# Weighted refined decoupling estimates and application to Falconer distance
set problem

Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang

###### Abstract.

We prove some weighted refined decoupling estimates. As an application, we
give an alternative proof of the following result on Falconer’s distance set
problem by the authors in a companion work: if a compact set
$E\subset\mathbb{R}^{d}$ has Hausdorff dimension larger than
$\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 4$, then there is a
point $x\in E$ such that the pinned distance set $\Delta_{x}(E)$ has positive
Lebesgue measure. Aside from this application, the weighted refined decoupling
estimates may be of independent interest.

##  1\. Introduction

In this paper, we prove some weighted refined decoupling estimates (see
Theorems 1.1 and 1.2) and discuss their application to Falconer’s distance set
problem.

###  1.1. Weighted refined decoupling estimates

Here is the setup for refined decoupling estimates.

Suppose that $S\subset\mathbb{R}^{d}$ is a compact and strictly convex $C^{2}$
hypersurface with Gaussian curvature $\sim 1$.

For any $\epsilon>0$, suppose there exists $0<\beta\ll\epsilon$ satisfying the
following. Suppose that the $R^{-1}$-neighborhood of $S$ is partitioned into
$R^{-1/2}\times...\times R^{-1/2}\times R^{-1}$ blocks $\theta$. For each
$\theta$, let $\mathbb{T}_{\theta}$ be a set of finitely overlapping tubes of
dimensions $R^{1/2+\beta}\times\cdots\times R^{1/2+\beta}\times R$ with long
axis perpendicular to $\theta$, let $G(\theta)\in\mathbb{S}^{d-1}$ denote this
direction, and let $\mathbb{T}=\cup_{\theta}\mathbb{T}_{\theta}$. Each
$T\in\mathbb{T}$ belongs to $\mathbb{T}_{\theta}$ for a single $\theta$, and
we let $\theta(T)$ denote this $\theta$. We say that $f$ is microlocalized to
$(T,\theta(T))$ if $f$ is essentially supported in $2T$ and $\hat{f}$ is
essentially supported in $2\theta(T)$.

Here is our first main result on weighted refined decoupling estimates.

######  Theorem 1.1.

Suppose that $f=\sum_{T\in\mathbb{W}}f_{T}$, where
$\mathbb{W}\subset\mathbb{T}$ and each $f_{T}$ is microlocalized to
$(T,\theta(T))$. Let $Y$ be a union of $R^{1/2}$-cubes in $B^{d}_{R}$ each of
which intersects at most $M$ tubes $T\in\mathbb{W}$. Denote
$p_{d}=\frac{2(d+1)}{d-1}$. Then the following refined decoupling inequalities
hold.

(a) Let $p\geq 2$. Then

(1.1)
$\|f\|_{L^{p}(Y)}\lesssim_{\epsilon}R^{\gamma_{d}(p)+\epsilon}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}},$

where $\gamma_{d}(p)=0$ when $2\leq p\leq p_{d}$, and
$\gamma_{d}(p)=\frac{d-1}{4}-\frac{d+1}{2p}$ when $p\geq p_{d}$.

(b) Let $p\leq p_{d}$, $\alpha\leq d$. Let $H:Y\to[0,1]$ be a function
satisfying that $\int_{Q}H(x)\,dx\lesssim R^{\alpha/2}$ for any $R^{1/2}$-cube
$Q$ in $Y$. Then

(1.2)
$\|f\|_{L^{p}(Y;Hdx)}\lesssim_{\epsilon}R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{d}})+\epsilon}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

The study of inequalities of this type originated from the work [5], where
linear and bilinear refined Strichartz estimates are established and applied
to resolve Carleson’s pointwise convergence problem of Schrödinger solutions
in dimension $2+1$. Later, multilinear refined Strichartz estimates are proved
in [6] and play an important role in the final resolution of the pointwise
convergence problem in all dimensions [11]. Refined decoupling inequalities
are stronger versions of linear refined Strichartz estimates; they first
appeared in [16] and played a key role in recent study of the Falconer
distance set problem [16, 8]. See [16] for a comparison between
Bourgain–Demeter’s decoupling theorem [3] and the refined decoupling theorem.

In the case that $2\leq p\leq p_{d}$, Theorem 1.1(a) and the refined
decoupling theorem in [16, Theorem 4.2] are equivalent. The main novelty is
Theorem 1.1(b), which says that if we take the weighted $L^{p}$-norm
$\|f\|_{L^{p}(Y;Hdx)}$ on the left-hand side of the decoupling inequality,
where $H$ is a weight that satisfies the ball condition with exponent $\alpha$
at scale $R^{1/2}$, then we have an extra gain
$R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{d}})}$ when $\alpha<d$ and
$p<p_{d}$.

We can also extend Theorem 1.1 to intermediate dimensions. Let $2\leq m\leq
d$. Denote $p_{m}=\frac{2(m+1)}{m-1}$. Let $R^{-1/2}\leq r\leq 1$, and
$f=\sum_{T\in\mathbb{W}}f_{T}$, where each $f_{T}$ is microlocalized to
$(T,\theta(T))$. We say that $f$ has $(r,m)$-concentrated frequencies if there
is some $m$-dimensional subspace $V$ such that

$\textrm{Angle}(G(\theta(T)),V)\leq r,\quad\forall T\in\mathbb{W}\,.$

Obviously, from the definition, all $f$ trivially has $(1,m)$-concentrated
frequencies and also $(r,d)$-concentrated frequencies for any $r>0$.

######  Theorem 1.2.

Suppose that $f=\sum_{T\in\mathbb{W}}f_{T}$, where
$\mathbb{W}\subset\mathbb{T}$ and each $f_{T}$ is microlocalized to
$(T,\theta(T))$. Let $Y$ be a union of $R^{1/2}$-cubes in $B^{d}_{R}$ each of
which intersects at most $M$ tubes $T\in\mathbb{W}$. Then the following
refined decoupling inequalities hold.

(a) Let $2\leq m\leq d$, $p\geq 2$. Suppose that $f$ has
$(R^{-1/2},m)$-concentrated frequencies. Then

(1.3)
$\|f\|_{L^{p}(Y)}\lesssim_{\epsilon}R^{\gamma_{m}(p)+\epsilon}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}},$

where $\gamma_{m}(p)=0$ when $2\leq p\leq p_{m}$, and
$\gamma_{m}(p)=\frac{m-1}{4}-\frac{m+1}{2p}$ when $p\geq p_{m}$.

(b) Let $2\leq m\leq d$, $p\leq p_{m}$, $\alpha\leq d$. Suppose that $f$ has
$(R^{-1/2},m)$-concentrated frequencies. Let $H:Y\to[0,1]$ be a function
satisfying that $\int_{Q}H(x)\,dx\lesssim R^{\alpha/2}$ for any $R^{1/2}$-cube
$Q$ in $Y$. Then

(1.4)
$\|f\|_{L^{p}(Y;Hdx)}\lesssim_{\epsilon}R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})+\epsilon}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

(c) Let $2\leq m\leq d$, $p_{d}\leq p\leq p_{m}$, $\alpha\leq d$. Suppose that
$f$ has $(r,m)$-concentrated frequencies, where $R^{-1/2}\leq r\leq 1$. Let
$H:Y\to[0,1]$ be a function satisfying that
$\int_{Q^{\prime}}H(x)\,dx\lesssim(\frac{1}{r})^{\alpha}$ for any
$\frac{1}{r}$-cube $Q^{\prime}$ in $Y$. Then

(1.5)
$\|f\|_{L^{p}(Y;Hdx)}\lesssim_{\epsilon}R^{\epsilon}r^{(d-\alpha)(\frac{1}{p}-\frac{1}{p_{m}})}(r^{2}R)^{\frac{d-1}{4}-\frac{d+1}{2p}}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

Note that Theorem 1.1(a)(b) is a special case of Theorem 1.2(a)(b) with $m=d$.
Morally speaking, Theorem 1.2(a)(b) says that if $G(\theta)$’s are
concentrated around a subspace, then one can apply decoupling in a lower
dimensional space. Theorem 1.2(c) is obtained by a two-step decoupling
process, combining Theorem 1.2(b) to first decouple frequencies into $r$-caps
and Theorem 1.1(a) to further decouple $r$-caps into smaller $R^{-1/2}$-caps.

