Fourier Restriction to a Hyperbolic Cone

TL;DR

This paper proves the Fourier restriction conjecture for a hyperbolic cone in four-dimensional space by applying bilinear restriction theorems and bilinear-to-linear techniques.

Contribution

It establishes the conjectured restriction estimates for a hyperbolic cone in -dimensional space, advancing the understanding of Fourier analysis on such hypersurfaces.

Findings

Confirmed the Fourier restriction conjecture for hyperbolic cones in D.

Applied bilinear restriction theorems to hyperbolic cross sections.

Extended bilinear-to-linear methods to this geometric setting.

Abstract

Using a bilinear restriction theorem of Lee and a bilinear-to-linear argument of Stovall, we obtain the conjectured range of Fourier restriction estimates for a conical hypersurface in $\mathbb{R}^4$ with hyperbolic cross sections.

Equations

$$\displaystyle\Gamma:=\bigg{\{}\bigg{(}\zeta,\sigma,\frac{\zeta_{1}\zeta_{2}}{\sigma}\bigg{)}:\zeta\in[-1,1]^{2},~{}\sigma\in[1,2]\bigg{\}}$$

$$\mathcal{E}f(x,x^{\prime },t):={\iint }_{[-1,1{]}^2\times [1,2]}{e}^{i(x,x^{\prime },t)\cdot (\zeta ,\sigma ,\frac{{\zeta }_1{\zeta }_2}{\sigma })}f(\zeta ,\sigma )d\zeta d\sigma$$

$$\Vert \mathcal{E}f\mathcal{E}g{\Vert }_q\lesssim \Vert f{\Vert }_2\Vert g{\Vert }_2$$

$$\Vert \mathcal{E}{\chi }_{\Omega }{\Vert }_{2q}\lesssim |\Omega {|}^{\frac{1}{q^{\prime }}}$$

$$\Omega (K):=\{(\zeta ,\sigma )\in \Omega :{\mathcal{H}}^1({\pi }_{1,3}^{-1}({\zeta }_1,\sigma )\cap \Omega )\sim 2^{-K}\}.$$

$$([-1,1{]}^2\times [1,2]{)}^2=\bigcup _{j,k}\bigcup _{\begin{array}{c}\tau ,\kappa \in {\mathcal{T}}_{j,k}\\ \tau \sim \kappa \end{array}}\tilde{\tau }\times \tilde{\kappa }.$$

$$\displaystyle\|\mathcal{E}\chi_{\Omega^{\prime}}\|_{2q}^{2}\lesssim\sum_{j,k}\bigg{(}\sum_{\begin{subarray}{c}\tau,\kappa\in\mathcal{T}_{j,k}\\ \tau\sim\kappa\end{subarray}}\|\mathcal{E}\chi_{\Omega^{\prime}\cap\tilde{\tau}}\mathcal{E}\chi_{\Omega^{\prime}\cap\tilde{\kappa}}\|_{q}^{q}\bigg{)}^{\frac{1}{q}}.$$

$$\Vert \mathcal{E}{\chi }_{{\Omega }^{\prime }\cap \tilde{\tau }}\mathcal{E}{\chi }_{{\Omega }^{\prime }\cap \tilde{\kappa }}{\Vert }_q\lesssim 2^{-(j+k)(1-\frac{2}{q})}|{\Omega }^{\prime }\cap \tilde{\tau }{|}^{\frac{1}{2}}|{\Omega }^{\prime }\cap \tilde{\kappa }{|}^{\frac{1}{2}}\le 2^{-(j+k)(1-\frac{2}{q})}|\Omega \cap \tilde{\tau }{|}^{\frac{1}{2}}|\Omega \cap \tilde{\kappa }{|}^{\frac{1}{2}}$$

$$\Vert \mathcal{E}{\chi }_{{\Omega }^{\prime }}{\Vert }_{2q}^2$$

$$\lesssim \sum _{j,k}{2}^{-(j+k)(1-\frac{2}{q})}\underset{\tau \in {\mathcal{T}}_{j,k}}{\max }|\Omega \cap 10\tilde{\tau }{|}^{1-\frac{1}{q}}|\Omega {|}^{\frac{1}{q}}.$$

$$\underset{\tau \in {\mathcal{T}}_{j,k}}{\max }|\Omega \cap 10\tilde{\tau }|\lesssim \min \{2^{-J},2^{-j}\}\min \{2^{-K},2^{-k}\}.$$

$$\Omega =\bigcup _{0<\delta \lesssim {\epsilon }^{1/5}}{\Omega }_{\delta },$$

$${\Omega }_{\alpha }^1$$

$${\alpha }^A{2}^{-J_{\alpha }}\lesssim 2^{-J}.$$

$$\Vert \mathcal{E}{\chi }_{{\Omega }^{\prime }}{\Vert }_{2q}^2\lesssim \sum _{j,k}{2}^{-(j+k)(1-\frac{2}{q})}\underset{\tau \in {\mathcal{T}}_{j,k}}{\max }|{\Omega }_{\alpha }^1\cap 10\tilde{\tau }{|}^{1-\frac{1}{q}}|\Omega {|}^{\frac{1}{q}}.$$

$$|{\Omega }_{\alpha }^1\cap 10\tilde{\tau }|$$

$$\lesssim {\alpha }^A\min \{2^{-J_{\alpha }},2^{-j}\}\min \{2^{-K},2^{-k}\}$$

$$\sum _{\begin{array}{c}j\le J_{\alpha }\\ k\le K\end{array}}{2}^{-(j+k)(1-\frac{2}{q})}({\alpha }^A{2}^{-J_{\alpha }-K}{)}^{1-\frac{1}{q}}|\Omega {|}^{\frac{1}{q}}$$

