2}^{\prime}

and

\xi-\xi^{\prime}=\xi^{\prime\prime}-\xi^{\prime\prime\prime}+O(S^{-1/2}).

Then by the mean value theorem, $\xi^{2}-(\xi^{\prime})^{2}=2\xi_{1}(\xi-\xi^{\prime})$ for some $\xi<\xi_{1}<\xi^{\prime}$ and $(\xi^{\prime\prime})^{2}-(\xi^{\prime\prime\prime})^{2}=2\xi_{2}(\xi^{\prime\prime}-\xi^{\prime\prime\prime})$ for some $\xi^{\prime\prime}<\xi_{2}<\xi^{\prime\prime\prime}$ . Since $(\xi_{1},\xi_{1}^{2})\in\tau$ and $(\xi_{2},\xi_{2}^{2})\in\tau^{\prime}$ , we also know that $|\xi_{1}-\xi_{2}|\geq D$ . Putting everything together, we have

	$\displaystyle\|\xi^{2}-(\xi^{\prime})^{2}-((\xi^{\prime\prime})^{2}-(\xi^{\prime\prime\prime})^{2})\|$	$\displaystyle=2\|\xi_{1}(\xi-\xi^{\prime})-\xi_{2}(\xi^{\prime\prime}-\xi^{\prime\prime\prime})\|$
		$\displaystyle\geq 2\|\xi_{1}-\xi_{2}\|\|\xi-\xi^{\prime}\|-cS^{-1/2}\geq(2C-c)S^{-1/2}$

if either $\text{dist}((\xi,\xi^{2}),(\xi^{\prime},(\xi^{\prime})^{2}))$ or $\text{dist}((\xi^{\prime\prime},(\xi^{\prime\prime})^{2}),(\xi^{\prime\prime\prime},(\xi^{\prime\prime\prime})^{2}))$ is larger than $CD^{-1}S^{-1/2}$ . Thus for a suitably large $C$ , the heights will have difference larger than the allowed $O(S^{-1/2})$ -neighborhood imposed by (9). The conclusion is that

	$\displaystyle\sum_{\begin{subarray}{c}\theta_{i}\cap\tau\not=\emptyset\\ \theta^{\prime}_{i}\cap\tau^{\prime}\not=\emptyset\end{subarray}}\int_{\mathbb{R}^{2}}f_{\theta_{1}}\overline{f}_{\theta_{2}}f_{\theta_{2}^{\prime}}\overline{f}_{\theta_{1}^{\prime}}\varphi_{B}$	$\displaystyle=\sum_{\begin{subarray}{c}\theta_{1}\cap\tau\not=\emptyset\\ \theta^{\prime}_{1}\cap\tau^{\prime}\not=\emptyset\end{subarray}}\,\,\sum_{\begin{subarray}{c}d(\theta_{1},\theta_{2})\leq CD^{-1}S^{-1/2}\\ d(\theta_{1}^{\prime},\theta_{2}^{\prime})\leq CD^{-1}S^{-1/2}\end{subarray}}\int_{\mathbb{R}^{2}}f_{\theta_{1}}\overline{f}_{\theta_{2}}f_{\theta_{2}^{\prime}}\overline{f}_{\theta_{1}^{\prime}}\varphi_{B}$
		$\displaystyle\lesssim D^{-2}\int_{\mathbb{R}^{2}}(\sum_{\theta}\|f_{\theta}\|^{2})^{2}\varphi_{B}.$

Using the locally constant property and summing over a finitely overlapping cover of $\mathbb{R}^{2}$ by $S^{1/2}-$ balls $B^{\prime}$ with centers $c_{B^{\prime}}$ , we have

	$\displaystyle\int_{\mathbb{R}^{2}}(\sum_{\theta}\|f_{\theta}\|^{2})^{2}\varphi_{B}$	$\displaystyle\leq\sum_{B^{\prime}}\|B\|\\|\sum_{\theta}\|f_{\theta}\|^{2}\\|_{L^{\infty}(B^{\prime})}^{2}\\|\varphi_{B}\\|_{L^{\infty}(B^{\prime})}$
		$\displaystyle\leq\|B\|\Big{(}\sum_{B^{\prime}}\\|\sum_{\theta}\|f_{\theta}\|^{2}\\|_{L^{\infty}(B^{\prime})}\\|\varphi_{B}^{1/2}\\|_{L^{\infty}(B^{\prime})}\Big{)}^{2}$
		$\displaystyle\lesssim\|B\|\Big{(}\sum_{B^{\prime}}\sum_{\theta}\|f_{\theta}\|^{2}*w_{S^{1/2}}(c_{B^{\prime}})\\|\varphi_{B}^{1/2}\\|_{L^{\infty}(B^{\prime})}\Big{)}^{2}$
		$\displaystyle\lesssim\|B\|^{-1}\Big{(}\int_{\mathbb{R}^{2}}\sum_{\theta}\|f_{\theta}\|^{2}*w_{S^{1/2}}(y)\varphi_{B}^{1/2}(y)dy\Big{)}^{2}$
		$\displaystyle\lesssim\|B\|^{-1}\Big{(}\int_{B}\sum_{\theta}\|f_{\theta}\|^{2}*w_{S^{1/2}}(y)dy\Big{)}^{2}$
		$\displaystyle\leq\int_{B}\Big{(}\sum_{\theta}\|f_{\theta}\|^{2}*w_{S^{1/2}}\Big{)}^{2}$

where we used that $w_{S^{1/2}}*\varphi_{B}^{1/2}(y)\lesssim w_{S^{1/2}}*\chi_{B}(y)$ in the second to last line.