######  Remark 1.3.

(i). In our application to the Falconer distance set problem, the weight
function $H$ satisfies the ball condition at all scales $\geq 1$. But in our
proof of Theorem 1.2, we only need the ball condition at a single scale: scale
$R^{1/2}$ for part (b) and scale $\frac{1}{r}$ for part (c).

(ii). Though Theorem 1.2 is stated for all $\alpha\leq d$, better result
exists when $\alpha<d-m$. For $d-m\leq\alpha<d-\frac{m+1}{2}$, there may also
be room to further improve Theorem 1.2. We don’t explore these directions in
the current paper as they will not help in our application to Falconer’s
distance problem.

For large $\alpha$, the gain
$R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})}$ when $\alpha<d$ and
$p<p_{m}$ in Theorem 1.2(b) is sharp.

######  Theorem 1.4.

Let $2\leq m\leq d$, $d-\frac{m+1}{2}\leq\alpha\leq d$. Then there are $f,Y$
and $H$ satisfying the conditions of Theorem 1.2(b) such that for any $p>0$
there holds

(1.6) $\|f\|_{L^{p}(Y;Hdx)}\gtrsim
R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

In fact, the weight function $H$ here satisfies the ball condition at all
scales up to $R^{1/2}$:

$\int_{\tilde{Q}}H(x)dx\lesssim s^{\alpha},\quad\forall s\text{-cube
}\tilde{Q},\,\forall 0<s\leq R^{1/2};$

moreover, if in addition $\alpha\geq m$, then $H$ satisfies the ball condition
at all scales up to $R$.

######  Remark 1.5.

In our example for Theorem 1.4, $(M,|\mathbb{W}|)$ is a fixed special pair of
values. We expect it to be a difficult and interesting question to further
enquire how sharp the gain
$R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})}$ is for other given
choices of $(M,|\mathbb{W}|)$.

###  1.2. Application to Falconer’s distance set problem

Now let us see a classical question in geometric measure theory introduced by
Falconer [13] in the early 80s. Let $E\subset\mathbb{R}^{d}$ be a compact set,
its _distance set_ $\Delta(E)$ is defined by

$\Delta(E):=\\{|x-y|:x,y\in E\\}\,.$

######  Conjecture.

[Falconer] Let $d\geq 2$ and $E\subset\mathbb{R}^{d}$ be a compact set. Then

${\dim_{H}}(E)>\frac{d}{2}\Rightarrow|\Delta(E)|>0.$

Here $|\cdot|$ denotes the Lebesgue measure and ${\dim_{H}}(\cdot)$ is the
Hausdorff dimension.

Falconer’s conjecture remains open in all dimensions as of today. It has
attracted a great amount of attention in the past decades. To name a few
landmarks: in 1985, Falconer [13] showed that $|\Delta(E)|>0$ if
${\dim_{H}}(E)>\frac{d}{2}+\frac{1}{2}$. Bourgain [2] was the first to lower
the threshold $\frac{d}{2}+\frac{1}{2}$ in dimensions $d=2,d=3$ and to use the
theory of Fourier restriction in the Falconer problem. The thresholds were
further improved by Wolff [20] to $\frac{4}{3}$ in the case $d=2$, and by
Erdoğan [12] to $\frac{d}{2}+\frac{1}{3}$ when $d\geq 3$. These records were
only very recently rewritten:

$\begin{cases}\frac{5}{4},&d=2,\quad\qquad\text{(Guth--Iosevich--Ou--Wang
\cite[cite]{[\@@bibref{}{guth2020falconer}{}{}]})}\\\
\frac{9}{5},&d=3,\quad\qquad\text{(Du--Guth--Ou--Wang--Wilson--Zhang
\cite[cite]{[\@@bibref{}{DGOWWZ}{}{}]})}\\\
\frac{d}{2}+\frac{1}{4}+\frac{1}{8d-4},&d\geq 3,\quad\qquad\text{(Du--Zhang
\cite[cite]{[\@@bibref{}{du2019sharp}{}{}]})}\\\
\frac{d}{2}+\frac{1}{4},&d\geq 4\text{ even},\quad\text{(Du--Iosevich--Ou--
Wang--Zhang \cite[cite]{[\@@bibref{}{du2021improved}{}{}]})}.\end{cases}$

In this paper, we prove the following result on Falconer’s conjecture using
weighted refined decoupling estimates. Similar to [16, 8], we in fact prove a
slightly stronger version regarding the pinned distance set.

######  Theorem 1.6.

Let $d\geq 3$ and $E\subset\mathbb{R}^{d}$ be a compact set. Suppose that

${\rm dim}_{H}(E)>\begin{cases}\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4},&d\geq
4,\\\ \frac{3}{2}+\frac{1}{4}+\frac{17-12\sqrt{2}}{4},&d=3.\end{cases}$

Then, there is a point $x\in E$ such that $|\Delta_{x}(E)|>0$, where

$\Delta_{x}(E):=\\{|x-y|:\,y\in E\\}.$

Theorem 1.6 improves the thresholds in [7, 11, 8] in all dimensions $d\geq 3$.
The work [7] uses the polynomial partitioning method developed by Guth [14,
15] and refined Strichartz estimates from [5, 6]. The main ingredients in [11]
are broad-narrow analysis, multilinear refined Strichartz estimates,
Bourgain–Demeter’s $l^{2}$ decoupling theorem, and a delicate induction on
scales argument. In the current paper, we adapt the good tube/bad tube and
refined decoupling method pioneered by [16] for dimension $d=2$ and continued
in [8] for even dimensions $d$. In both papers, Orponen’s radial projection
theorem [18] plays a key role. However, the argument does not perform well for
odd dimensions $d$, and the result of [11] provides a better bound for
distance sets. The reason is that Orponen’s radial projection theorem only
works for sets with dimension $>d-1$, where $d$ is the dimension of the
ambient space. To overcome this issue, [8] projected the set onto a generic
$(\frac{d}{2}+1)$-dimensional subspace of $\mathbb{R}^{d}$ (assuming $d$ is
even). While this orthogonal projection trick works well in even dimensions,
for odd dimensions we are forced to project to a $(\frac{d+1}{2})$-dimensional
subspace instead, which creates some loss. To avoid this loss, a natural
approach is to avoid the initial orthogonal projection; but then, we need a
radial projection theorem that works for sets of dimension $\leq d-1$.

One new ingredient in this paper is a radial projection result, Theorem 4.1,
by the third author [19]. For each $\delta$-tube $T$, let $r(T)\in[\delta,1]$
be the thickness of the smallest heavy plate containing $T$ (see Section 4.1
for the precise definition). We can remove more bad parts (see Section 4.2)
using Theorem 4.1 and give a new threshold (4.1) for bad tubes depending on
$r(T)$. To deal with the varying values of $r(T)$, we apply weighted refined
decoupling estimates in Theorem 1.2(c). In the case that $r(T)=\delta$, the
threshold (4.1) is the same as the one obtained from combining Orponen’s
radial projection theorem and orthogonal projections; however, Theorem 1.2
gives an extra gain when $r(T)$ is small. As $r(T)$ increases, the threshold
(4.1) gets much better.

######  Remark 1.7.

In a companion work [10], we provide an alternative proof of Theorem 1.6 (in
fact, in [10] we can establish the dimensional threshold
$\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$ in all dimensions $d\geq 3$).
Compared with [10], in the current paper the construction of the good part
$\mu_{1,g}$ is simpler and more intuitive so that the control of the bad part
is much easier than that in [10]. On the other hand, the $L^{2}$ estimate for
the good part is slightly complex, for which we need the new weighted refined
decoupling.

### Outline

In Section 2, we prove refined decoupling estimates in Theorem 1.2. In Section
3, we present sharp examples for Theorem 1.2 in the case of large fractal
dimensions to prove Theorem 1.4. In Section 4, we discuss an application of
weighted refined decoupling estimates to Falconer distance set problem - proof
of Theorem 1.6.

### Notations.

Throughout the article, we write $A\lesssim B$ if $A\leq CB$ for some absolute
constant $C$; $A\sim B$ if $A\lesssim B$ and $B\lesssim A$;
$A\lesssim_{\varepsilon}B$ if $A\leq C_{\varepsilon}B$; $A\lessapprox B$ if
$A\leq C_{\varepsilon}R^{\varepsilon}B$ for any $\varepsilon>0,R>1$.