$$\lesssim {\alpha }^{A(\frac{2}{q}-1)}{2}^{-(J+K)(2-\frac{3}{q})}|\Omega {|}^{\frac{1}{q}}$$

$$\sim {\alpha }^{A(\frac{2}{q}-1)}|\Omega {|}^{\frac{2}{q^{\prime }}}.$$

$$\sum _{\begin{array}{c}j\le J_{\alpha }\\ k>K\end{array}}{2}^{-j(1-\frac{2}{q})}{2}^{-k(2-\frac{3}{q})}{\alpha }^{A(1-\frac{1}{q})}{2}^{-J_{\alpha }(1-\frac{1}{q})}|\Omega {|}^{\frac{1}{q}}$$

$$\lesssim {\alpha }^{A(\frac{2}{q}-1)}|\Omega {|}^{\frac{2}{q^{\prime }}}.$$

$$\Vert \mathcal{E}{\chi }_{{\Omega }^{\prime }}{\Vert }_{2q}\lesssim {\alpha }^{A(\frac{1}{q}-\frac{1}{2})}|\Omega {|}^{\frac{1}{q^{\prime }}}.$$

$$\frac{|I\cap S_{\alpha }|}{|I|}\ge {\eta }^B,$$

$$T_{\alpha ,\eta }:=\{{\zeta }_1\in S_{\alpha }:|I_{\alpha ,\eta }({\zeta }_1)|\ge {\eta }^B{2}^{-J_{\alpha }}\}.$$

$${\Omega }_{\alpha ,\eta }^2:={\Omega }_{\alpha }^1\cap {\pi }_{1,3}^{-1}({\pi }_1^{-1}(S_{\alpha ,\eta })).$$

$${\Omega }_{\epsilon ,\eta }^2:=\bigcup _{\epsilon \le \alpha \le 1}{\tilde{\Omega }}_{\alpha ,\eta }^2,$$

$$\Omega =\bigcup _{0<\alpha \le 1}{\Omega }_{\alpha }^1=\bigcup _{0<\alpha \le \epsilon }\bigcup _{0<\eta \le \alpha }{\Omega }_{\alpha ,\eta }^2.$$

$$\frac{|I\cap S_{\alpha }|}{|S_{\alpha }|}=\frac{|I\cap S_{\alpha }|}{|I|}\cdot \frac{|I|}{|S_{\alpha }|}\gtrsim {\eta }^{2B}.$$

$$|I|\lesssim {\eta }^{-B}{2}^{-J_{\alpha }}\lesssim {\eta }^{-B}{\alpha }^{-A}{2}^{-J}\le {\eta }^{-B-A}{2}^{-J}$$

$$\Vert \mathcal{E}{\chi }_{{\Omega }^{\prime }}{\Vert }_{2q}^2\lesssim \sum _{j,k}{2}^{-(j+k)(1-\frac{2}{q})}\underset{\tau \in {\mathcal{T}}_{j,k}}{\max }|{\Omega }^{\prime }\cap 10\tilde{\tau }{|}^{1-\frac{1}{q}}|\Omega {|}^{\frac{1}{q}}.$$

$$|{\Omega }^{\prime }\cap 10\tilde{\tau }|$$

$$\lesssim {\alpha }^A\min \{|{\pi }_1({\pi }_{1,3}({\Omega }^{\prime }))|,|{\pi }_1({\pi }_{1,3}(10\tilde{\tau }))|\}\min \{2^{-K},2^{-k}\}$$

Full text

1. Introduction

In this article, we resolve the Fourier restriction problem for the conical hypersurface

[TABLE]

in $\mathbb{R}^{4}$. In this case, the problem asks, for which exponents $p,q$ is the extension (adjoint restriction) operator

[TABLE]

of strong type $(p,2q)$? The restriction problem for the light cone in $\mathbb{R}^{4}$ was solved by Wolff [7], while for other conical hypersurfaces, such as those with negatively curved cross sections, it has remained open. In the case of $\Gamma$, nearly optimal results are known: Greenleaf [1] proved that $\mathcal{E}$ is of strong type $(p,2q)$ for $p\geq q^{\prime}$ and $q\geq 2$, and Lee [2] extended that range to $q>3/2$ and $p>q^{\prime}$. The main result of this article is the boundedness of $\mathcal{E}$ on the scaling line $p=q^{\prime}$ for $3/2<q<2$, solving the remaining part of the restriction problem for $\Gamma$.

Theorem 1.1**.**

The operator $\mathcal{E}$ is of strong type $(q^{\prime},2q)$ for $3/2<q<2$.

Because $\Gamma$ is (a compact piece of) a cone whose cross sections are hyperbolic paraboloids, the slicing argument in [3] shows that a strong type $(p,2q)$ restriction estimate for the hyperbolic paraboloid in $\mathbb{R}^{3}$ implies the corresponding result for $\Gamma$. Therefore, by [4] (and the references therein), the estimate in Theorem 1.1 is known for $q>13/8$ and holds conditionally for smaller $q$, pending further progress on restriction to the hyperbolic paraboloid. The superior bilinear restriction theory for $\Gamma$, in relation to that of the hyperbolic paraboloid, allows us to prove Theorem 1.1 unconditionally.