∎

5. Proof of Theorem 4

Theorem 4 follows from the following proposition and a broad-narrow argument in §5.2. First we prove a version of Theorem 4 where $U_{\alpha}$ is replaced by a “broad” version of $U_{\alpha}$ .

5.1. The broad version of Theorem 4

Let $\delta>0$ be a parameter we will choose in the broad/narrow analysis. The notation $\ell(\tau)=s$ means that $\tau$ is an approximate $s\times s^{2}$ block which is part of a partition of $\mathcal{N}_{s^{2}}(\mathbb{P}^{1})$ . For two non-adjacent blocks $\tau,\tau^{\prime}$ satisfying $\ell(\tau)=\ell(\tau^{\prime})=R^{-\delta}$ , define the broad version of $U_{\alpha}$ to be

(10)

\text{Br}_{\alpha}(\tau,\tau^{\prime})=\{x\in\mathbb{R}^{2}:\alpha\sim|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2},\,\,(|f_{\tau}(x)|+|f_{\tau^{\prime}}(x)|)\leq R^{O(\delta)}\alpha\}\}.

Proposition 1.

Suppose that $f$ satisfies the hypotheses of Theorem 4 and has an $(R,\varepsilon)$ -normalized distribution function $\lambda(\cdot)$ . Then

\displaystyle|\text{\emph{Br}}_{\alpha}(\tau,\tau^{\prime})|

\displaystyle\leq C_{\varepsilon,\delta}R^{\varepsilon}R^{O(\delta)}\begin{cases}\frac{1}{\alpha^{4}}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}\quad&\text{if}\quad\alpha^{2}>\frac{\lambda(1)^{2}}{\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)}\\ \frac{\lambda(1)^{2}}{\alpha^{6}}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}\quad&\text{if}\quad\alpha^{2}\leq\frac{\lambda(1)^{2}}{\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)}\end{cases}.

Proof of Proposition 1.

Bounding $|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H|$ : Using bilinear restriction, given here by Theorem 15, we have

\alpha^{4}|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H|\lesssim\sum_{\begin{subarray}{c}\ell(\tau)=\ell(\tau)=R^{-\delta}\\ d(\tau,\tau^{\prime})\gtrsim R^{-\delta}\end{subarray}}\int_{U_{\alpha}\cap H}|f_{\tau}|^{2}|f_{\tau^{\prime}}|^{2}\lesssim R^{O(\delta)}\int_{\mathcal{N}_{R^{1/2}}(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H)}(\sum_{\theta}|f_{\theta}|^{2}*w_{R^{1/2}})^{2}.

By the locally constant property and the pointwise inequality $w_{R^{1/2}}*w_{\theta}\lesssim w_{\theta}$ for each $\theta$ , we have that $\sum_{\theta}|f_{\theta}|^{2}*w_{R^{1/2}}\lesssim G(x)$ . Then

(11)

\displaystyle\int_{\mathcal{N}_{R^{1/2}}(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H)}|G(x)|^{2}dx\leq\sum_{\begin{subarray}{c}Q_{R^{1/2}}:\\ Q_{R^{1/2}}\cap(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H)\not=\emptyset\end{subarray}}|Q_{R^{1/2}}|\|G\|_{L^{\infty}(Q_{R^{1/2}}\cap(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H))}^{2}

For each $x\in H$ , $G(x)\leq 2|G^{h}(x)|$ . Also note the equality $G^{h}(x)=\sum_{s}G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}(x)$ where the sum is over dyadic $s$ in the range $\lceil{R^{\beta}}\rceil^{-1}\lesssim s\lesssim R^{-1/2}$ . This is because the Fourier support of $G^{h}$ is contained in $\cup_{\theta}(\theta-\theta)\setminus B_{c\lceil R^{\beta}\rceil^{-1}}$ for a sufficiently small $c>0$ . By dyadic pigeonholing, there is some dyadic $s$ , $\lceil{R^{\beta}}\rceil^{-1}\lesssim s\lesssim R^{-1/2}$ , so that the upper bound in (11) is bounded by

(\log R)\sum_{\begin{subarray}{c}Q_{R^{1/2}}:\\ Q_{R^{1/2}}\cap(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H)\not=\emptyset\end{subarray}}|Q_{R^{1/2}}|\|G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}\|_{L^{\infty}(Q_{R^{1/2}}\cap(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H))}^{2}.

By the locally constant property, the above displayed expression is bounded by

(\log R)\sum_{\begin{subarray}{c}Q_{R^{1/2}}:\\ Q_{R^{1/2}}\cap(\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H)\end{subarray}}\int_{\mathbb{R}^{2}}|G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}|^{2}w_{Q_{R^{1/2}}}\lesssim(\log R)\int_{\mathbb{R}^{2}}|G*\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\eta}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-624.0pt]{4.30554pt}{624.0pt}}}{}}{0.5ex}}\stackon[1pt]{\eta}{\scalebox{-1.0}{\tmpbox}}_{\sim s}|^{2}.

Use Lemma 13 to upper bound the above integral to finish bounding $|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H|$ .

Bounding $|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}|$ : First write the trivial inequality

\alpha^{4}|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}|\leq\sum_{\begin{subarray}{c}\ell(\tau)=\ell(\tau)=R^{-\delta}\\ d(\tau,\tau^{\prime})\gtrsim R^{-\delta}\end{subarray}}\int_{\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}\cap\{|f_{\tau}f_{\tau^{\prime}}|^{1/2}\sim\alpha\}}|f_{\tau}|^{2}|f_{\tau^{\prime}}|^{2}.