For a large parameter $R$, ${\rm RapDec}(R)$ denotes those quantities that are
bounded by a huge (absolute) negative power of $R$, i.e. ${\rm RapDec}(R)\leq
C_{N}R^{-N}$ for arbitrarily large $N>0$. Such quantities are negligible in
our argument.

For subsets $E_{1},E_{2}\subset\mathbb{R}^{d}$,
$\operatorname{dist}(E_{1},E_{2})$ is their Euclidean distance.

For $A\subset X\times Y$ and $x\in X$, define the slice $A|_{x}=\\{y\in
Y:(x,y)\in A\\}$. Similar definition for $A|_{y}$, when $y\in Y$.

We say a measure $\mu$ in $\mathbb{R}^{d}$ is an $\alpha$-dimensional measure
with constant $C_{\mu}$ if it is a probability measure satisfying that

$\mu(B(x,t))\leq C_{\mu}t^{\alpha},\qquad\forall x\in\mathbb{R}^{d},\,\forall
t>0.$

An $(r,m)$-plate $H$ in $\mathbb{R}^{d}$ is the $r$-neighborhood of an
$m$-dimensional affine plane in the cube $[-10,10]^{d}$. More precisely,

$H=\\{z\in[-10,10]^{d}:\operatorname{dist}(z,P_{H})<r\\},$

where $P_{H}$ is an $m$-dimensional affine plane, which is called the central
plane of $H$. A $C$-scaling of $H$ is

$CH=\\{z\in[-10,10]^{d}:\operatorname{dist}(z,P_{H})<Cr\\}.$

We say that an $(r,m)$-plate $H$ is $\gamma$-concentrated on $\mu$ if
$\mu(H)\geq\gamma$.

Let $\mathcal{E}_{r,m}$ be a set of $(r,m)$-plates with the following
properties:

  * •

Each $(\frac{r}{2},m)$-plate intersecting $B(0,1)$ lies in at least one plate
of $\mathcal{E}_{r,m}$;

  * •

For $s\geq r$, every $(s,m)$-plate contains
$\lesssim\left(\frac{s}{r}\right)^{(m+1)(d-m)}$ many $(r,m)$-plates of
$\mathcal{E}_{r,m}$.

For example, when $m=1$ and $d=2$, we can simply pick $\sim r^{-1}$ many
$r$-tubes in each of an $r$-net of directions. This generalizes to higher $m$
and $d$ via a standard $r$-net argument, see [19, Section 2.2] for the details
of its construction.

######  Acknowledgements.

XD is supported by NSF DMS-2107729 (transferred from DMS-1856475), NSF
DMS-2237349 and Sloan Research Fellowship. YO is supported by NSF DMS-2142221
and NSF DMS-2055008. KR is supported by a NSF GRFP fellowship. RZ is supported
by NSF DMS-2207281 (transferred from DMS-1856541), NSF DMS-2143989 and the
Sloan Research Fellowship.

##  2\. Weighted refined decoupling estimates - Proof of Theorem 1.2

In this section, we prove Theorem 1.2 for the truncated paraboloid. The proof
can be generalized for any compact and strictly convex $C^{2}$ hypersurface
with Gaussian curvature $\sim 1$ by standard arguments. In particular, in the
proof of part (a), we use the fact that under the assumption of
$(R^{-1/2},m)$-concentration, one can apply Bourgain–Demeter’s
$l^{2}$-decoupling in dimension $m$. See [15, Lemma 9.3] and [5, Lemma 7.4]
for justification of this fact in the case of the truncated paraboloid, and
one can follow [3, Section 7] to generalize this fact to hypersurfaces as in
the above.

First, we present a slightly simplified proof of Theorem 1.1(a) based on that
in [16].

_Proof of Theorem 1.1(a)._ Without loss of generality, we can assume that

(2.1) $\|f\|_{L^{p}(Q)}\sim\textrm{ constant for all $R^{1/2}$-cubes $Q\subset
Y$}.$

Now we decompose $f$ as follows. We cover $\mathcal{S}$ with larger blocks
$\tau$ of dimensions $R^{-1/4}\times\cdots\times R^{-1/4}\times R^{-1/2}$. For
each $\tau$ we cover $B^{d}_{R}$ with cylinders $\Box$ with radius $R^{3/4}$
and length $R$, with the long axis perpendicular to $\tau$. Each cylinder
$\Box$ is associated to a unique $\tau$, which we denote by $\tau(\Box)$. Then
we define

$\mathbb{W}_{\Box}:=\\{T\in\mathbb{W}:\theta(T)\subset\tau(\Box)\textrm{ and
}T\cap B_{R}\subset\Box\\}$

and define $f_{\Box}:=\sum_{T\in\mathbb{W}_{\Box}}f_{T}.$ Note that
$\widehat{f}_{\Box}$ is essentially supported in $\tau(\Box)$.

Next, write each $\Box$ as a union of cylinders running parallel to the long
axis of $\Box$, with radius $R^{1/2}$ and length $R^{3/4}$. Let
$Y_{\Box,M^{\prime}}$ be the union of those cylinders that each intersect
$\sim M^{\prime}$ of the tubes $T\in\mathbb{W}_{\Box}$.

Now we dyadically pigeonhole $M^{\prime}$ so that

(2.2) $\|f\|_{L^{p}(Q)}\lessapprox\left\|\sum_{\Box:\,Q\subset
Y_{\Box,M^{\prime}}}f_{\Box}\right\|_{L^{p}(Q)}$

for a fraction $\approx 1$ of $Q\subset Y$. We fix this value of $M^{\prime}$,
and from now on we abbreviate $Y_{\Box}=Y_{\Box,M^{\prime}}$.

Denote the collection of cylinders $\Box$ by $\mathbb{B}$. We dyadically
pigeonhole the cubes $Q\subset Y$ according to the number of
$\Box\in\mathbb{B}$ so that $Q\subset Y_{\Box}$. We get a subset
$Y^{\prime}\subset Y$ so that for each cube $Q\subset Y^{\prime}$, $Q\subset
Y_{\Box}$ for $\sim M^{\prime\prime}$ choices of $\Box\in\mathbb{B}$, and $Q$
obeys (2.2). Moreover, by dyadic pigeonholing, we have
$|Y^{\prime}|\approx|Y|$. Since each cube $Q\subset Y$ has approximately equal
$L^{p}$ norm, we also get $\|f\|_{L^{p}(Y^{\prime})}\approx\|f\|_{L^{p}(Y)}$.

We also note that

(2.3) $M^{\prime}M^{\prime\prime}\lesssim M,$

because a cube $Q\subset Y^{\prime}$ belongs to $Y_{\Box}$ for $\sim
M^{\prime\prime}$ different $\Box$, and if $Q\subset Y_{\Box}$, then it
belongs to $T$ for $\sim M^{\prime}$ different $T\in\mathbb{W}_{\Box}$.