Terminology and notation. A positive constant is admissible if it depends only on $q$. We write $A\lesssim B$ or $A=O(B)$ to mean $A\leq CB$ for some admissible constant $C$, which is allowed to change from line to line. We denote the one-dimensional Hausdorff measure by $\mathcal{H}^{1}$. We write $\log$ for the base $2$ logarithm. An interval of the form $[n2^{-j},(n+1)2^{-j})$ for some $j,n\in\mathbb{Z}$ is dyadic, and $\mathcal{I}_{j}$ denotes the set of dyadic intervals of length $2^{-j}$. The product of two dyadic intervals is a tile, and $\mathcal{T}_{j,k}$ denotes the set of $2^{-j}\times 2^{-k}$ tiles. Given $\tau\in\mathcal{T}_{j,k}$, we set $\tilde{\tau}:=\tau\times[1,2]$. We denote by $\pi_{i,3}$ and $\pi_{1}$, respectively, the projections $(\zeta,\sigma)\mapsto(\zeta_{i},\sigma)$ and $(\zeta_{i},\sigma)\mapsto\zeta_{i}$, for $i=1,2$ and $(\zeta,\sigma)\in\mathbb{R}^{2}\times\mathbb{R}$. If $\pi$ is one of these projections and $S$ a subset of the domain of $\pi$, the $\pi$-projection of $S$ refers to the set $\pi(S)$, and a $\pi$-fiber of $S$ is any set of the form $\pi^{-1}(\pi(s))\cap S$ with $s\in S$. Horizontal and vertical refer to the directions in $\mathbb{R}^{2}$ parallel to the standard basis vectors ${\mathbf{e}_{1}}$ and ${\mathbf{e}_{2}}$, respectively. Finally, the extension of a set refers to the Fourier extension of the set’s characteristic function.

Outline of the proof. We adapt an argument of Stovall [4] which showed that, for $3/2<q<2$, the extension operator associated to the hyperbolic paraboloid in $\mathbb{R}^{3}$ is of strong type $(q^{\prime},2q)$, provided an appropriate $L^{p_{0}}\times L^{p_{0}}\rightarrow L^{q_{0}}$ bilinear restriction inequality holds for some $q_{0}<q$ and $p_{0}/2<q_{0}<p_{0}^{\prime}$. A bilinear estimate suitable for running Stovall’s argument on the hypersurface $\Gamma$ is already known:

Theorem 1.2** (Lee [2]).**

Let $\tau,\kappa\subseteq[-1,1]^{2}$ be squares with unit separation in both the horizontal and vertical directions. If $q>3/2$, then

[TABLE]

for all bounded measurable functions $f$ and $g$ supported in $\tau\times[1,2]$ and $\kappa\times[1,2]$, respectively.

To prove Theorem 1.1, it suffices to show that $\mathcal{E}$ is of restricted strong type $(q^{\prime},2q)$ for every $3/2<q<2$. Thus, we aim to prove that

[TABLE]

for an arbitrary measurable set $\Omega\subseteq[-1,1]^{2}\times[1,2]$. In Section 2, we use Theorem 1.2 and a bilinear-to-linear argument of Vargas [5] to show that sets having roughly constant $\pi_{1,3}$- (or $\pi_{2,3}$-) fiber length obey (1.1). In Section 3, we solve a related inverse problem: For which sets $\Omega$ of constant fiber length can the inequality in (1.1) be reversed? Oversimplified, our answer is that $\Omega$ must be a box of the form $\tilde{\tau}$; proving (1.1) then becomes a matter of bounding the extension of a union of boxes, which we do in Section 4. Our real answer, however, is quantitative: We show that $\Omega$ is approximately a union of boxes, where the number of boxes in the union and the tightness of the approximation are controlled by the constant $\operatorname{C}(\Omega)$, defined thus:

Definition 1.3**.**

For measurable sets $\Omega_{1}\subseteq\Omega_{2}\subseteq[-1,1]^{2}\times[1,2]$, let $\operatorname{C}(\Omega_{1},\Omega_{2})$ denote the smallest number $\varepsilon$, either dyadic, zero, or infinite, such that $\|\mathcal{E}\chi_{\Omega_{1}^{\prime}}\|_{2q}\leq\varepsilon|\Omega_{2}|^{1/q^{\prime}}$ for every measurable set $\Omega_{1}^{\prime}\subseteq\Omega_{1}$, and let $\operatorname{C}(\Omega_{1}):=\operatorname{C}(\Omega_{1},\Omega_{1})$.

Finally, in Section 5, we start with a generic set $\Omega$, decompose it into sets $\Omega(K)$ of fiber length roughly $2^{-K}$, sorted thence according to the value of $\operatorname{C}(\Omega(K))$, and apply the restriction estimates of Sections 3 and 4 to obtain (1.1).

While much of our argument resembles Stovall’s in [4], we include full details for the convenience of the reader.

Acknowledgments. The author thanks Sanghyuk Lee for introducing him to this problem and Betsy Stovall for her advice. This work was supported by National Science Foundation grants DMS-1653264 and DMS-1147523.

2. Extensions of sets of constant fiber length

In this section, we prove a scaling line restriction estimate for characteristic functions of sets of constant $\pi_{1,3}$-fiber length, arguing à la Vargas [5]. By symmetry, the same estimate then holds for sets of constant $\pi_{2,3}$-fiber length.

Definition 2.1**.**

Given a measurable set $\Omega\subseteq[-1,1]^{2}\times[1,2]$ and an integer $K\geq 0$, let

[TABLE]

Proposition 2.2**.**

Suppose that $\Omega=\Omega(K)$ for some $K$. Then $\operatorname{C}(\Omega)\lesssim 1$.

Proof.