By the definition of $\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}$ and Lemma 12, for each $x\in\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}$ we have

	$\displaystyle\|f_{\tau}(x)f_{\tau^{\prime}}(x)\|$	$\displaystyle\leq\|f_{\tau}(x)\|\|f_{\tau^{\prime}}(x)-f_{\tau^{\prime}}^{k+1}(x)\|+\|f_{\tau}(x)-f_{\tau}^{k+1}(x)\|\|f_{\tau^{\prime}}^{k+1}(x)\|+\|f_{\tau}^{k+1}(x)f_{\tau^{\prime}}^{k+1}(x)\|$
		$\displaystyle\lesssim C_{\varepsilon}R^{O(\delta)}R^{-M\delta}\alpha^{2}+\|f_{\tau}^{k+1}(x)f_{\tau^{\prime}}^{k+1}(x)\|.$

For $M$ large enough in the definition of pruning (depending on the implicit universal constant from the broad/narrow analysis which determines the set $\text{Br}_{\alpha}(\tau,\tau^{\prime})$ ) so that $R^{O(\delta)}R^{-M\delta}\leq R^{-\delta}$ and for $R$ large enough depending on $\varepsilon$ and $\delta$ , we may bound each integral by

\int_{\{\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}\cap\{|f_{\tau}f_{\tau^{\prime}}|^{1/2}\sim\alpha\}}|f_{\tau}|^{2}|f_{\tau^{\prime}}|^{2}\lesssim\int_{\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}}|f_{\tau}^{k+1}|^{2}|f_{\tau^{\prime}}^{k+1}|^{2}.

Repeat analogous bilinear restriction, high-dominated from the definition of $\Omega_{k}$ , and locally-constant steps from the argument bounding $\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap H$ to obtain

\displaystyle\alpha^{4}|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}|

\displaystyle\lesssim R^{O(\delta)}\int_{\mathbb{R}^{2}}|g_{k}^{h}|^{2}.

Use Lemma 14 and Lemma 9 to bound the above integral, obtaining

	$\displaystyle\alpha^{4}\|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap\Omega_{k}\|$	$\displaystyle\lesssim(\log R)^{4}\int_{\mathbb{R}^{2}}\|g_{k}^{h}\|^{2}$
		$\displaystyle\lesssim R^{O(\delta)}R^{O(\varepsilon)}\frac{\lambda(1)^{2}}{\alpha^{2}}\sum_{\tau_{k+1}}\int_{\mathbb{R}^{2}}\|f_{\tau_{k+1}}^{k+1}\|^{2}.$

Use $L^{2}$ -orthogonality and that $|f_{\tau_{m}}^{m}|\leq|f_{\tau_{m}}^{m+1}|$ for each $m$ to bound each integral above:

\displaystyle\int_{\mathbb{R}^{2}}|f_{\tau_{k+1}}^{k+1}|^{2}

\displaystyle\leq\int_{\mathbb{R}^{2}}|f_{\tau_{k+1}}^{k+2}|^{2}\leq C\sum_{\tau_{k+2}\subset\tau_{k+1}}\int_{\mathbb{R}^{2}}|f_{\tau_{k+2}}^{k+2}|^{2}\leq\cdots\leq C^{\varepsilon^{-1}}\sum_{\gamma\subset\tau_{k+1}}\int_{\mathbb{R}^{2}}|f_{\gamma}|^{2}.

We are done with this case because

\displaystyle\frac{\lambda(1)^{2}}{\alpha^{2}}

\displaystyle\leq\begin{cases}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\quad&\text{if}\quad\alpha^{2}>\frac{\lambda(1)^{2}}{\underset{s}{max}\lambda(s^{-1}R^{-1})\lambda(s)}\\ \frac{\lambda(1)^{2}}{\alpha^{2}}\quad&\text{if}\quad\alpha^{2}\leq\frac{\lambda(1)^{2}}{\max_{s}\lambda(s^{-1}R^{-1})\lambda(s)}\end{cases}.

Bounding $|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap L|$ : Repeat the pruning step from the previous case to get

\alpha^{6}|\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap L|\lesssim\sum_{\begin{subarray}{c}\ell(\tau)=\ell(\tau)=R^{-\delta}\\ d(\tau,\tau^{\prime})\gtrsim R^{-\delta}\end{subarray}}\int_{\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap L\cap\{|f_{\tau}f_{\tau^{\prime}}|^{1/2}\sim\alpha\}}|f_{\tau}^{1}f_{\tau^{\prime}}^{1}|^{2}|f_{\tau}f_{\tau^{\prime}}|.

Use Cauchy-Schwartz and the locally constant lemma for the bound $|f_{\tau}^{1}f_{\tau^{\prime}}^{1}|\lesssim R^{O(\varepsilon)}G_{0}$ and recall that by Lemma 11, $G_{0}\leq C_{\varepsilon}R^{\varepsilon}\lambda(1)$ . Then

\displaystyle R^{O(\varepsilon)}\sum_{\begin{subarray}{c}\ell(\tau)=\ell(\tau)=R^{-\delta}\\ d(\tau,\tau^{\prime})\gtrsim R^{-\delta}\end{subarray}}\int_{\text{Br}_{\alpha}(\tau,\tau^{\prime})\cap L}|G_{0}|^{2}|f_{\tau}f_{\tau^{\prime}}|

\displaystyle\leq R^{O(\varepsilon)}\lambda(1)^{2}\sum_{\ell(\tau)=R^{-\delta}}\int_{\mathbb{R}^{2}}|f_{\tau}|^{2}\lesssim R^{O(\varepsilon)}\lambda(1)^{2}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}.

Using the same upper bound for $\frac{\lambda(1)^{2}}{\alpha^{2}}$ as in the previous case finishes the proof.