Note that an $R^{1/2}$-cube $Q$ lies in one cylinder $\Box$ associated to each
cap $\tau$. So by applying Bourgain–Demeter’s $l^{2}$-decoupling [3] at scale
$R^{1/2}$ to the RHS of (2.2), for each $Q\subset Y^{\prime}$ we get

(2.4) $\|f\|_{L^{p}(Q)}\lessapprox
R^{\frac{1}{2}\gamma_{d}(p)}\left(\sum_{\Box:Q\subset
Y_{\Box}}\|f_{\Box}\|_{L^{p}(Q)}^{2}\right)^{1/2}.$

The next ingredient is induction on scales. After parabolic rescaling, the
function $f_{\Box}$ with the decomposition
$f_{\Box}=\sum_{T\in\mathbb{W}_{\Box}}f_{T}$ on the subset $Y_{\Box}$ is
equivalent to the setup of the theorem at scale $R^{1/2}$ instead of scale
$R$. So by induction on the radius, we get a version of our main inequality
for each function $f_{\Box}$:

(2.5) $\|f_{\Box}\|_{L^{p}(Y_{\Box})}\lesssim
R^{\frac{1}{2}(\gamma_{d}(p)+\epsilon)}(M^{\prime})^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}_{\Box}}\|f_{T}\|_{L^{p}}^{p}\right)^{1/p}.$

Now, combining all these ingredients (2.3), (2.4) and (2.5), we are ready to
estimate $\|f\|_{L^{p}(Y)}$:

$\displaystyle\|f\|^{p}_{L^{p}(Y)}$ $\displaystyle\lessapprox\sum_{Q\subset
Y^{\prime}}\|f\|^{p}_{L^{p}(Q)}$ $\displaystyle\lessapprox
R^{\frac{p}{2}\gamma_{d}(p)}\sum_{Q\subset
Y^{\prime}}\left(\sum_{\Box:Q\subset
Y_{\Box}}\|f_{\Box}\|_{L^{p}(Q)}^{2}\right)^{p/2}$ $\displaystyle\lesssim
R^{\frac{p}{2}\gamma_{d}(p)}(M^{\prime\prime})^{\frac{p}{2}-1}\sum_{Q\subset
Y^{\prime}}\sum_{\Box:Q\subset Y_{\Box}}\|f_{\Box}\|_{L^{p}(Q)}^{p}$
$\displaystyle\lesssim
R^{\frac{p}{2}\gamma_{d}(p)}(M^{\prime\prime})^{\frac{p}{2}-1}\sum_{\Box}\|f_{\Box}\|_{L^{p}(Y_{\Box})}^{p}$
$\displaystyle\lesssim
R^{p\gamma_{d}(p)+\frac{p\epsilon}{2}}(M^{\prime}M^{\prime\prime})^{\frac{p}{2}-1}\sum_{\Box}\sum_{T\in\mathbb{W}_{\Box}}\|f_{T}\|_{L^{p}}^{p}$
$\displaystyle\lesssim
R^{p\gamma_{d}(p)+\frac{p\epsilon}{2}}M^{\frac{p}{2}-1}\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}.$

Taking account of $\lessapprox$ throughout, we get

$\|f\|_{L^{p}(Y)}\lesssim
R^{\gamma_{d}(p)+\frac{3\epsilon}{4}}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

This closes the induction and finishes the proof of Theorem 1.1(a).

_Proof of Theorem 1.2(a)_. The proof is almost identical to that of Theorem
1.1(a). The only difference is that, when applying Bourgain–Demeter’s
$l^{2}$-decoupling at scale $R^{1/2}$, one uses the decoupling in dimension
$m$ (instead of $d$) because of the $(R^{-1/2},m)$-concentration assumption
(see [15, Lemma 9.3] and [5, Lemma 7.4] for justifications of similar
statements). Also, note that after parabolic rescaling, the function
$f_{\Box}$ with the decomposition $f_{\Box}=\sum_{T\in\mathbb{W}_{\Box}}f_{T}$
on the subset $Y_{\Box}$ is equivalent to the setup of the theorem at scale
$R^{1/2}$: the $f_{\Box}$ after rescaling has $(R^{-1/4},m)$-concentrated
frequencies.

_Proof of Theorem 1.2(b)_. Now we prove part (b) using the case $p=p_{m}$ of
part (a). Without loss of generality, we can assume that

$|f_{T}|\sim\textrm{ constant for all $T\in\mathbb{W}$},$

and each $R^{1/2}$-cube $Q$ in $Y$ intersects $\sim M$ tubes $T\in\mathbb{W}$.
Denote the number of $R^{1/2}$-cubes in $Y$ by $N$, and let $W=|\mathbb{W}|$.
Considering the incidence between $R^{1/2}$-cubes in $Y$ and tubes
$T\in\mathbb{W}$, we get

$NM\lessapprox WR^{1/2}.$

By the assumption that $|f_{T}|\sim$ constant for all $T\in\mathbb{W}$, we
also get

(2.6)
$\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p_{m}}}^{p_{m}}\right)^{\frac{1}{p_{m}}}\lessapprox\left(\frac{1}{WR^{(d+1)/2}}\right)^{\frac{1}{p}-\frac{1}{p_{m}}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

Combining all these ingredients together with the assumption that
$\int_{Q}H(x)\,dx\lesssim R^{\alpha/2}$ for any $R^{1/2}$-cube $Q$ in $Y$, and
applying the case $p=p_{m}$ of part (a) we get the following for any $p\leq
p_{m}$:

$\displaystyle\|f\|_{L^{p}(Y;Hdx)}$
$\displaystyle\leq\left(\int_{Y}H\,dx\right)^{\frac{1}{p}-\frac{1}{p_{m}}}\|f\|_{L^{p_{m}}(Y)}$
$\displaystyle\lessapprox\left(\int_{Y}H\,dx\right)^{\frac{1}{p}-\frac{1}{p_{m}}}M^{\frac{1}{2}-\frac{1}{p_{m}}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p_{m}}}^{p_{m}}\right)^{\frac{1}{p_{m}}}$
$\displaystyle\lessapprox
M^{\frac{1}{2}-\frac{1}{p}}\left(\frac{NR^{\alpha/2}M}{WR^{(d+1)/2}}\right)^{\frac{1}{p}-\frac{1}{p_{m}}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}$
$\displaystyle\lesssim
R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}},$

as desired.

_Proof of Theorem 1.2(c)_. We prove part (c) by combining two steps of refined
decoupling inequalities from (b) and Theorem 1.1(a).

Let $R^{-1/2}\leq r\leq 1$. Let $\tilde{Y}$ be a union of $r^{-2}$-cubes in
$B_{R}$ such that each $r^{-2}$-cube $Q_{1}$ in $\tilde{Y}$ intersects some
$R^{1/2}$-cube $Q$ in $Y$. Without loss of generality, we can assume that

(2.7) $\|f\|_{L^{p}(Q_{1};Hdx)}\sim\textrm{ constant for all $r^{-2}$-cubes
$Q_{1}\subset\tilde{Y}$},$

and

(2.8) $\|f\|_{L^{p}(Y;Hdx)}\lessapprox\|f\|_{L^{p}(\tilde{Y};Hdx)}.$

Now we decompose $f$ as follows. We cover $\mathcal{S}$ with blocks $\tau$ of
dimensions $r\times\cdots\times r\times r^{2}$. For each $\tau$ we cover
$B^{d}_{R}$ with cylinders $\Box$ with radius $rR$ and length $R$, with the
long axis perpendicular to $\tau$. Each cylinder $\Box$ is associated to a
unique $\tau$, which we denote by $\tau(\Box)$. Then we define
$\mathbb{W}_{\Box}:=\\{T\in\mathbb{W}:\theta(T)\subset\tau(\Box)\textrm{ and
}T\cap B_{R}\subset\Box\\}$ and define
$f_{\Box}:=\sum_{T\in\mathbb{W}_{\Box}}f_{T}.$ Note that $\widehat{f}_{\Box}$
is essentially supported in $\tau(\Box)$. Denote the collection of boxes
$\Box$ by $\mathbb{B}$.

Next, write each $\Box$ as a union of cylinders $\Box_{1}$ running parallel to
the long axis of $\Box$, with radius $R^{1/2}$ and length $r^{-1}R^{1/2}$. Let
$Y_{\Box,M_{2}}$ be the union of those cylinders that each intersect $\sim
M_{2}$ of the tubes $T\in\mathbb{W}_{\Box}$.

For each $r^{-2}$-cube $Q_{1}$ in $\tilde{Y}$, let

$f|_{Q_{1}}=\sum_{T_{1}\in\mathbb{T}[Q_{1}]}f_{T_{1}}$

be the wave packet decomposition of $f|_{Q_{1}}$ at scale $r^{-2}$. Each tube
$T_{1}\in\mathbb{T}[Q_{1}]$ has radius roughly $r^{-1}$ and length $r^{-2}$.
Note that each $T_{1}$ is contained in a unique cylinder $\Box_{1}$ with
radius $R^{1/2}$ and length $r^{-1}R^{1/2}$, which runs in the same direction
as $T_{1}$. We denote this $\Box_{1}$ by $\Box_{1}(T_{1})$. And this
$\Box_{1}(T_{1})$ is contained in a unique $\Box\in\mathbb{B}$, which runs in
the same direction as $T_{1}$. We denote this $\Box$ by $\Box(T_{1})$.