Let $\Omega^{\prime}\subseteq\Omega$ be measurable. Given $\tau,\kappa\in\mathcal{T}_{j,k}$, we write $\tau\sim\kappa$ if $\tau$ and $\kappa$ are separated by a distance of $\sim 2^{-j}$ in the horizontal direction and $\sim 2^{-k}$ in the vertical direction. Up to a set of measure zero, we have

[TABLE]

Consequently, by the triangle inequality and Lemma 6.1 in [6] (using that $q<2$),

[TABLE]

By rescaling, Theorem 1.2 implies that

[TABLE]

for $\tau,\kappa\in\mathcal{T}_{j,k}$ with $\tau\sim\kappa$. Given $\tau\in\mathcal{T}_{j,k}$, there are admissibly many $\kappa$ such that $\tau\sim\kappa$, and for each such $\kappa$, we have (say) $10\tau\supseteq\kappa$. Thus,

[TABLE]

Let $J$ be an integer such that $|\pi_{1,3}(\Omega)|\sim 2^{-J}$. Then, by Fubini’s theorem, $|\Omega|\sim 2^{-J-K}$ and

[TABLE]

We split the right-hand side of (2) into four parts: summation over $j,k$ satisfying (i) $j\leq J$, $k\leq K$; (ii) $j\leq J$, $k>K$; (iii) $j>J$, $k\leq K$; (iv) $j>J$, $k>K$. Each part is estimated simply by applying (2.2) and summing a geometric series. We obtain the desired bound in this way. ∎

3. An inverse problem related to Proposition 2.2

In this section, we answer quantitatively the following question: If $\Omega$ extremizes the inequality in Proposition 2.2, what structure must $\Omega$ have?

Proposition 3.1**.**

Suppose that $\Omega=\Omega(K)$ for some $K$, let $J$ be an integer such that $|\Omega|\sim 2^{-J-K}$, and let $\varepsilon:=\operatorname{C}(\Omega)$. Up to a set of measure zero, there exists a decomposition

[TABLE]

where the union is taken over dyadic numbers, such that

1. (i)

$\operatorname{C}(\Omega_{\delta},\Omega)\lesssim\delta^{1/3}$, and 2. (ii)

$\Omega_{\delta}\subseteq\bigcup_{\tau\in\mathcal{T}_{\delta}}\tilde{\tau}$, where $\mathcal{T}_{\delta}\subseteq\mathcal{T}_{J,K}$ with $\#\mathcal{T}_{\delta}\lesssim\delta^{-C_{0}}$ for some admissible constant $C_{0}$.

Proof of Proposition 3.1.

The construction of the sets $\Omega_{\delta}$ consists of five steps. We will begin by dividing $\Omega$ into sets $\Omega_{\alpha}^{1}$ whose $\pi_{1,3}$-projections have constant $\pi_{1}$-fiber length $\alpha$, respectively. That simple step enables us to adapt then the decomposition scheme employed in [4]. We divide each $\Omega_{\alpha}^{1}$ into sets $\Omega_{\alpha,\eta}^{2}$ whose respective projections to the $\zeta_{1}$-axis are contained in $\eta^{-1}$ intervals in $\mathcal{I}_{J}$. In our third step, we divide each $\Omega_{\alpha,\eta}^{2}$ into sets $\Omega_{\alpha,\eta,\rho}^{3}$ of constant $\pi_{2,3}$-fiber length $\rho\eta^{-1}2^{-J}$. To each $\Omega_{\alpha,\eta,\rho}^{3}$ we may then apply variants of the first two steps wherein the roles of the coordinates $\zeta_{1},\zeta_{2}$ are reversed. Indeed, were $\pi_{1,3}$ replaced by $\pi_{2,3}$ in Definition 2.1, each $\Omega_{\alpha,\eta,\rho}^{3}$ would be of the form $\Omega_{\alpha,\eta,\rho}^{3}(J+\log(\rho^{-1}\eta))$. In the end, we obtain sets $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$ whose respective projections to the $\zeta_{2}$-axis are contained in $\delta^{-1}$ intervals in $\mathcal{I}_{K}$. For fixed $\delta$, we define $\Omega_{\delta}$ to be the union of the sets $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$, of which there will be at most $(\log\delta^{-1})^{4}$ by construction. Appearing in the argument below, there are of course constants and minor technical adjustments missing from this summary.

Step 1. For each dyadic number $0<\alpha\leq 1$, define

[TABLE]

where $A$ is an admissible constant to be chosen momentarily.

Lemma 3.2**.**

For every $0<\alpha\leq 1$, we have $\operatorname{C}(\Omega_{\alpha}^{1},\Omega)\lesssim\alpha$.

Proof of Lemma 3.2.

Let $\Omega^{\prime}\subseteq\Omega_{\alpha}^{1}$ be measurable, and let $J_{\alpha}$ be an integer such that $|\pi_{1,3}(\Omega_{\alpha}^{1})|\sim\alpha^{A}2^{-J_{\alpha}}$. We record the bound

[TABLE]

Following the proof of Proposition 2.2, we have

[TABLE]

By Fubini’s theorem,

[TABLE]

for every $\tau\in\mathcal{T}_{j,k}$. As in the proof of Proposition 2.2, we split the right-hand side of (3.2) into four parts: summation over $j,k$ satisfying (i) $j\leq J_{\alpha}$, $k\leq K$; (ii) $j\leq J_{\alpha}$, $k>K$; (iii) $j>J_{\alpha}$, $k\leq K$; (iv) $j>J_{\alpha}$, $k>K$. Using (3) and (3.1), we bound the sum corresponding to (i) by

[TABLE]

Using the same steps, the sum corresponding to (ii) is at most

[TABLE]

The sums corresponding to (iii) and (iv) can be handled in essentially the same way, leading to the estimate

[TABLE]

We conclude the proof by setting $A:=(\frac{1}{q}-\frac{1}{2})^{-1}$. ∎

Step 2. For each $0<\alpha\leq 1$, let $S_{\alpha}:=\pi_{1}(\pi_{1,3}(\Omega_{\alpha}^{1}))$, and note that $|S_{\alpha}|\sim 2^{-J_{\alpha}}$ with $J_{\alpha}$ as in the proof of Lemma 3.2. Given a dyadic number $0<\eta\leq\alpha$ and a Lebesgue point $\zeta_{1}$ of $S_{\alpha}$, let $I_{\alpha,\eta}(\zeta_{1})$ be the maximal dyadic interval $I$ such that $\zeta_{1}\in I$ and