∎

5.2. Bilinear reduction

We will present a broad/narrow analysis to show that Proposition 1 implies Theorem 4. In order to apply Proposition 1, we must reduce to the case that $f$ has an $(R,\varepsilon)$ -normalized distribution function $\lambda(\cdot)$ . We demonstrate this through a series of pigeonholing steps.

Proposition 1 implies Theorem 4.

We will pigeonhole the $f_{\gamma}$ so that roughly, for any $s$ -arc $\omega$ of the parabola, the number

\#\{\gamma:\gamma\cap\omega\not=\emptyset,\quad f_{\gamma}\not=0\}

is either $0$ or relatively constant among $s$ -arcs $\omega$ . For the initial step, write

\{\tau_{N}:\exists\gamma\,\text{s.t.}\,f_{\gamma}\not=0,\,\,\gamma\subset\tau_{N}\}=\sum_{1\leq\lambda\lesssim R^{\beta}R^{-\varepsilon}}\Lambda_{N}(\lambda)

where $\lambda$ is a dyadic number, $\{\tau_{N}:\#\gamma\subset\tau_{N}\sim\lambda\}$ , $\#\gamma\subset\tau_{N}$ means $\#\{\gamma\subset\tau_{N}:f_{\gamma}\not=0\}$ , and $\#\gamma\subset\tau_{N}\sim\lambda$ means $\lambda\leq\#\gamma\subset\tau_{N}<2\lambda$ . Since there are $\lesssim\log R$ many $\lambda$ in the sum, there exists some $\lambda_{N}$ such that

|\{x:|f(x)|>\alpha\}|\leq C(\log R)|\{x:C(\log R)|\sum_{\tau_{N}\in\Lambda_{N}(\lambda_{N})}f_{\tau_{N}}(x)|>\alpha\}|.

Continuing in this manner, we have

\{\tau_{k}:\exists\tau_{k+1}\in\Lambda_{k+1}(\lambda_{k+1})\,\text{s.t.}\,\tau_{k+1}\subset\tau_{k}\}=\sum_{1\leq\lambda\leq r_{k}}\Lambda_{k}(\lambda)

where $\Lambda_{k}(\lambda)=\{\tau_{k}:\exists\tau_{k+1}\in\Lambda_{k+1}(\lambda_{k+1})\,\text{s.t.}\,\tau_{k+1}\subset\tau_{k}\quad\text{and}\quad\#\gamma\subset\tau_{k}\sim\lambda\}$ and for some $\lambda_{k}$ ,

	$\displaystyle\|\{x:(C(\log R))^{N-k}\|$	$\displaystyle\sum_{\tau_{k+1}\in\Lambda_{k+1}(\lambda_{k+1})}f_{\tau_{k+1}}(x)\|>\alpha\}\|$
		$\displaystyle\qquad\qquad\leq C(\log R)\|\{x:(C(\log R))^{N-k+1}\|\sum_{\tau_{k}\in\Lambda_{k}(\lambda_{k})}f_{\tau_{k}}(x)\|>\alpha\}\|.$

Continue this process until we have found $\tau_{1}$ , $\lambda_{1}$ so that

\displaystyle|\{x:|f(x)|>\alpha\}|\leq C^{\varepsilon^{-1}}(\log R)^{O(\varepsilon^{-1})}|\{x:C^{\varepsilon^{-1}}(\log R)^{O(\varepsilon^{-1})}|\sum_{\tau_{1}\in\Lambda_{1}(\lambda_{1})}f_{\tau_{1}}(x)|>\alpha\}|.

The function $\sum_{\tau_{1}\in\Lambda_{1}(\lambda_{1})}f_{\tau_{1}}$ now satisfies the hypotheses of Theorem 4 and the property that $\#\gamma\subset\tau_{k}\sim\lambda_{k}$ or $\#\gamma\subset\tau_{k}=0$ for all $k$ , $\tau_{k}$ . It follows that the associated distribution function $\lambda(\cdot)$ of $\sum_{\tau_{1}\in\Lambda_{1}(\lambda_{1})}f_{\tau_{1}}$ is $(R,\varepsilon)$ -normalized since

\lambda_{m}\sim\#\gamma\subset\tau_{m}=\sum_{\tau_{k}\subset\tau_{m}}\#\gamma\subset\tau_{k}\sim(\#\tau_{k}\subset\tau_{m})(\lambda_{k})

where we only count the $\gamma$ or $\tau_{k}$ for which $f_{\gamma}$ or $f_{\tau_{k}}$ is nonzero. Now we may apply Proposition 1. Note that since $\log R\leq\varepsilon^{-1}R^{\varepsilon}$ for all $R\geq 1$ , the accumulated constant from this pigeonholing process satisfies $C^{\varepsilon^{-1}}(\log R)^{O(\varepsilon^{-1})}\leq C_{\varepsilon}R^{\varepsilon}$ . It thus suffices to prove Theorem 4 assuming that $f$ is $(R,\varepsilon)$ -normalized.