Now we dyadically pigeonhole $M_{2}$ so that

(2.9)
$\|f\|_{L^{p}(Q_{1};Hdx)}\lessapprox\left\|\sum_{T_{1}\in\mathbb{T}[Q_{1}]:\,\Box_{1}(T_{1})\subset
Y_{\Box(T_{1}),M_{2}}}f_{T_{1}}\right\|_{L^{p}(Q_{1};Hdx)}$

for a fraction $\approx 1$ of $Q_{1}\subset\tilde{Y}$. We fix this value of
$M_{2}$, and from now on we abbreviate $Y_{\Box}=Y_{\Box,M_{2}}$.

Next, write each $Q_{1}\cap Y$ as a union of $r^{-1}$-cubes $Q^{\prime}$. Let
$Y_{Q_{1},M_{1}}$ be the union of those $r^{-1}$-cubes that each intersect
$\sim M_{1}$ of the tubes $T_{1}\in\mathbb{T}[Q_{1}]$ with
$\Box_{1}(T_{1})\subset Y_{\Box(T_{1})}$.

Now we dyadically pigeonhole $M_{1}$ so that

(2.10)
$\|f\|_{L^{p}(Q_{1};Hdx)}\lessapprox\left\|\sum_{T_{1}\in\mathbb{T}[Q_{1}]:\,\Box_{1}(T_{1})\subset
Y_{\Box(T_{1})}}f_{T_{1}}\right\|_{L^{p}(Y_{Q_{1},M_{1}};Hdx)}$

for a fraction $\approx 1$ of $Q_{1}\subset\tilde{Y}$. We fix this value of
$M_{1}$, and from now on we abbreviate $Y_{Q_{1}}=Y_{Q_{1},M_{1}}$.

Let $\tilde{Y}^{\prime}$ be the collections of $Q_{1}$ satisfying (2.10).
Since $\|f\|_{L^{p}(Q_{1};Hdx)}\sim\textrm{ constant }$ for all
$Q_{1}\subset\tilde{Y}$, we get
$\|f\|_{L^{p}(Y;Hdx)}\approx\|f\|_{L^{p}(\tilde{Y};Hdx)}\approx\|f\|_{L^{p}(\tilde{Y}^{\prime};Hdx)}$.

We also note that

(2.11) $M_{1}M_{2}\lesssim M,$

because an $r^{-1}$ cube $Q^{\prime}\subset Y_{Q_{1}}$ intersects $\sim M_{1}$
of the tubes $T_{1}\in\mathbb{T}[Q_{1}]$ with $\Box_{1}(T_{1})\subset
Y_{\Box(T_{1})}$, and each $\Box_{1}(T_{1})\subset Y_{\Box(T_{1})}$ intersects
$\sim M_{2}$ of the tubes $T\in\mathbb{W}_{\Box(T_{1})}$.

Let $p_{d}\leq p\leq p_{m}$. By applying part (b) to the RHS of (2.10), for
each $r^{-2}$-cube $Q_{1}$ in $\tilde{Y}^{\prime}$, we have

(2.12) $\|f\|_{L^{p}(Q_{1};Hdx)}\lessapprox
r^{(d-\alpha)(\frac{1}{p}-\frac{1}{p_{m}})}M_{1}^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T_{1}\in\mathbb{T}[Q_{1}]:\,\Box_{1}(T_{1})\subset
Y_{\Box(T_{1})}}\|f_{T_{1}}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

Also, after parabolic rescaling, the function $f_{\Box}$ with the
decomposition $f_{\Box}=\sum_{T\in\mathbb{W}_{\Box}}f_{T}$ on the subset
$Y_{\Box}$ is equivalent to the setup of Theorem 1.1(a) at scale $r^{2}R$
instead of scale $R$. So by applying Theorem 1.1(a) in the case $p\geq p_{d}$,
we get

(2.13)
$\|f_{\Box}\|_{L^{p}(Y_{\Box})}\lessapprox(r^{2}R)^{\frac{d-1}{4}-\frac{d+1}{2p}}M_{2}^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}_{\Box}}\|f_{T}\|_{L^{p}}^{p}\right)^{1/p}.$

Now we are ready to estimate $\|f\|_{L^{p}(Y;Hdx)}$ by combining (2.11),
(2.12) and (2.13):

$\displaystyle\|f\|^{p}_{L^{p}(Y;Hdx)}$
$\displaystyle\approx\sum_{Q_{1}\subset\tilde{Y}^{\prime}}\|f\|^{p}_{L^{p}(Q_{1};Hdx)}$
$\displaystyle\lessapprox\sum_{Q_{1}\subset\tilde{Y}^{\prime}}r^{(d-\alpha)(1-\frac{p}{p_{m}})}M_{1}^{\frac{p}{2}-1}\left(\sum_{T_{1}\in\mathbb{T}[Q_{1}]:\,\Box_{1}(T_{1})\subset
Y_{\Box(T_{1})}}\|f_{T_{1}}\|_{L^{p}}^{p}\right)$ $\displaystyle\sim
r^{(d-\alpha)(1-\frac{p}{p_{m}})}M_{1}^{\frac{p}{2}-1}\sum_{\Box}\sum_{Q_{1}\subset\tilde{Y}^{\prime}}\sum_{\underset{\Box_{1}(T_{1})\subset
Y_{\Box(T_{1})},\,\Box(T_{1})=\Box}{T_{1}\in\mathbb{T}[Q_{1}]}}\|f_{T_{1}}\|_{L^{p}}^{p}$
$\displaystyle\lesssim
r^{(d-\alpha)(1-\frac{p}{p_{m}})}M_{1}^{\frac{p}{2}-1}\sum_{\Box}\|f_{\Box}\|^{p}_{L^{p}(Y_{\Box})}$
$\displaystyle\lessapprox
r^{(d-\alpha)(1-\frac{p}{p_{m}})}(r^{2}R)^{\frac{(d-1)p}{4}-\frac{d+1}{2}}(M_{1}M_{2})^{\frac{p}{2}-1}\sum_{\Box}\left(\sum_{T\in\mathbb{W}_{\Box}}\|f_{T}\|_{L^{p}}^{p}\right)$
$\displaystyle\lesssim
r^{(d-\alpha)(1-\frac{p}{p_{m}})}(r^{2}R)^{\frac{(d-1)p}{4}-\frac{d+1}{2}}M^{\frac{p}{2}-1}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)\,.$

This concludes the proof of (c).

##  3\. A sharp example in the case of large fractal dimensions - Proof of
Theorem 1.4

We consider the following example obtained by adapting the one in [1] to
intermediate dimensions. Similar adaptions can be found in [9, 4].

Let $c=1/1000$ be a fixed small constant, $0<\kappa<1/2$, and $2\leq m\leq d$.
Denote

$x=(x_{1},\cdots,x_{d})=(x^{\prime},x^{\prime\prime},x_{d})\in B^{d}(0,R)\,,$
$\xi=(\xi_{1},\cdots,\xi_{d-1})=(\xi^{\prime},\xi^{\prime\prime})\in
B^{d-1}(0,1)\,,$

where

$x^{\prime}=(x_{1},\cdots,x_{d-m}),\quad
x^{\prime\prime}=(x_{d-m+1},\cdots,x_{d-1}),$
$\xi^{\prime}=(\xi_{1},\cdots,\xi_{d-m}),\quad\xi^{\prime\prime}=(\xi_{d-m+1},\cdots,\xi_{d-1}).$

For simplicity, we denote $B^{d}(0,r)$ by $B^{d}_{r}$, and write the interval
$(-r,r)$ as $I_{r}$. Let $g(\xi)=\raisebox{3.01385pt}{$\chi$}_{\Omega}(\xi)$,
where the set $\Omega$ is defined by

(3.14) $\Omega:=\left[B^{d-m}_{cR^{-1/2}}\times\left(2\pi
R^{-\kappa}\mathbb{Z}^{m-1}+B^{m-1}_{cR^{-1}}\right)\right]\cap
B^{d-1}(0,1)\,.$

Take $f(x)=Eg(x)$, the Fourier extension of $g$ over the truncated paraboloid:

(3.15)
$f(x):=Eg(x)=\frac{1}{(2\pi)^{d/2}}\int_{B^{d-1}(0,1)}e^{i(x^{\prime}\cdot\xi^{\prime}+x^{\prime\prime}\cdot\xi^{\prime\prime}+x_{d}|\xi^{\prime}|^{2}+x_{d}|\xi^{\prime\prime}|^{2})}g(\xi)\,d\xi.$

Next, we define a set $\Lambda$ in $B^{d}(0,R)$ by

(3.16)
$\Lambda:=\left[B^{d-m}_{cR^{1/2}}\times\left(R^{\kappa}\mathbb{Z}^{m-1}+B^{m-1}_{c}\right)\times\left(\frac{1}{2\pi}R^{2\kappa}\mathbb{Z}+I_{c}\right)\right]\cap
B^{d}_{R}\,.$

And define $Y$ and $H$ by

(3.17) $Y:=B^{d-m}_{cR^{1/2}}\times B^{m}_{R}\quad\text{ and }\quad
H:=\raisebox{3.01385pt}{$\chi$}_{\Lambda}.$

From the definition, it follows that

(3.18) $|\Omega|\sim R^{(\kappa-1)(m-1)-(d-m)/2}\,.$

and

(3.19) $|\Lambda|\sim
R^{(d-m)/2+(1-\kappa)(m-1)+1-2\kappa}=R^{(d+m)/2-\kappa(m+1)}\,.$

And it is straightforward (for example, see [4, proof of Lemma 3.1]) to check

(3.20)
$x^{\prime}\cdot\xi^{\prime}+x^{\prime\prime}\cdot\xi^{\prime\prime}+x_{d}|\xi^{\prime}|^{2}+x_{d}|\xi^{\prime\prime}|^{2}\in
2\pi\mathbb{Z}+(-\frac{1}{100},\frac{1}{100})\,,$

provided that $\xi\in\Omega$ and $x\in\Lambda$.

######  Claim 3.1.

For $d-\frac{m+1}{2}\leq\alpha\leq d$, we can take

(3.21) $\kappa:=\frac{d-\alpha}{2(m+1)}$

such that $\kappa\leq 1/4$, and $f,Y$ and $H$ defined as in the above satisfy
the conditions of Theorem 1.2(b), and for any $p>0$ there holds

(3.22) $\|f\|_{L^{p}(Y;Hdx)}\gtrsim
R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}.$

Indeed, from the constructions, we have the following properties:

  * •

$f$ has $(R^{-1/2},m)$-concentrated frequencies;

  * •

By (3.20), $|f(x)|\sim|\Omega|,\forall x\in\Lambda$;

  * •

The Fourier support of $f$ are partitioned into $\sim R^{\kappa(m-1)}$ many
parabolic caps $\theta$, each of radius $R^{-1/2}$;

  * •

For each fixed $\theta$, let $f_{\theta}:=(\hat{f}|_{\theta})^{\vee}$. Then,
by a reason similar to (3.20), we get $|f_{\theta}|\sim R^{-(d-m)/2-(m-1)}$ on
$Y$;

  * •

Take $\mathbb{W}:=\\{T\in\mathbb{T}:T\subset Y\\}$. Then for each
$R^{1/2}$-cube in $Y$, there are $M\sim R^{\kappa(m-1)}$ many $T\in\mathbb{W}$
passing through it;

  * •

For each $R^{1/2}$-cube $Q$ in $Y$,

$\int_{Q}H(x)\,dx=|Q\cap\Lambda|\sim R^{d/2-(m+1)\kappa}=R^{\alpha/2}\,,$

by the choice of $\kappa$ in (3.21).

Therefore,

$\|f\|_{L^{p}(Y;Hdx)}\sim|\Omega|\cdot|\Lambda|^{1/p}\,,$

and

$\begin{split}&M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}\sim
M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{\theta}\|f_{\theta}\|_{L^{p}(Y)}^{p}\right)^{\frac{1}{p}}\\\
&\sim
R^{\frac{\kappa(m-1)}{2}}R^{-\frac{d-m}{2}-(m-1)}R^{\frac{d+m}{2p}}\,,\end{split}$

and thus

$\frac{\|f\|_{L^{p}(Y;Hdx)}}{M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}}\sim
R^{\kappa(\frac{m-1}{2}-\frac{m+1}{p})}=R^{\frac{1}{2}(\alpha-d)(\frac{1}{p}-\frac{1}{p_{m}})}\,,$

as desired.

Moreover, by direct computation, one can verify the ball condition at all
scales up to $R^{1/2}$:

$\int_{\tilde{Q}}H(x)dx=|\tilde{Q}\cap\Lambda|\lesssim s^{\alpha},\quad\forall
s\text{-cube }\tilde{Q}\text{ in }Y,\forall 0<s\leq R^{1/2};$

and also at all scales up to $R$ if in addition $\alpha\geq m$.

This completes the proof of Theorem 1.4.

##  4\. Application to Falconer distance set problem

As an application of Theorem 1.2, in this section we give a proof of Theorem
1.6 which is different from that in [10].

We first recall the following new radial projection theorem, which follows
from [19, Theorem 1.13].

######  Theorem 4.1.

Let $m\in\\{1,2,\cdots,d-1\\}$, $m-1<\alpha\leq m$, and fix
$\eta,\varepsilon>0$, and two $\alpha$-dimensional measures $\mu_{1},\mu_{2}$
with constants $C_{\mu_{1}},C_{\mu_{2}}$ supported on $E_{1},E_{2}\subset
B(0,1)$ respectively. There exists $\gamma>0$ depending on
$\eta,\varepsilon,\alpha,m$ such that the following holds. Fix $\delta<r<1$.
Let $A$ be the set of pairs $(x,y)\in E_{1}\times E_{2}$ satisfying that $x$
and $y$ lie in some $\delta^{\eta}$-concentrated $(r,m)$-plate on
$\mu_{1}+\mu_{2}$. Then there exists a set $B\subset E_{1}\times E_{2}$ with
$\mu_{1}\times\mu_{2}(B)\leq\delta^{\gamma}$ such that for every $x\in E_{1}$
and $\delta$-tube $T$ through $x$, we have

$\mu_{2}(T\setminus(A|_{x}\cup
B|_{x}))\lesssim\frac{\delta^{\alpha}}{r^{\alpha-(m-1)}}\delta^{-\varepsilon}.$

The implicit constant may depend on
$\eta,\varepsilon,\alpha,m,C_{\mu_{1}},C_{\mu_{2}}$.

Essentially, this theorem says that, up to a small loss, one can assume that
the wave packets associated with the fractal measure supported on the set of
interest all have small mass. We explore this idea in detail in the next
subsections.

###  4.1. Outline of the proof of Theorem 1.6

The setup of the proof is in the same line as [16, 8]. To begin with, let
$E\subset\mathbb{R}^{d}$ be a compact set with positive $\alpha$-dimensional
Hausdorff measure, with $\frac{d}{2}<\alpha<\frac{d+1}{2}$. Without loss of
generality, assume that $E$ is contained in the unit ball, and there are
subsets $E_{1},E_{2}\subset E$, each with positive $\alpha$-dimensional
Hausdorff measure, and ${\rm dist}(E_{1},E_{2})\gtrsim 1$. Then there exist
$\alpha$-dimensional probability measures $\mu_{1}$ and $\mu_{2}$ supported on
$E_{1}$ and $E_{2}$ respectively, according to the classical Frostman lemma.

To relate the measures to the distance set, we consider their pushforward
measures under the distance map. For a fixed point $x\in E_{2}$, let
$d^{x}:E_{1}\to\mathbb{R}$ be the pinned distance map given by
$d^{x}(y):=|x-y|$. Then, the pushforward measure $d^{x}_{\ast}(\mu_{1})$,
defined as

$\int_{\mathbb{R}}\psi(t)\,d^{x}_{\ast}(\mu_{1})(t)=\int_{E_{1}}\psi(|x-y|)\,d\mu_{1}(y),$

is a natural probability measure that is supported on $\Delta_{x}(E_{1})$.