[TABLE]

where $B$ is an admissible constant to be chosen later; such an interval exists by the Lebesgue differentiation theorem. Without loss of generality, we assume that $S_{\alpha}$ is equal to its set of Lebesgue points. Let

[TABLE]

If $\alpha<\varepsilon$, define $S_{\alpha,\alpha}:=T_{\alpha,\alpha}$ and $S_{\alpha,\eta}:=T_{\alpha,\eta}\setminus T_{\alpha,2\eta}$ for $\eta<\alpha$, and let

[TABLE]

For $\varepsilon\leq\alpha\leq 1$, define $S_{\alpha,\varepsilon}:=T_{\alpha,\varepsilon}$ and $S_{\alpha,\eta}:=T_{\alpha,\eta}\setminus T_{\alpha,2\eta}$ for $\eta<\varepsilon$. For $\eta\leq\varepsilon$, let

[TABLE]

where $\tilde{\Omega}_{\alpha,\eta}^{2}:=\Omega_{\alpha}^{1}\cap\pi_{1,3}^{-1}(\pi_{1}^{-1}(S_{\alpha,\eta}))$.

Remark 3.3**.**

We note that $\Omega_{\alpha,\eta}^{2}\subseteq\Omega_{\alpha}^{1}$ for $\alpha<\varepsilon$ and $\tilde{\Omega}_{\alpha,\eta}^{2}\subseteq\Omega_{\alpha}^{1}$ for $\varepsilon\leq\alpha\leq 1$, while in general $\Omega_{\varepsilon,\eta}^{2}$ is not contained in $\Omega_{\varepsilon}^{1}$. We do have

[TABLE]

Lemma 3.4**.**

For every $0<\eta\leq\alpha\leq\varepsilon$, the set $\Omega_{\alpha,\eta}^{2}$ is contained in a union of $O(\eta^{-3B-A-1})$ boxes of the form $\tilde{\tau}$, with $\tau\in\mathcal{T}_{J,0}$, and satisfies $\operatorname{C}(\Omega_{\alpha,\eta}^{2},\Omega)\lesssim\eta^{1/2}$.

Proof of Lemma 3.4.

We argue first under the hypothesis that $\alpha<\varepsilon$, then indicate the changes needed when $\alpha=\varepsilon$. By its definition, $S_{\alpha,\eta}$ is covered by dyadic intervals $I$ of length $|I|\gtrsim\eta^{B}|S_{\alpha}|$, in each of which $S_{\alpha}$ has density obeying (3.4). The density of each such $I$ in $S_{\alpha}$ is

[TABLE]

Therefore, if $\mathcal{C}$ is a minimal-cardinality covering of $S_{\alpha,\eta}$ by these $I$ (consisting necessarily of pairwise disjoint intervals), then $\#\mathcal{C}\lesssim\eta^{-2B}$. Moreover, (3.4) and (3.1) imply that

[TABLE]

for every $I\in\mathcal{C}$. Thus, $S_{\alpha,\eta}$ is covered by $O(\eta^{-3B-A})$ intervals in $\mathcal{I}_{J}$. Since $\alpha<\varepsilon$, it immediately follows that $\Omega_{\alpha,\eta}^{2}$ is contained in a union of $O(\eta^{-3B-A})$ boxes of the form claimed.

We turn to the restriction estimate. If $\eta=\alpha$, the result follows from Lemma 3.2 and Remark 3.3. Thus, we may assume that $\eta<\alpha$. We proceed by optimizing the proof of Proposition 2.2, as in [4]. Let $\Omega^{\prime}\subseteq\Omega_{\alpha,\eta}^{2}$ be measurable. From the proof of (2), we see that

[TABLE]

Fix $\tau\in\mathcal{T}_{j,k}$. By Fubini’s theorem and the definition of $\Omega_{\alpha}^{1}$ (with $\alpha<\varepsilon$), we have

[TABLE]

For certain $j$, the definition of $\Omega_{\alpha,\eta}^{2}$ leads to a better estimate. We claim that if $|j-J_{\alpha}|<\frac{B}{4}\log\eta^{-1}$, then

[TABLE]

Fix such a $j$. Note that $10\tau$ is contained in a union of four tiles $\kappa$ in $\mathcal{T}_{j-4,k-4}$, so it suffices to prove (3.7) with $\kappa$ in place of $10\tau$. Let $\kappa=:I_{j-4}\times I_{k-4}$, where $I_{j-4}\in\mathcal{I}_{j-4}$ and $I_{k-4}\in\mathcal{I}_{k-4}$. We have

[TABLE]

provided $\eta$ is sufficiently small. Suppose that $I_{j-4}\cap S_{\alpha,\eta}\neq\emptyset$. Then there exists $\zeta_{1}\in I_{j-4}$ such that $\zeta_{1}\notin T_{\alpha,2\eta}$, whence

[TABLE]

Consequently, by the maximality of $I_{\alpha,2\eta}(\zeta_{1})$ and the fact that $2^{-j}\leq\eta^{-\frac{B}{4}}2^{-J_{\alpha}}$, we have

[TABLE]

Thus, by Fubini’s theorem,

[TABLE]

as claimed.

Now, to bound (3.5), we split the sum into eight parts determined by the conditions (a) $k\leq K$, (b) $k>K$ and (i) $j\leq J_{\alpha}-\frac{B}{4}\log\eta^{-1}$, (ii) $J_{\alpha}-\frac{B}{4}\log\eta^{-1}<j\leq J_{\alpha}$, (iii) $J_{\alpha}<j<J_{\alpha}+\frac{B}{4}\log\eta^{-1}$, (iv) $J_{\alpha}+\frac{B}{4}\log\eta^{-1}\leq j$. In each case, we use (3.7) if it applies, otherwise (3). Summing geometric series and using (3.1) and the fact that $|\Omega|\sim 2^{-J-K}$, it is straightforward to deduce the bound

[TABLE]

where $B^{\prime}$ is an admissible constant determined by $B$. We may choose $B$ so that $B^{\prime}=1$; this better-than-required exponent will be utilized in the next paragraph.

