An improved restriction estimate in $\mathbb{R}^{3}$

The scale-$R$ wave packet decomposition is

(2.1)

$$f=\sum_{\theta}f_{\theta}=\sum_{\theta}(\psi_{\theta}f)=\sum_{\theta,v}(\eta_{v})^{\vee}\ast(\psi_{\theta}f),$$

where $\{\theta\}$ is a collection of finitely overlapping $R^{-1/2}$-balls in $B^{2}(0,1)$ and $\psi_{\theta}$ is a bump function associated to $\theta$; $v\in R^{\frac{1+\delta}{2}}\mathbb{Z}^{2}$ and $\{\eta_{v}\}$ is a smooth partition of unity of $\mathbb{R}^{2}$ with compact Fourier support. Such wave packet decomposition can be found in [Wan22] Section 2 (see also [Gut16] Section 2).

Here is a key feature of wave packets: Given a function $f$ in $\mathbb{R}^{n}$, if $\widehat{f}$ is supported in on a unit ball, then $|f|{\bf{1}}_{B}$ is essentially constant (in the sense of averaging) for any unit ball $B\subset\mathbb{R}^{n}$. That is, one would expect $\|f{\bf{1}}_{B}\|_{p}\approx\|f{\bf{1}}_{B}\|_{\infty}$ for any $1\leq p\leq\infty$ (see for instance, [Wan22] Lemma 2.6, 2.7). This leads to the following observation: $|Ef_{\theta}{\bf{1}}_{T}|$ is essentially constant on every scale-$R$ tube $T$ in the form

(2.2)

$$T=\{x=(x^{\prime},x_{3})\in B_{R}:|x^{\prime}+2x_{3}c_{\theta}+v|\leq R^{1/2+\delta}\},$$

where $c_{\theta}$ is the center of $\theta$, and $v$ is any point in $\mathbb{R}^{2}$ (for example, $v\in R^{\frac{1+\delta}{2}}\mathbb{Z}^{2}$).

What follows is a pruning of scale-$R$ wave packets of the function $f$. Let us introduce $O_{\varepsilon}(1)$ scales:

(2.3)

$$\rho_{1}=R^{1/2+\delta},\rho_{2}=R^{1/2+2\delta},\ldots,\rho_{\ell}=R$$

with $\ell\sim 1/(2\delta)=O_{\varepsilon}(1)$. The pruning of $f$ will be proceeded from the smallest scale $\rho_{1}$ to the biggest scale $\rho_{\ell}$. First let us prune wave packets at the smallest scale $\rho_{1}=R^{1/2+\delta}$. By dyadic pigeonholing, we can find a dyadic number $\lambda_{1}$ so that

1. (1)

   There is a corresponding scale-$R$ directional set $\Theta_{\lambda_{1}}[R]$, and for each $\theta\in\Theta_{\lambda_{1}}[R]$, a scale-$R$ tube set $\mathbb{T}_{\lambda_{1},\theta}[R]$. Denote also $\mathbb{T}_{\lambda_{1}}[R]=\bigcup_{\theta}\mathbb{T}_{\lambda_{1},\theta}[R]$.

2. (2)

   Uniformly for each $\theta\in\Theta_{\lambda_{1}}[R]$,

   (2.4)

   $$|\mathbb{T}_{\lambda_{1},\theta}[R]|\sim\lambda_{1}.$$

3. (3)

   For each $T\in\mathbb{T}_{\lambda_{1}}[R]$, $\|f_{T}\|_{2}$ are about the same up to a constant multiple.

4. (4)

   The $L^{p}$ norm of the sum of wave packets $\sum_{T\in\mathbb{T}_{\lambda_{1}}}Ef_{T}$ dominates the $L^{p}$ norm of $Ef$, in the sense that $\|\sum_{T\in\mathbb{T}_{\lambda_{1}}[R]}Ef_{T}\|_{L^{p}(B_{R})}\gtrapprox\|Ef\|_{L^{p}(B_{R})}$.

Define $f_{\rho_{1}}=\sum_{T\in\mathbb{T}_{\lambda_{1}}[R]}f_{T}$ be the pruned function at the smallest scale $\rho_{1}$. This is the pruning at the first step.

Next, suppose there is a pruned function $f_{\rho_{j-1}}$ at scale $\rho_{j-1}$, we would like to prune it at scale $\rho_{j}$. By dyadic pigeonholing, there are dyadic numbers $\kappa_{1}=\kappa_{1}(j),\kappa_{2}=\kappa_{2}(j)$, and a collection of $\rho_{j}\times\rho_{j}\times R$ fat tubes $\widetilde{\mathcal{T}}[\rho_{j}]$ so that

1. (1)

   Any two $\widetilde{T}_{1},\widetilde{T}_{2}\in\widetilde{\mathcal{T}}[\rho_{j}]$ either are parallel, or make an angle $\gtrsim\rho_{j}/R$.