Now we present a broad-narrow argument adapted to our set-up. Write $K=R^{\delta}$ for some $\delta>0$ which will be chosen later. Since $|f(x)|\leq\sum_{\ell(\tau)=K^{-1}}|f_{\tau}(x)|$ , there is a universal constant $C>0$ so that $|f(x)|>K^{C}\max_{\begin{subarray}{c}\ell(\tau)=\ell(\tau^{\prime})=K^{-1}\\ \tau,\tau^{\prime}\,\,\text{nonadj.}\end{subarray}}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2}$ implies $|f(x)|\leq C\max_{\ell(\tau)=K^{-1}}|f_{\tau}(x)|$ . If $|f(x)|\leq K^{C}\max_{\begin{subarray}{c}\ell(\tau)=\ell(\tau^{\prime})=K^{-1}\\ \tau,\tau^{\prime}\,\,\text{nonadj.}\end{subarray}}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2}$ and $K^{C}\max_{\begin{subarray}{c}\ell(\tau)=\ell(\tau^{\prime})=K^{-1}\\ \tau,\tau^{\prime}\,\,\text{nonadj.}\end{subarray}}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2}\leq C\max_{\ell(\tau)=K^{-1}}|f_{\tau}(x)|$ , then $|f(x)|\leq C\max_{\ell(\tau)=K^{-1}}|f_{\tau}(x)|$ . Using this reasoning, we obtain the first step in the broad-narrow inequality

\displaystyle|f(x)|

\displaystyle\leq C\max_{\ell(\tau)=K^{-1}}|f_{\tau}(x)|+K^{C}\max_{\begin{subarray}{c}\ell(\tau)=\ell(\tau^{\prime})=K^{-1}\\ \tau,\tau^{\prime}\,\text{nonadj.}\\ C\underset{\ell(\tau_{0})=K^{-1}}{\max}|f_{\tau_{0}}(x)|\leq K^{C}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2}\end{subarray}}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2}.

Iterate the inequality $m$ times (for the first term) where $K^{m}\sim R^{1/2}$ to bound $|f(x)|$ by

	$\displaystyle\|f(x)\|$	$\displaystyle\lesssim C^{m}\max_{\ell(\tau)=R^{-1/2}}\|f_{\tau}(x)\|$
		$\displaystyle\qquad+C^{m}K^{C}\sum_{\begin{subarray}{c}R^{-1/2}<\Delta<1\\ \Delta\in K^{\mathbb{N}}\end{subarray}}\max_{\begin{subarray}{c}\ell(\tilde{\tau})\sim\Delta\end{subarray}}\max_{\begin{subarray}{c}\ell(\tau)=\ell(\tau^{\prime})\sim K^{-1}\Delta\\ \tau,\tau^{\prime}\subset\tilde{\tau},\,\,\text{nonadj.}\\ C\underset{\begin{subarray}{c}\ell(\tau_{0})=K^{-1}\Delta\\ \tau_{0}\subset\tilde{\tau}\end{subarray}}{\max}\|f_{\tau_{0}}(x)\|\leq K^{C}\|f_{\tau}(x)f_{\tau^{\prime}}(x)\|^{1/2}\end{subarray}}\|f_{\tau}(x)f_{\tau^{\prime}}(x)\|^{1/2}.$

Recall that our goal is to bound the size of the set

U_{\alpha}=\{x\in\mathbb{R}^{2}:\alpha\leq|f(x)|\}.

By the triangle inequality and using the notation $\theta$ for blocks $\tau$ with $\ell(\tau)=R^{-1/2}$

(12)

\displaystyle|U_{\alpha}|\leq|\{x\in\mathbb{R}^{2}:\alpha\lesssim C^{m}\max_{\theta}|f_{\theta}(x)|\}|+\sum_{\begin{subarray}{c}R^{-1/2}<\Delta<1\\ \Delta\in K^{\mathbb{N}}\end{subarray}}\sum_{\begin{subarray}{c}\ell(\tilde{\tau})\sim\Delta\\ \ell(\tau)=\ell(\tau^{\prime})\sim K^{-1}\Delta\\ \tau,\tau^{\prime}\subset\tilde{\tau},\,\,\text{nonadj.}\end{subarray}}|U_{\alpha}(\tau,\tau^{\prime})|

where $U_{\alpha}(\tau,\tau^{\prime})$ is the set

\{x\in\mathbb{R}^{2}:\alpha\lesssim(\log R)C^{m}K^{C}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2},\,\,C(|f_{\tau}(x)|+|f_{\tau^{\prime}}(x)|)\leq K^{C}|f_{\tau}(x)f_{\tau^{\prime}}(x)|^{1/2}\}.

The first term in the upper bound from (12) is bounded trivially by $\frac{\lambda(R^{-1/2})^{2}}{\alpha^{4}}\sum_{\gamma}\|f_{\gamma}\|_{2}^{2}$ . By the assumption that $\|f_{\gamma}\|_{\infty}\lesssim 1$ for every $\gamma$ , we know that $|f_{\tau}|\lesssim R^{\beta}$ for any $\tau$ . Also assume without loss of generality that $\alpha>1$ (otherwise Theorem 4 follows from $L^{2}$ -orthogonality). This means that there are $\sim\log R$ dyadic values of $\alpha^{\prime}$ between $\alpha$ and $R^{\beta}$ so by pigeonholing, there exists $\alpha^{\prime}\in[\alpha/(C^{m}K^{C}),R^{\beta}]$ so that

|U_{\alpha}(\tau,\tau^{\prime})|\lesssim(\log R+\log(C^{m}K^{C}))|\text{Br}_{\alpha^{\prime}}(\tau,\tau^{\prime})|

where the set $\text{Br}_{\alpha^{\prime}}(\tau,\tau^{\prime})$ is defined in (10). By parabolic rescaling, there exists an affine transformation $T$ so that $f_{\tau}\circ T=g_{\underline{\tau}}$ and $f_{\tau^{\prime}}\circ T=g_{\underline{\tau}^{\prime}}$ where $\underline{\tau}$ and $\underline{\tau^{\prime}}$ are $\sim K^{-1}$ -separated blocks in $\mathcal{N}_{\Delta^{-2}R^{-1}}(\mathbb{P}^{1})$ . Note that the functions $g_{\underline{\tau}}$ and $g_{\underline{\tau}^{\prime}}$ inherit the property of being $(\Delta^{2}R,\varepsilon)$ -normalized in the sense required to apply Proposition 1 in each of the following cases.