Suppose now that $\alpha=\varepsilon$. For $\eta<\varepsilon$, the preceding arguments work equally well with $\Omega_{\alpha,\eta}^{2}$ replaced by $\tilde{\Omega}_{\alpha^{\prime},\eta}^{2}$, where $\varepsilon\leq\alpha^{\prime}\leq 1$. In particular, each such $\tilde{\Omega}_{\alpha^{\prime},\eta}^{2}$ is contained in a union of $O(\eta^{-3B-A})$ boxes $\tilde{\tau}$, with $\tau\in\mathcal{T}_{J,0}$, and satisfies $\operatorname{C}(\tilde{\Omega}_{\alpha^{\prime},\eta}^{2},\Omega)\lesssim\eta$. The case $\eta=\varepsilon$ is similar, but with the bound $\operatorname{C}(\tilde{\Omega}_{\alpha^{\prime},\varepsilon}^{2},\Omega)\lesssim\varepsilon$ following directly from the definition of $\varepsilon$. Since the number of sets $\tilde{\Omega}_{\alpha^{\prime},\eta}^{2}$ is $O(\log\varepsilon^{-1})=O(\eta^{-1/2})$ and their union is $\Omega_{\varepsilon,\eta}^{2}$, the lemma holds for $\alpha=\varepsilon$ as well. ∎

Step 3. For dyadic $0<\eta\leq\alpha\leq\varepsilon$ and $0<\rho\lesssim\eta^{1/5}$, define

[TABLE]

where $C$ is an admissible constant to be chosen later. Lemma 3.4 implies that $\mathcal{H}^{1}(\pi_{2,3}^{-1}(\zeta_{2},\sigma)\cap\Omega_{\alpha,\eta}^{2})\lesssim\eta^{-3B-A-1}2^{-J}$ for every $(\zeta,\sigma)\in\Omega_{\alpha,\eta}^{2}$. Thus,

[TABLE]

Lemma 3.5**.**

For every $0<\eta\leq\alpha\leq\varepsilon$ and $0<\rho\lesssim\eta^{1/5}$, we have $\operatorname{C}(\Omega_{\alpha,\eta,\rho}^{3},\Omega)\lesssim\rho$.

Proof of Lemma 3.5.

If $\rho^{5C}\eta^{-3B-A-1-C}\geq\rho^{2C}$, then by Lemma 3.4, we have

[TABLE]

for $C$ chosen sufficiently large. Thus, we may assume that $\rho^{5C}\eta^{-3B-A-1-C}\leq\rho^{2C}$. Given a measurable set $\Omega^{\prime}\subseteq\Omega_{\alpha,\eta,\rho}^{3}$ and $\tau\in\mathcal{T}_{j,k}$, the set $\Omega^{\prime}\cap 10\tilde{\tau}$ has $\pi_{1,3}$- and $\pi_{2,3}$-fibers of length at most $\min\{2^{-K},2^{-k}\}$ and $\min\{\rho^{2C}2^{-J},2^{-j}\}$, respectively, and it has $\pi_{1,3}$- and $\pi_{2,3}$-projections of measure at most $\min\{2^{-J},2^{-j}\}$ and $2^{-k}$, respectively. Therefore, by Fubini’s theorem,

[TABLE]

Following [4], we define

[TABLE]

Each $(j,k)$ belongs to some $R_{i}$, $1\leq i\leq 4$, so by (3.5) and (3.8), we have

[TABLE]

Summing these geometric series leads to the bound $\|\mathcal{E}\chi_{\Omega^{\prime}}\|_{2q}\lesssim\rho^{C^{\prime}}|\Omega|^{1/q^{\prime}}$, where $C^{\prime}$ is an admissible constant determined by $C$; increasing $C$ if necessary, we can make $C^{\prime}\geq 1$. ∎

As indicated above, the final two steps of our construction are variants of the first two, wherein the roles of the coordinates $\zeta_{1},\zeta_{2}$ are reversed. Below, we briefly explain how the argument in Steps 1 and 2 transfers, without rewriting all the details. In short, $\Omega_{\alpha,\eta,\rho}^{3}$ has constant $\pi_{2,3}$-fiber length by construction and thus may replace $\Omega$, and $\rho$ may replace $\varepsilon$ by Lemma 3.5.

Step 4. For each dyadic number $0<\beta\leq 1$, define

[TABLE]

Lemma 3.6**.**

For every $0<\beta\leq 1$, $0<\eta\leq\alpha\leq\varepsilon$, and $0<\rho\lesssim\eta^{1/5}$, we have $\operatorname{C}(\Omega_{\alpha,\eta,\rho,\beta}^{4},\Omega)\lesssim\beta$.

Proof of Lemma 3.6.

Since $\Omega_{\alpha,\eta,\rho}^{3}$ has constant $\pi_{2,3}$-fiber length, we can imitate the proof of Lemma 3.2 to show that $\beta\gtrsim\operatorname{C}(\Omega_{\alpha,\eta,\rho,\beta}^{4},\Omega_{\alpha,\eta,\rho}^{3})\geq\operatorname{C}(\Omega_{\alpha,\eta,\rho,\beta}^{4},\Omega)$. ∎

Step 5. For each $0<\beta\leq 1$, let $S_{\alpha,\eta,\rho,\beta}:=\pi_{1}(\pi_{2,3}(\Omega_{\alpha,\eta,\rho,\beta}^{4}))$, and let $K_{\alpha,\eta,\rho,\beta}$ be an integer such that $|S_{\alpha,\eta,\rho,\beta}|\sim 2^{-K_{\alpha,\eta,\rho,\beta}}$. Given a dyadic number $0<\delta\leq\beta$ and a Lebesgue point $\zeta_{2}$ of $S_{\alpha,\eta,\rho,\beta}$, let $I_{\alpha,\eta,\rho,\beta,\delta}(\zeta_{2})$ be the maximal dyadic interval $I$ such that $\zeta_{2}\in I$ and