The idea is that we will construct another complex-valued measure $\mu_{1,g}$
that is the _good_ part of $\mu_{1}$ with respect to $\mu_{2}$, and study its
pushforward under the map $d^{x}$. To set things up, we recall the following
decomposition of a function into microlocalized pieces, which has been used in
[16, 8]. We will eventually be choosing the following small parameters with
the dependence

$0<\beta\ll\gamma\ll\eta\ll\varepsilon\ll\epsilon.$

Fix a large parameter $R_{0}$, to be determined later, and consider a sequence
of scales $R_{j}=2^{j}R_{0}$, $\forall j\geq 1$. In $\mathbb{R}^{d}$, cover
the annulus $R_{j-1}\leq|\omega|\leq R_{j}$ by rectangular blocks $\tau$ with
dimensions approximately $R_{j}^{1/2}\times\cdots\times R_{j}^{1/2}\times
R_{j}$, with the long direction of each block $\tau$ being the radial
direction. Choose a smooth partition of unity subordinate to this cover such
that

$1=\psi_{0}+\sum_{j\geq 1,\tau}\psi_{j,\tau},$

where $\psi_{0}$ is supported in the ball $B(0,2R_{0})$.

Let $\beta>0$ be a sufficiently small constant that we will choose later
(depending on $\eta,\varepsilon$). For each $(j,\tau)$, cover the unit ball in
$\mathbb{R}^{d}$ with tubes $T$ of dimensions approximately
$R_{j}^{-1/2+\beta}\times\cdots\times R_{j}^{-1/2+\beta}\times 2$ with the
long axis parallel to the long axis of $\tau$. The covering has uniformly
bounded overlap, each $T$ intersects at most $C(d)$ other tubes. We denote the
collection of all these tubes as $\mathbb{T}_{j,\tau}$. Let $\eta_{T}$ be a
smooth partition of unity subordinate to this covering, so that for each
choice of $j$ and $\tau$, $\sum_{T\in\mathbb{T}_{j,\tau}}\eta_{T}$ is equal to
$1$ on the ball of radius $2$ and each $\eta_{T}$ is smooth.

For each $T\in\mathbb{T}_{j,\tau}$, define an operator

$M_{T}f:=\eta_{T}(\psi_{j,\tau}\hat{f})^{\vee},$

which, morally speaking, maps $f$ to the part of it that has Fourier support
in $\tau$ and physical support in $T$. Define also
$M_{0}f:=(\psi_{0}\hat{f})^{\vee}$. We denote
$\mathbb{T}_{j}=\cup_{\tau}\mathbb{T}_{j,\tau}$ and $\mathbb{T}=\cup_{j\geq
1}\mathbb{T}_{j}$. Hence, for any $L^{1}$ function $f$ supported on the unit
ball, one has the decomposition

$f=M_{0}f+\sum_{T\in\mathbb{T}}M_{T}f+\text{RapDec}(R_{0})\|f\|_{L^{1}}.$

Fix parameters $\eta,\varepsilon>0$ (these parameters will be chosen depending
on $\epsilon$ in Proposition 4.5). Let $2T$ denote the concentric dilation of
$T$ of twice the radius. For each $j\geq 1$ fixed, let
$\delta=2R_{j}^{-\frac{1}{2}+\beta}$ and consider a dyadic sequence of scales
$r\in[C_{0}\delta,1]$. Here, $C_{0}$ is a large constant depending on the
separation of the sets $E_{1},E_{2}$. For each fixed $r$, recall that
$\mathcal{E}_{r,m}$ denotes a collection of essentially distinct
$(r,m)$-plates such that every $(r/2,m)$-plate is contained in some element of
$\mathcal{E}_{r,m}$. We further let $\mathcal{H}_{r}$ denote the subcollection
of all $\delta^{\eta}$-concentrated $(r,m)$-plates of $\mathcal{E}_{r,m}$ on
$\mu_{1}+\mu_{2}$. Here, $m\in\mathbb{Z}$ is the unique integer satisfying
$m-1<\alpha\leq m$.

For each tube $T\in\mathbb{T}_{j}$, define

$r(T):=\min\\{{\rm dyadic}\,\,r\in[C_{0}\delta,1]:\,\exists
H\in\mathcal{H}_{r}\,\,{\rm s.t.}\,2T\subset H\\}.$

Here, $r(T)$ captures the most efficient choice of the scale of heavy plate
that $T$ is contained in.

We say a tube $T\in\mathbb{T}_{j}$ is _bad_ if

(4.1)
$\mu_{2}(4T)\geq\frac{\delta^{\alpha-2\varepsilon}}{r(T)^{\alpha-(m-1)}}.$

Here, $\varepsilon$ is a fixed parameter and will be chosen in Proposition 4.5
below. A tube $T$ is _good_ if it is not bad, and we define

$\mu_{1,g}:=M_{0}\mu_{1}+\sum_{T\in\mathbb{T},T\textrm{ good}}M_{T}\mu_{1}.$

We point out that $\mu_{1,g}$ is only a complex valued measure, and is
essentially supported in the $R_{0}^{-1/2+\beta}$-neighborhood of $E_{1}$ with
a rapidly decaying tail away from it.

Theorem 1.6 will follow from the following two main estimates in the exact
same way as in [8]. We omit the details.

######  Proposition 4.2.

Let $d\geq 3$, and $0<\alpha\leq d-1$. There exists a choice of $\beta>0$ and
sufficiently large $R_{0}$ in the construction of $\mu_{1,g}$ in the above,
such that there is a subset $E_{2}^{\prime}\subset E_{2}$ so that
$\mu_{2}(E_{2}^{\prime})\geq 1-\frac{1}{1000}$ and for each $x\in
E_{2}^{\prime}$,

$\|d^{x}_{*}(\mu_{1})-d^{x}_{*}(\mu_{1,g})\|_{L^{1}}<\frac{1}{1000}.$

######  Proposition 4.3.

Let $d\geq 3$ and

$\alpha>\begin{cases}\frac{d}{2}+\frac{1}{4}-\frac{1}{4(2d+1)},&d\geq 4,\\\
\frac{3}{2}+\frac{1}{4}+\frac{17-12\sqrt{2}}{4},&d=3,\end{cases}$

then there exist choices of $\varepsilon$ and $R_{0}$ so that for sufficiently
small $\beta$ in terms of $\alpha$ in the construction of $\mu_{1,g}$ in the
above,

$\int_{E_{2}}\|d^{x}_{*}(\mu_{1,g})\|_{L^{2}}^{2}d\mu_{2}(x)<+\infty.$

We will prove these two propositions in the next two subsections. Before that,
let us briefly explain the key new ideas here. In contrast to [16, 8], where a
similar framework were used to study the Falconer distance problem, our
definition of good tubes here involves a new parameter $r(T)$, which captures
the size of the smallest heavy plate containing tube $T$. Thanks to the new
radial projection result (Theorem 4.1), we are able to show that the bad tubes
can be safely removed (Proposition 4.2). To prove Proposition 4.3, we make use
of the weighted refined decoupling estimates to deal with varying values of
$r(T)$.

In the extreme case that $r(T)\sim R_{j}^{-1/2}$, the threshold for the
good/bad tubes is the same as the one in [8], however, in this case, from the
fact that $T$ is contained in a thin $(R_{j}^{-1/2},m)$-plate, we get extra
gain by applying the weighted refined decoupling estimate in Theorem 1.2. At
the other end of the spectrum, where $r(T)=1$, even though the weighted
decoupling estimate is not going to help, the threshold for the good/bad tubes
becomes much better than the one in [8], allowing us to obtain improvement for
the Falconer problem in this case as well.

###  4.2. Removal of the bad region via the radial projection theorem

In this section, we apply the radial projection theorem (Theorem 4.1) to prove
Proposition 4.2 .

From the exact same deduction as in the proof of [8, Proposition 2.1] (more
precisely, Lemma 3.1), one has the bound

$\|d^{x}_{*}(\mu_{1})-d^{x}_{*}(\mu_{1,g})\|_{L^{1}}\lesssim\sum_{j\geq
1}R_{j}^{\beta d}\mu_{1}({\rm Bad}_{j}(x))+{\rm RapDec}(R_{0}),$

where

${\rm Bad_{j}}(x):=\bigcup_{T\in\mathbb{T}_{j}:\,x\in 2T\textrm{ and $T$ is
bad}}2T,\quad\forall j\geq 1.$

We also denote

$\begin{split}\text{Bad}_{j}:=&\\{(y,x)\in E_{1}\times
E_{2}:\,y\in\text{Bad}_{j}(x)\\}\\\ =&\\{(y,x)\in E_{1}\times
E_{2}:\,\exists\text{ bad }T\in\mathbb{T}_{j}\text{ s.t. }x,y\in
2T\\}.\end{split}$

The goal is then to obtain decay for $\mu_{1}({\rm Bad}_{j}(x))$, $\forall
j\geq 1$. We have the following estimate, from which Proposition 4.2 follows.