Case 1: Suppose that for some $\beta^{\prime}\in[\frac{1}{2},1]$ , $\Delta^{-1}R^{-\beta}=(\Delta^{2}R)^{-\beta^{\prime}}$ . Then for each $\gamma\in\mathcal{P}(R,\beta)$ , $f_{\gamma}\circ T=g_{\underline{\gamma}}$ for some $\underline{\gamma}\in\mathcal{P}(\Delta^{2}R,\beta^{\prime})$ . Applying Proposition 1 with functions $g_{\underline{\tau}}$ and $g_{\underline{\tau}^{\prime}}$ and level set parameter $\alpha^{\prime}$ leads to the inequality

	$\displaystyle\|\text{Br}_{\alpha^{\prime}}(\tau,\tau^{\prime})\|\leq K^{C}\alpha^{\prime}\}\|\leq C_{\varepsilon,\delta}R^{\varepsilon}C^{m}K^{O(1)}\times$
	$\displaystyle\qquad\begin{cases}\frac{1}{(\alpha^{\prime})^{4}}\underset{R^{-\beta}<s<R^{-1/2}}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma\subset\tilde{\tau}}\\|f_{\gamma}\\|_{2}^{2}\quad&\text{if}\quad(\alpha^{\prime})^{2}>\frac{\lambda(\Delta)^{2}}{\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)}\\ \frac{\lambda(\Delta)^{2}}{(\alpha^{\prime})^{6}}\sum_{\gamma\subset\tilde{\tau}}\\|f_{\gamma}\\|_{2}^{2}\quad&\text{if}\quad(\alpha^{\prime})^{2}\leq\frac{\lambda(\Delta)^{2}}{\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)}\end{cases}$

Case 2: Now suppose that $\Delta^{-1}R^{-\beta}<(\Delta^{2}R)^{-1}$ . Let $\tilde{\theta}$ be $\Delta^{-1}R^{-1}\times R^{-1}$ blocks and let $\underline{\tilde{\theta}}$ be $(\Delta^{2}R)^{-1}\times(\Delta^{2}R)^{-1}$ blocks so that $f_{\tilde{\theta}}\circ T=g_{\underline{\tilde{\theta}}}$ . Let $B=\max_{\tilde{\theta}}|f_{\tilde{\theta}}|$ and divide everything by $B$ in order to satisfy the hypotheses $\|g_{\underline{\tilde{\theta}}}\|_{\infty}/B\leq 1$ for all $\underline{\tilde{\theta}}$ . Let $\tilde{\lambda}(s):=\lambda(\Delta s)/\lambda(\Delta^{-1}R^{-1})$ count the number of $\underline{\tilde{\theta}}$ intersecting an $s$ -arc. In the case $(\alpha^{\prime})^{2}>\frac{\tilde{\lambda}(1)B^{2}}{\max_{s}\tilde{\lambda}(s^{-1}(\Delta^{2}R)^{-1})\tilde{\lambda}(s)}$ (with the maximum taken over $(\Delta^{2}R)^{-1}<s<(\Delta^{2}R)^{-1/2}$ ), use Proposition 1 with functions $g_{\underline{\tau}}/B$ and $g_{\underline{\tau}^{\prime}}/B$ and level set parameter $\alpha^{\prime}/B$ to get the inequality

|\text{Br}_{\alpha^{\prime}}(\tau,\tau^{\prime})|\leq C_{\varepsilon,\delta}R^{\varepsilon}C^{m}K^{O(1)}\frac{B^{4}}{(\alpha^{\prime})^{4}}\underset{(\Delta^{2}R)^{-1}<s<(\Delta^{2}R)^{-1/2}}{\max}\tilde{\lambda}(s^{-1}(\Delta^{2}R)^{-1})\tilde{\lambda}(s)\sum_{\tilde{\theta}\subset\tilde{\tau}}\|f_{\tilde{\theta}}\|_{2}^{2}/B^{2}.

Note that since $B\leq\lambda(\Delta^{-1}R^{-1})$ ,

B^{2}\underset{(\Delta^{2}R)^{-1}<s<(\Delta^{2}R)^{-1/2}}{\max}\tilde{\lambda}(s^{-1}(\Delta^{2}R)^{-1})\tilde{\lambda}(s)\leq\underset{\Delta^{-1}R^{-1}<s<R^{-1/2}}{\max}{\lambda}(s^{-1}R^{-1}){\lambda}(s)

and

\frac{\tilde{\lambda}(1)^{2}B^{2}}{\underset{s}{\max}\tilde{\lambda}(s^{-1}(\Delta^{2}R)^{-1})\tilde{\lambda}(s)}\leq\frac{\lambda(\Delta)^{2}\lambda(\Delta^{-1}R^{-1})^{2}}{\underset{\Delta^{-1}R^{-1}<s<R^{-1/2}}{\max}{\lambda}(s^{-1}R^{-1}){\lambda}(s)}\leq\lambda(\Delta^{-1}R^{-1})\lambda(\Delta).