[TABLE]

As before, we may assume that $S_{\alpha,\eta,\rho,\beta}$ is equal to its set of Lebesgue points. Let

[TABLE]

If $\beta<\rho$, define $S_{\alpha,\eta,\rho,\beta,\beta}:=T_{\alpha,\eta,\rho,\beta,\beta}$ and $S_{\alpha,\eta,\rho,\beta,\delta}:=T_{\alpha,\eta,\rho,\beta,\delta}\setminus T_{\alpha,\eta,\rho,\beta,2\delta}$ for $\delta<\beta$, and let

[TABLE]

If $\rho\leq\beta\leq 1$, define $S_{\alpha,\eta,\rho,\beta,\rho}:=T_{\alpha,\eta,\rho,\beta,\rho}$ and $S_{\alpha,\eta,\rho,\beta,\delta}:=T_{\alpha,\eta,\rho,\beta,\delta}\setminus T_{\alpha,\eta,\rho,\beta,2\delta}$ for $\delta<\rho$. For $\delta\leq\rho$, let

[TABLE]

where $\tilde{\Omega}_{\alpha,\eta,\rho,\beta,\delta}^{5}:=\Omega_{\alpha,\eta,\rho,\beta}^{4}\cap\pi_{2,3}^{-1}(\pi_{1}^{-1}(S_{\alpha,\eta,\rho,\beta,\delta}))$.

Admittedly, the subscripts have become awkward. However, all we have done is repeated Step 2, replacing $\Omega_{\alpha}^{1}$ and $\varepsilon$ by $\Omega_{\alpha,\eta,\rho,\beta}^{4}$ and $\rho$, respectively, and projecting onto the $\zeta_{2}$-axis instead of the $\zeta_{1}$-axis. We note that

[TABLE]

Lemma 3.7**.**

For every $0<\eta\leq\alpha\leq\varepsilon$ and $0<\delta\leq\beta\leq\rho\lesssim\eta^{1/5}$, the set $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$ is contained in a union of $O(\delta^{-18B-6A-5C-6})$ boxes of the form $\tilde{\tau}$, with $\tau\in\mathcal{T}_{J,K}$, and satisfies $\operatorname{C}(\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5},\Omega)\lesssim\delta^{1/2}$.

Proof of Lemma 3.7.

Let $K_{\alpha,\eta,\rho}$ be an integer such that $|\pi_{2,3}(\Omega_{\alpha,\eta,\rho}^{3})|\sim 2^{-K_{\alpha,\eta,\rho}}$. Imitating the proof of Lemma 3.4, we can show that $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$ is covered by $O(\delta^{-3B-A-1})$ boxes of the form $\tilde{\tau}$, where $\tau\in\mathcal{T}_{0,K_{\alpha,\eta,\rho}}$. Since $\Omega_{\alpha,\eta,\rho}^{3}$ has $\pi_{2,3}$-fibers of length $\rho^{5C}\eta^{-3B-A-1-C}2^{-J}$ and volume at most $2^{-J-K}$, it follows that $2^{-K_{\alpha,\eta,\rho}}\lesssim\rho^{-5C}2^{-K}$. Thus, $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$ is covered by $O(\rho^{-5C}\delta^{-3B-A-1})=O(\delta^{-3B-A-5C-1})$ boxes $\tilde{\tau}$, with $\tau\in\mathcal{T}_{0,K}$. Since $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}\subseteq\Omega_{\alpha,\eta}^{2}$ and $\eta\gtrsim\delta^{5}$, Lemma 3.4 now implies that $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$ is covered by $O(\delta^{-18B-6A-5C-6})$ boxes $\tilde{\tau}$, with $\tau\in\mathcal{T}_{J,K}$.

To obtain the restriction estimate, we can adapt the proof of Lemma 3.4. ∎

Finally, we are equipped to finish the proof of Proposition 3.1. We have

[TABLE]

where

[TABLE]

Since for fixed $\delta$ there are $O((\log\delta^{-1})^{4})$ sets $\Omega_{\alpha,\eta,\rho,\beta,\delta}^{5}$, properties (i) and (ii) in the proposition follow from Lemma 3.7.

∎

4. Extensions of near unions of boxes

For each $K$, let $J(K)$ be an integer such that $|\Omega(K)|\sim 2^{-J(K)-K}$. For each dyadic number $\varepsilon$, let $\mathcal{K}(\varepsilon)$ denote the collection of all integers $K\geq 0$ for which $\varepsilon=\operatorname{C}(\Omega(K))$. For each $K\in\mathcal{K}(\varepsilon)$, Proposition 3.1 gives a decomposition $\Omega(K)=\bigcup_{0<\delta\lesssim\varepsilon^{1/5}}\Omega(K)_{\delta}$ such that for each $\delta$, we have $\Omega(K)_{\delta}\subseteq\bigcup_{\tau\in\mathcal{T}(K)_{\delta}}\tilde{\tau}$ for some $\mathcal{T}(K)_{\delta}\subseteq\mathcal{T}_{J(K),K}$ with $\#\mathcal{T}(K)_{\delta}\lesssim\delta^{-C_{0}}$.

Lemma 4.1**.**

For every $0<\delta\lesssim\varepsilon^{1/5}$, we have

[TABLE]

Proof of Lemma 4.1.