2. (2)

   Each $\widetilde{T}\in\widetilde{\mathcal{T}}[\rho_{j}]$ contains $\sim\kappa_{1}$ many $\rho_{j-1}\times\rho_{j-1}\times R$ fat tubes in $\widetilde{\mathcal{T}}[\rho_{j-1}]$.

3. (3)

   For each directional cap $\theta^{\prime}$ with $d(\theta^{\prime})=\rho_{j}/R^{1+\delta}$, there are either $\sim\kappa_{2}$ parallel $\rho_{j}\times\rho_{j}\times R$ fat tubes pointing to this direction, or no tubes at all.

4. (4)

   The above two items imply that for any $\widetilde{T}\in\widetilde{\mathcal{T}}[\rho_{j}]$ with direction $\theta^{\prime}$,

   (2.5)

   $$\sum_{T\subset\widetilde{T}}\|f_{T}\|_{2}^{2}\lesssim\kappa_{2}^{-1}\|f_{\theta^{\prime}}\|_{2}^{2}.$$

5. (5)

   $\|\sum_{T\subset\bigcup_{\widetilde{T}\in\widetilde{\mathcal{T}}[\rho_{j}]}}Ef_{T}\|_{L^{p}(B_{R})}\gtrapprox\|Ef\|_{L^{p}(B_{R})}$.

Define $f_{\rho_{j}}=\sum_{T\subset\bigcup_{\widetilde{T}\in\widetilde{\mathcal{T}}[\rho_{j}]}}f_{T}$ be the pruned function at scale $\rho_{j}$. We remark that although the pruning at scale $\rho_{j}$ may destroy some uniform structure at scale $\rho_{j^{\prime}}$ with $j^{\prime}<j$ (for example, the fact that at scale $\rho_{j^{\prime}}$ there are $\sim\kappa_{2}(j^{\prime})$ parallel $\rho_{j^{\prime}}\times\rho_{j^{\prime}}\times R$ fat tubes may no longer be true. Instead, there will be $\lesssim\kappa_{2}(j^{\prime})$ parallel fat tubes), the upper bound estimate (2.5) still remains true for all $j^{\prime}<j$, namely, $\sum_{T\subset\widetilde{T}}\|f_{T}\|_{2}^{2}\lesssim\kappa_{2}(j^{\prime})^{-1}\|f_{\theta^{\prime}}\|_{2}^{2}$ for every $\theta^{\prime}$ with $d(\theta^{\prime})=\rho_{j^{\prime}}/R^{1+\delta}$ and every fat tube $\widetilde{T}\in\widetilde{\mathcal{T}}[\rho_{j^{\prime}}]$ that points to $\theta^{\prime}$.

The pruned function $f_{\rho_{l}}$ at the biggest scale $\rho_{\ell}$ is the one we will carefully study in the rest of the paper. For simplicity, we still denote $f=f_{\rho_{l}}$ by an abuse of notation. Let us emphasize some properties of the new $f$:

1. (1)

   For each scale $\rho_{j}$ in (2.3) the following is true: For each $\rho_{j}/R^{1+\delta}$ directional cap $\theta^{\prime}$, there are $\lesssim\kappa_{2}$ parallel $\rho_{j}\times\rho_{j}\times R$ fat tubes $\widetilde{T}$, each of which contains the same amount (up to a constant multiple) of thin tubes from $\mathbb{T}_{\lambda_{1}}[R]$, where $\mathbb{T}_{\lambda_{1}}[R]$ was defined near (2.4).

2. (2)

   We have for each $\rho_{j}/R^{1+\delta}$ directional cap $\theta^{\prime}$ and $\kappa_{2}=\kappa_{2}(j)$,

   (2.6)

   $$\sum_{T\subset\widetilde{T}}\|f_{T}\|_{2}^{2}\lesssim\kappa_{2}^{-1}\|f_{\theta^{\prime}}\|_{2}^{2}.$$

The broad-narrow reduction was introduced in [BG11]. Specifically, partition the unit ball $B^{2}(0,1)$ into cubes $\{\tau\}$ such that $d(\tau)=K^{-1}$, where $K$ is a large number but is also small compared to $R$, for example, $K=(\log R)^{100\varepsilon^{-100}}$. Here is the formal definition of broadness (2-broad).