Then in the case $(\alpha^{\prime})^{2}\leq\frac{\tilde{\lambda}(1)B^{2}}{\max_{s}\tilde{\lambda}(s^{-1}(\Delta^{2}R)^{-1})\tilde{\lambda}(s)}$ , compute directly that

	$\displaystyle(\alpha^{\prime})^{4}\|\{x\in\mathbb{R}^{2}:\alpha^{\prime}\sim\|f_{\tau}(x)f_{\tau^{\prime}}(x)\|^{1/2},\,\,(\|f_{\tau}(x)\|+\|f_{\tau^{\prime}}(x)\|)\leq K^{C}\alpha^{\prime}\}\|$
	$\displaystyle\quad\lesssim\lambda(\Delta^{-1}R^{-1})\lambda(\Delta)\int_{\mathbb{R}^{2}}(\|f_{\tau}\|^{2}+\|f_{\tau^{\prime}}\|^{2})\lesssim\underset{\Delta^{-1}R^{-1}<s<R^{-1/2}}{\max}{\lambda}(s^{-1}R^{-1}){\lambda}(s)\sum_{\gamma\subset\tilde{\tau}}\\|f_{\gamma}\\|_{2}^{2}.$

Using also that $\sum_{\tilde{\theta}\subset\tilde{\tau}}\|f_{\tilde{\theta}}\|_{2}^{2}\leq\sum_{\gamma\subset\tilde{\tau}}\|f_{\gamma}\|_{2}^{2}$ , the bound for Case 2 is

	$\displaystyle\|\{x\in\mathbb{R}^{2}:\alpha^{\prime}\sim\|f_{\tau}(x)f_{\tau^{\prime}}(x)\|^{1/2},\,\,(\|f_{\tau}(x)\|+\|f_{\tau^{\prime}}(x)\|)\leq K^{C}\alpha^{\prime}\}\|$
	$\displaystyle\quad\leq C_{\varepsilon,\delta}R^{\varepsilon}C^{m}K^{O(1)}\frac{1}{(\alpha^{\prime})^{4}}\underset{R^{-\beta}<s<R^{-1/2}}{max}{\lambda}(s^{-1}(\Delta^{2}R)^{-1}){\lambda}(s)\sum_{\gamma\subset\tilde{\tau}}\\|f_{\gamma}\\|_{2}^{2}.$

It follows from (12) and the combined Case 1 and Case 2 arguments above that

	$\displaystyle\|U_{\alpha}\|\leq C_{\varepsilon,\delta}R^{\varepsilon}C^{m}K^{O(1)}\times$
	$\displaystyle\qquad\begin{cases}\frac{1}{\alpha^{4}}\underset{R^{-\beta}<s<R^{-1/2}}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}\quad&\text{if}\quad\alpha>\frac{\lambda(1)^{2}}{\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)}\\ \frac{\lambda(1)^{2}}{\alpha^{6}}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}\quad&\text{if}\quad\alpha^{2}\leq\frac{\lambda(1)^{2}}{\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)}\end{cases}.$

Recall that $K^{m}\sim R^{-1/2}$ and $K=R^{\delta}$ so that $C_{\varepsilon,\delta}R^{\varepsilon}C^{m}K^{O(1)}\leq C_{\varepsilon,\delta}R^{\varepsilon}C^{O(\delta^{-1})}R^{O(1)\delta}$ . Choosing $\delta$ small enough so that $R^{O(1)\delta}\leq R^{\varepsilon}$ finishes the proof.

∎

References

[Apo76] Tom M. Apostol. Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.
[BD15] Jean Bourgain and Ciprian Demeter. The proof of the $l^{2}$ decoupling conjecture. Ann. of Math. (2), 182(1):351–389, 2015.
[Bou93] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal., 3(2):107–156, 1993.
[Bou17a] J. Bourgain. Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc., 30(1):205–224, 2017.
[Bou17b] Jean Bourgain. Decoupling inequalities and some mean-value theorems. J. Anal. Math., 133:313–334, 2017.
[Cor77] Antonio Cordoba. The Kakeya maximal function and the spherical summation multipliers. American Journal of Mathematics, 99(1):1–22, 1977.
[Dem20] Ciprian Demeter. Fourier restriction, decoupling, and applications, volume 184 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020.
[DGW20] Ciprian Demeter, Larry Guth, and Hong Wang. Small cap decouplings. Geom. Funct. Anal., 30(4):989–1062, 2020. With an appendix by D. R. Heath-Brown.
[GMW20] Larry Guth, Dominique Maldague, and Hong Wang. Improved decoupling for the parabola. arXiv preprint arXiv:2009.07953, 2020.

	$\displaystyle\alpha^{p}\|U_{\alpha}\|$	$\displaystyle\leq C_{\varepsilon}R^{\varepsilon}\alpha^{p-4}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle\sim C_{\varepsilon}R^{\varepsilon}\frac{\alpha^{p-4}}{\lambda(1)^{\frac{p}{q}-1}}\underset{s}{\max}\lambda(s^{-1}R^{-1})\lambda(s)(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}$
		$\displaystyle\lesssim C_{\varepsilon}R^{\varepsilon}\frac{\lambda(1)^{p-4}}{\lambda(1)^{\frac{p}{q}-1}}\underset{s}{\max}(R^{\beta}s^{-1}R^{-1})(R^{\beta}s)(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}$
		$\displaystyle\lesssim C_{\varepsilon}R^{\varepsilon}\lambda(1)^{p(1-\frac{1}{q})-3}R^{2\beta-1}(\sum_{\gamma}\\|f_{\gamma}\\|_{p}^{q})^{\frac{p}{q}}.$

	$\displaystyle\alpha^{p}\|U_{\alpha}\|$	$\displaystyle\leq\lambda(1)^{p-6}\alpha^{6}\|U_{\alpha}\|\leq\lambda(1)^{p-6}C_{\varepsilon}R^{\varepsilon}R^{6\beta(1-\frac{1}{q})-(1+\beta)}\lambda(1)^{\frac{6}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle=\Big{(}\frac{\lambda(1)}{R^{\beta}}\Big{)}^{(p-6)(1-\frac{1}{q})}C_{\varepsilon}R^{\varepsilon}R^{p\beta(1-\frac{1}{q})-(1+\beta)}\lambda(1)^{\frac{p}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}$
		$\displaystyle\leq C_{\varepsilon}R^{\varepsilon}R^{p\beta(1-\frac{1}{q})-(1+\beta)}\lambda(1)^{\frac{p}{q}-1}\sum_{\gamma}\\|f_{\gamma}\\|_{2}^{2}.$