Let $A$ be an admissible constant to be chosen later, and divide $\mathcal{K}(\varepsilon)$ into $O(\log\delta^{-1})$ subsets $\mathcal{K}$ such that each is $A\log\delta^{-1}$-separated. It suffices to prove that

[TABLE]

for each $\mathcal{K}$. Since $q<2$, we have

[TABLE]

where $D(\mathcal{K}^{4}):=\{\boldsymbol{K}\in\mathcal{K}^{4}:K_{1}=\cdots=K_{4}\}$. To control the latter sum, we have the following lemma.

Lemma 4.2**.**

For all $K,K^{\prime}\in\mathcal{K}$, we have

[TABLE]

for some admissible constant $c_{0}$.

Proof of Lemma 4.2.

By the Cauchy–Schwarz inequality and Proposition 2.2,

[TABLE]

For $J:=J(K)$ and $J^{\prime}:=J(K^{\prime})$, we have

[TABLE]

whenever either (i) $K=K^{\prime}$, (ii) $J=J^{\prime}$, (iii) $J<J^{\prime}$ and $K<K^{\prime}$, or (iv) $J>J^{\prime}$ and $K>K^{\prime}$; in these cases, (4.2) follows immediately.

Thus, by symmetry, it suffices to prove (4.2) for $K<K^{\prime}$ and $J>J^{\prime}$. By the bound $\#(\mathcal{T}(K)_{\delta}\times\mathcal{T}(K^{\prime})_{\delta})\lesssim\delta^{-2C_{0}}$ and the separation condition on $\mathcal{K}$ (with $A$ sufficiently large), it suffices to prove that

[TABLE]

for all $\tau\in\mathcal{T}(K)_{\delta}$, $\kappa\in\mathcal{T}(K^{\prime})_{\delta}$, and some admissible constant $c$.

Fix two such tiles $\tau,\kappa$, and note that $\tau$ must be taller than $\kappa$ and $\kappa$ wider than $\tau$. By translation, we may assume that the $\zeta_{2}$- and $\zeta_{1}$-axes intersect the centers of $\tau$ and $\kappa$, respectively. Define

[TABLE]

so that

[TABLE]

By the two-parameter Littlewood–Paley square function estimate and fact that $q<2$, we have

[TABLE]

where $\tilde{\tau}_{k}:=\tau_{k}\times[1,2]$ and $\tilde{\kappa}_{j}:=\kappa_{j}\times[1,2]$. We first sum the terms with $k=K^{\prime}$. By the Cauchy–Schwarz inequality and Proposition 2.2, we have

[TABLE]

Since $\kappa$ has width $2^{-{J^{\prime}}}$, there are at most two nonempty $\kappa_{j}$ with $j\leq J^{\prime}$. This fact and the bound

[TABLE]

imply that $\sum_{j=0}^{J}|\tilde{\kappa}_{j}|^{\frac{q}{q^{\prime}}}\lesssim|\kappa|^{\frac{q}{q^{\prime}}}$. Since $|\tilde{\tau}_{K^{\prime}}|\lesssim 2^{-(K^{\prime}-K)}|\tilde{\tau}|$, $|\tilde{\tau}|\sim|\Omega(K)|$, and $|\tilde{\kappa}|\sim|\Omega(K^{\prime})|$, we altogether have

[TABLE]

which is acceptable. A similar argument shows that

[TABLE]

We now consider the terms with $k<K^{\prime}$ and $j<J$. In this case, $\tau_{k}$ is a subset of four tiles in $\mathcal{T}_{J,\max\{K,k\}}$ and $\kappa_{j}$ is a subset of four tiles in $\mathcal{T}_{\max\{J^{\prime},j\},K^{\prime}}$. Moreover, these tiles are separated by a distance of $2^{-k}$ and $2^{-j}$ in the vertical and horizontal directions, respectively. Thus, by Theorem 1.2 (rescaled, as in the proof of Proposition 2.2),

[TABLE]

Using (4.5) and the analogous bound for $|\tilde{\tau}_{k}|$, we now get

[TABLE]

By the relations $K<K^{\prime}$, $J>J^{\prime}$ and (4), we have now proved (4.3). ∎

Returning to the proof of Lemma 4.1, we consider the second sum in (4). Given $\boldsymbol{K}\in\mathcal{K}^{4}\setminus D(\mathcal{K}^{4})$, let $p(\boldsymbol{K})=(p_{i}(\boldsymbol{K}))_{i=1}^{4}$ be a permutation of $\boldsymbol{K}$ such that $|\Omega(p_{1}(\boldsymbol{K}))|$ is maximal among $|\Omega(K_{i})|$, $1\leq i\leq 4$, and such that $|K_{i}-K_{j}|\leq 2|p_{1}(\boldsymbol{K})-p_{2}(\boldsymbol{K})|$ for all $1\leq i,j\leq 4$. Then by the Cauchy–Schwarz inequality, Lemma 4.2, the separation condition on $\mathcal{K}$, the fact that $q^{\prime}<2q$, and choosing $A$ sufficiently large, we get

[TABLE]

∎

5. Proof of Theorem 1.1

In this final section, we prove our main result. We recall our setup: For $\Omega\subseteq[-1,1]^{2}\times[1,2]$ a measurable set, we have divided $\Omega$ into sets $\Omega(K)$ of constant fiber length $2^{-K}$, partitioned the indices $K$ into sets $\mathcal{K}(\varepsilon)$ according to the value of $\varepsilon:=\operatorname{C}(\Omega(K))$, and decomposed each $\Omega(K)$ into near unions of boxes $\Omega(K)_{\delta}$ for $0<\delta\lesssim\varepsilon^{1/5}$. Thus,

[TABLE]

(Actually, there may be $K$ such that $\operatorname{C}(\Omega(K))=0$; however, those terms contribute nothing to the left-hand side below.)

Proof of Theorem 1.1.

By the triangle inequality, Lemma 4.1, Proposition 3.1, and the fact that $q^{\prime}<2q$, we have

[TABLE]

proving (1.1). ∎