Eventually we will choose $A\leq K^{\varepsilon}$. Roughly speaking, the broad-narrow reduction allows us to focus on those points $x\in\mathbb{R}^{3}$ where considerably many $\{Ef_{\tau}(x)\}_{\tau}$ make major contribution to $Ef(x)$. A similar definition is given in [Wan22] Section 2.2. After the broad-narrow reduction (see for instance [Gut16]), we only need to consider the $\textup{BL}^{p}$-norm $\|Ef\|_{\textup{BL}^{p}(B_{R})}^{p}:=\|{\rm{Br}}_{A}Ef\|_{L^{p}(B_{R})}^{p}$.

Let us recall the polynomial partitioning introduced in [Gut16]. One can use Corollary 1.7 in [Gut16] with a given degree $d$ to partition the measure $\mu_{Ef}(B_{R})=\|Ef\|_{\textup{BL}^{p}(B_{R})}^{p}$. The outcomes are

1. (1)

   A polynomial $P$ of degree $\sim d$.

2. (2)

   A collection of disjoint cells $\mathcal{U}=\{U\}$ with $|\mathcal{U}|\sim d^{3}$ such that

   (2.8)

   $$B_{R}\setminus Z(Q)=\bigcup_{U\in\mathcal{U}}U.$$

3. (3)

   $\mu_{Ef}(U)=\|Ef\|_{\textup{BL}^{p}(U)}^{p}$ are about the same up to a constant multiple for all $U\in\mathcal{U}$.

4. (4)

   We can refined the polynomial partitioning a little bit to get an extra information on each $U\in\mathcal{U}$: $U$ is contained in an $Rd^{-1}$-ball in $B_{R}$. This refinement was obtained in [Wan22].

Now we introduce a wall $W$, which is a thin neighborhood of the variety $Z(Q)$:

(2.9)

$$W:=N_{R^{1/2+\delta}}(Z(Q)).$$

Define for each $U\in\mathcal{U}$ a smaller cell $O=U\setminus W$ and let $\mathcal{O}=\{O\}$. One advantage for looking at the smaller cell $O$ is that any tube of dimensions $R^{1/2+\delta}\times R^{1/2+\delta}\times R$ can only intersect at most $O(d)$ cells in $\mathcal{O}$.

The decomposition above leads to a partition

(2.10)

$$B_{R}=W\bigsqcup\Big{(}\bigsqcup_{O\in\mathcal{O}}O\Big{)}$$

and hence an estimate

(2.11)

$$\|Ef\|_{\textup{BL}^{p}(B_{R})}^{p}\lesssim\sum_{O\in\mathcal{O}}\|Ef\|_{\textup{BL}^{p}(O)}^{p}+\|Ef\|_{\textup{BL}^{p}(W)}^{p}.$$

If the first term in (2.11) dominates, then we say “we are in the cellular case”. Let $f_{O}$ be the sum of wave packets that intersect $O$. A crucial fact is

(2.12)

$$Ef_{O}(x)=Ef(x),\hskip 14.22636ptx\in O.$$

This yields, supposing that we are in the cellular case,

(2.13)

$$\|Ef\|_{\textup{BL}^{p}(B_{R})}^{p}\lesssim\sum_{O\in\mathcal{O}}\|Ef_{O}\|_{\textup{BL}^{p}(O)}^{p}.$$

If the second term in (2.11) dominates, then we say “we are in the algebraic case”. In this case we will divide wave packets into a transversal part and a tangential part. Introduce a collection of $R^{1-\delta}$-balls $B$ in $B_{R}$. For each $B$, we define $S$ as the portion of the wall inside $B$:

(2.14)

$$S=W\cap B,$$

and call it an “algebraic fat surface”. Then we define a tangential function $f_{S}$ and a transverse function $f_{S,trans}$ for each $B$. Roughly speaking, the tangential function $f_{S}$ contains all the wave packets $f_{T}$ that the $R$-tube $T$ is $R^{-1/2+2\delta}$-tangent to $Z(Q)$ at each point of $Z(Q)\cap B$, and the transverse function $f_{S,trans}$ contains all the remaining wave packet. Here is the formal definition of tangential tubes:

One can iteratively use the polynomial partitioning above for either cellular case and algebraic case to obtain a “tree” structure.¹¹1We do not break the algebraic case into transverse case and tangent case when performing the iteration. This is different from the iteration (induction) given in [Gut16]. Each node of this tree is either a cellular cell or an algebraic cell. Note that in either cellular or algebraic case, the diameter of each resulting cell is strictly decreasing (decrease by either $1/d$ or $R^{-\delta}$). The iteration will stop when the diameter of each cell is smaller than $R^{\delta}$. A cell that appears the last step of the iteration is called a “leaf”. Here is a summary of this iteration:

The algebraic cases in the iteration are very important to us. Note that for a leaf $O^{\prime}\in\mathcal{O}_{leaf}$, there are $n$ unique algebraic fat surfaces $S_{1},S_{2},\ldots,S_{n}$, so that $O^{\prime}$ is within a containing chain

(2.20)

$$O^{\prime}\subset S_{n}\subset S_{n-1}\subset\cdots\subset S_{1}\subset B_{R}.$$

Besides the outcomes from Lemma 2.4, we are in particular interested in some pointwise estimates similar to (2.12) and (2.17). Note that for any $1\leq j\leq n$ and any ancestor $O_{j}$ of $S_{j}$ that $S_{j}\subset O_{j}\subset S_{j-1}$, by (2.12) one has

(2.21)

$$Ef_{O_{j}}(x)=Ef_{S_{j-1},trans}(x),\hskip 14.22636ptx\in O_{j}.$$

Here we use the convention $S_{0}=B_{R}$ and $f_{S_{0}}=f$.

We know that $f_{O_{j}}$ is essentially a sum of scale-$r_{j}$ ($r_{j}$ is the scale of $O_{j}$) wave packets, which are determined by tubes from, say $\mathbb{T}_{O_{j}}$. The dimensions of the tubes in $\mathbb{T}_{O_{j}}$ are roughly $r_{j}^{1/2}\times r_{j}^{1/2}\times r_{j}$. For any subcollection $\mathbb{T}\subset\mathbb{T}_{O_{j}}$, define $f^{\mathbb{T}}$ as the function concentrated on wave packets from $\mathbb{T}$. Recall (2.17). The pointwise equality (2.21) further yields that

(2.22)

$$Ef_{O_{j}}^{\mathbb{T}}(x)=Ef^{\mathbb{T}}(x)-\Big{(}\sum_{l=1}^{j-1}Ef_{S_{l}}^{\mathbb{T}}(x)\Big{)},\hskip 14.22636ptx\in O_{j}.$$

If all $Ef_{S_{l}}^{\mathbb{T}}(x)$ are small, then similar to (2.12) we also have $Ef_{O_{j}}^{\mathbb{T}}(x)\sim Ef^{\mathbb{T}}(x)$. To find out when $Ef_{S_{l}}^{\mathbb{T}}(x)$ are small, we want to compare $Ef_{S_{l}}^{\mathbb{T}}$ with $Ef_{S_{l}}$. It is hard to get something meaningful for general $\mathbb{T}$, while if $\mathbb{T}$ is either a collection of transverse tubes or a collection of tangent tubes with respect to an algebraic fat surface $\widetilde{S}$, it was showed in [Wan22] Lemma 10.2 that

(2.23)

$$|Ef_{S_{l}}^{\mathbb{T}}(x)|\lesssim|Ef_{S_{l}}(x)|$$

for any $x\in O_{j}\cap\widetilde{S}$.

This crucial fact gives that for any fat surface $S_{t}$ and $B_{K}\subset S_{t}$ (see Definition 2.1 for $K$), if $\|Ef_{S_{l}}\|_{\textup{BL}^{p}(B_{K})}\lessapprox R^{-\Omega(\delta)}\|Ef_{S_{t}}\|_{\textup{BL}^{p}(B_{K})}$ for all $1\leq l<t$, then

(2.24)

$$\|Ef_{S_{t}}\|_{\textup{BL}^{p}(B_{K})}\sim\|Ef^{\mathbb{T}_{S_{t}}}\|_{\textup{BL}^{p}(B_{K})},$$

where $\mathbb{T}_{S_{t}}$ is the collection of tangent tube corresponding to the algebraic fat surface $S_{t}$ (note that $Ef_{S_{t}}=Ef_{S_{t}}^{\mathbb{T}_{S_{t}}}$ and see also Lemma 3.10 in [Wan22]). We remark that (2.23) and (2.24) also hold for $f$ replaced by $g$, where $g$ is a function similar to $f$ that will be given later in (4.6).

We know by definition that tubes from $\mathbb{T}_{S}$ are contained in the fat surface $S$ with some scale $r$. While more is true because we are looking at the broad norm $\textup{BL}^{p}(S)$ on $S$. Roughly speaking, the following lemma allows us to assume that tubes in $\mathbb{T}_{S}$ are contained in the $r^{1/2+O(\delta)}$-neighborhood of $r^{O(\delta)}$ planes.