	$\displaystyle\|f_{\tau_{m}}^{m}(x)-f_{\tau_{m}}^{m+1}(x)\|$	$\displaystyle=\|\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}\psi_{T_{\tau_{m}}}(x)f_{\tau_{m}}^{m+1}(x)\|=\sum_{T_{\tau_{m}}\in T_{\tau_{m}}^{b}}\|\psi_{T_{\tau_{m}}}^{1/2}(x)f_{\tau_{m}}^{m+1}(x)\|\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}R^{-M\delta}\frac{\alpha}{\lambda(1)}\lambda^{-1}\\|\psi_{T_{\tau_{m}}}f_{{\tau_{m}}}^{m+1}\\|_{L^{\infty}(\mathbb{R}^{2})}\\|\psi_{T_{\tau_{m}}}^{1/2}f_{{\tau_{m}}}^{m+1}\\|_{L^{\infty}(\mathbb{R}^{2})}\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}\\|\psi_{T_{\tau_{m}}}^{1/2}f_{{\tau_{m}}}^{m+1}\\|_{L^{\infty}(\mathbb{R}^{2})}^{2}\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{T_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}^{b}}\sum_{\tilde{T}_{{\tau_{m}}}}\\|\psi_{T_{\tau_{m}}}\|f_{{\tau_{m}}}^{m+1}\|^{2}\\|_{L^{\infty}(\tilde{T}_{{\tau_{m}}})}\psi_{T_{\tau_{m}}}^{1/2}(x)$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\sum_{T_{\tau_{m}},\tilde{T}_{\tau_{m}}\in\mathbb{T}_{\tau_{m}}}\\|\psi_{T_{\tau_{m}}}\\|_{L^{\infty}(\tilde{T}_{\tau_{m}})}\\|\|f_{{\tau_{m}}}^{m+1}\|^{2}\\|_{{L}^{\infty}(\tilde{T}_{{\tau_{m}}})}\psi_{T_{\tau_{m}}}^{1/2}(x).$

	$\displaystyle\|f_{\tau_{m}}^{m}(x)-f_{\tau_{m}}^{m+1}(x)\|$	$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}R_{m}^{3/2}\sum_{\tilde{T}_{\tau_{m}}\in T_{\tau_{m}}}w_{\tau_{m}}(x-c_{\tilde{T}_{\tau_{m}}})\\|\|f_{{\tau_{m}}}^{m+1}\|^{2}\\|_{{L}^{\infty}(\tilde{T}_{{\tau_{m}}})}$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}R_{m}^{3/2}\sum_{\tilde{T}_{\tau_{m}}\in T_{\tau_{m}}}w_{\tau_{m}}(x-c_{\tilde{T}_{\tau_{m}}})\|f_{{\tau_{m}}}^{m+1}\|^{2}*w_{\tau_{m}}(c_{\tilde{T}_{\tau_{m}}})$
		$\displaystyle\lesssim R^{-M\delta}\frac{\alpha}{\lambda(1)}\|f_{{\tau_{m}}}^{m+1}\|^{2}*w_{\tau_{m}}(x).$

	$\displaystyle\int_{\mathbb{R}^{2}}\|g_{k}^{h}\|^{2}$	$\displaystyle=\int_{\mathbb{R}^{2}}\|g_{k}-g_{k}^{\ell}\|^{2}$
		$\displaystyle=\int_{\mathbb{R}^{2}}\|\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}-\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}\widehat{\overline{f^{k+1}_{\tau_{k}}}})\widehat{w}_{\tau_{k}}\eta_{k}\|^{2}$
		$\displaystyle\leq\int_{\|\xi\|>cR_{k+1}^{-1/2}}\|\sum_{\tau_{k}}(\widehat{f_{\tau_{k}}^{k+1}}*\widehat{\overline{f_{\tau_{k}}^{k+1}}})\widehat{w}_{\tau_{k}}\|^{2}$

Sharp superlevel set estimates for small cap decouplings of the parabola

Abstract.

Theorem 1.

Conjecture 2.

Corollary 1.

1. Main results

Theorem 3.

Theorem 4.

Corollary 5 ((lq,Lp)(l^{q},L^{p}) small cap decoupling).

2. A sharp example

Lemma 6.

Proof.

Lemma 7.

Proof.

Proposition 8.

Proof.

3. Implications of Theorem 4

Proof of Theorem 3 from Theorem 4.

Proof of Corollary 5 from Theorem 4.

4. Tools to prove Theorem 4

4.1. A pruning step

Definition 1 (Pruning with respect to τk\tau_{k}).

Lemma 9 (Properties of fkf^{k}).

Proof.

Definition 2.

Lemma 10 (Locally constant property).

Proof of Lemma 10.

Definition 3 (Auxiliary functions).

Definition 4.

Definition 5.

4.2. High/low frequency lemmas

Lemma 11 (Low lemma).

Proof.

Lemma 12 (Pruning lemma).

Proof.

Definition 6.

Lemma 13 (High lemma I).

Proof.

Lemma 14 (High lemma II).

Proof.

4.3. Bilinear restriction

Theorem 15.

Proof.

5. Proof of Theorem 4

5.1. The broad version of Theorem 4

Proposition 1.

Proof of Proposition 1.

5.2. Bilinear reduction

Proposition 1 implies Theorem 4.

References

Corollary 5 ( $(l^{q},L^{p})$ small cap decoupling).

Definition 1 (Pruning with respect to $\tau_{k}$ ).

Lemma 9 (Properties of $f^{k}$ ).