######  Lemma 4.4.

There exist sufficiently large $R_{0}$ and sufficiently small $\beta>0$ such
that there is a subset $E_{2}^{\prime}\subset E_{2}$ so that
$\mu_{2}(E_{2}^{\prime})\geq 1-\frac{1}{1000}$ and for each $x\in
E_{2}^{\prime}$,

$\mu_{1}({\rm Bad}_{j}(x))\lesssim R_{j}^{-100\beta d},\quad\forall j\geq 1.$

###### Proof.

Fix a scale $j\geq 1$. Our goal is to show that

(4.2) $\mu_{1}\times\mu_{2}(\text{Bad}_{j})\lesssim R_{j}^{-200\beta d}.$

To see how this would imply the desired estimate, one first rewrites

$\mu_{1}\times\mu_{2}(\text{Bad}_{j})=\int\mu_{1}(\text{Bad}_{j}(x))\,d\mu_{2}(x).$

Then, one can find a set $F_{j}\subset E_{2}$ with $\mu_{2}(F_{j})\leq
R_{j}^{-50\beta d}$ such that $\mu_{1}(\text{Bad}_{j}(x))\leq R_{j}^{-150\beta
d}$ for $x\in E_{2}\setminus F_{j}$. We take
$E_{2}^{\prime}:=E_{2}\setminus\bigcup_{j\geq 1}F_{j}$. Observe that
$\mu_{2}(E_{2}^{\prime})\geq\mu(E_{2})-\sum_{j\geq 1}R_{j}^{-50\beta
d}>1-R_{0}^{-\beta}>1-\frac{1}{1000}$ if $R_{0}$ is sufficiently large, hence
the desired result would follow.

To prove (4.2), one recalls that $\delta=2R_{j}^{-\frac{1}{2}+\beta}$. For
each bad $T\in\mathbb{T}_{j}$, let $r(T)$ be the parameter as in the
definition of bad tubes. Since there are only
$\sim\log\frac{1}{\delta}\sim\log R_{j}$ many choices of $r(T)$, one can
assume that $r(T)$ are the same for bad $T\in\mathbb{T}_{j}$.

One first decomposes

$\text{Bad}_{j}=\text{Bad}^{1}_{j}\cup\text{Bad}^{2}_{j},$

where for $k=1,2$,

$\text{Bad}^{k}_{j}:=\\{(y,x)\in E_{1}\times E_{2}:\,\exists
T\in\mathcal{T}_{j}^{k}\text{ s.t. }x,y\in 2T\\},$

with

$\mathcal{T}_{j}^{1}:=\\{T\in\mathbb{T}_{j}:\,T\text{ is
bad},\,C_{0}\delta\leq r(T)\leq 10C_{0}\delta\\},$
$\mathcal{T}_{j}^{2}:=\\{T\in\mathbb{T}_{j}:\,T\text{ is
bad},\,r(T)>10C_{0}\delta\\}.$

The part $\text{Bad}_{j}^{1}$ is simpler and doesn’t require Theorem 4.1.
Indeed, all the bad tubes in this case satisfies that
$\mu_{2}(4T)\gtrsim\delta^{m-1-2\varepsilon}$. Since $m-1<\alpha\leq m$, one
can follow the exact same argument in [8, justification of (3.1)] to project
the sets $E_{1},E_{2}$ onto an $m$-dimensional subspace and apply the radial
projection theorem of Orponen there. We omit the details.

Next, to estimate $\text{Bad}_{j}^{2}(y)$, define $r=\frac{r(T)}{8}$ and apply
the new radial projection theorem (Theorem 4.1) with the $\delta$ and $r$
specified in the above, and with the roles of $x,y$ interchanged. Note that
the parameter $\eta$ is the same as the one in the definition of $r(T)$.

Then, one has a set $B\in E_{1}\times E_{2}$ and $\gamma>0$ such that
$\mu_{1}\times\mu_{2}(B)\leq\delta^{\gamma}$ and that for each $y\in E_{1}$
and $\delta$-tube $\tilde{T}$ through $y$, one has

$\mu_{2}(\tilde{T}\setminus(A|_{y}\cup
B|_{y}))\lesssim\frac{\delta^{\alpha}}{r^{\alpha-(m-1)}}\delta^{-\varepsilon}.$

Recall that here

$\begin{split}A:=\\{(y,x)\in&E_{1}\times E_{2}:\,\\\ &x,y\in H,\,\text{ for
some $\delta^{\eta}$-concentrated $(r,m)$-plate $H$}\\}.\end{split}$

For any fixed $y\in E_{1}$ and
$T\in\mathcal{T}^{2}_{j}(y):=\\{T\in\mathcal{T}_{j}^{2}:\,y\in 2T\\}$, we
claim that $2T\cap A|_{y}=\emptyset$. Indeed, suppose there is a point $x\in
A|_{y}$ such that $x,y\in 2T$. Then, one has that $x,y\in H$ for some
$\delta^{\eta}$-concentrated $(r,m)$-plate $H$. Since $y\in E_{1}$ and $x\in
E_{2}$ are separated, for sufficiently large constant $C_{0}$ (depending on
${\rm dist}(E_{1},E_{2})$), one has that $2T\subset 2H$. By the construction
of collections $\\{\mathcal{E}_{r,m}\\}_{r}$, $2H$ is contained in some
$H^{\prime}\in\mathcal{E}_{4r,m}$. Obviously, $H^{\prime}$ is
$\delta^{\eta}$-concentrated, hence one concludes that $2T$ is contained in
some $H^{\prime}\in\mathcal{H}_{\frac{r(T)}{2}}$, which contradicts the
definition of $r(T)$. (Note that the assumption $r(T)>10C_{0}\delta$ in this
part is to guarantee that $\frac{r(T)}{2}\geq C_{0}\delta$.)

By taking $\beta$ sufficiently small depending on $\gamma$, it suffices to
estimate $\mu_{1}\times\mu_{2}(\text{Bad}^{2}_{j}\setminus B)$. Write

$\mu_{1}\times\mu_{2}(\text{Bad}^{2}_{j}\setminus
B)=\int\mu_{2}(\text{Bad}^{2}_{j}(y)\setminus(B|_{y}))\,d\mu_{1}(y)$

where

$\text{Bad}^{2}_{j}(y):=\\{x\in
E_{2}:\,(y,x)\in\text{Bad}^{2}_{j}\\},\quad\forall y\in E_{1}.$

Hence

$\text{Bad}^{2}_{j}(y)\setminus(B|_{y})=\bigcup_{T\in\mathcal{T}_{j}^{2}(y)}2T\setminus
B|_{y}.$

For any $y\in E_{1}$ fixed, since each bad tube $T$ satisfies
$\mu_{2}(4T)\geq\frac{\delta^{\alpha-2\varepsilon}}{r(T)^{\alpha-(m-1)}}$ and
each point in $E_{2}$ is contained in at most $\sim R_{j}^{d\beta}$ many $4T$
with $y\in 2T$, one has that

$|\mathcal{T}_{j}(y)|\frac{\delta^{\alpha-2\varepsilon}}{r(T)^{\alpha-(m-1)}}\leq\int\sum_{T\in\mathcal{T}_{j}(y)}\chi_{4T}\,d\mu_{2}\lesssim
R_{j}^{d\beta}\int\chi_{\bigcup_{T\in\mathcal{T}_{j}(y)}4T}\,d\mu_{2}\lesssim
R_{j}^{\beta d},$

hence

$|\mathcal{T}_{j}(y)|\lesssim R_{j}^{\beta
d}\frac{r(T)^{\alpha-(m-1)}}{\delta^{\alpha-2\varepsilon}}.$

We now have the estimate

$\mu_{2}({\rm
Bad}_{j}^{2}(y)\setminus(B|_{y}))\lesssim\frac{\delta^{\alpha}}{r^{\alpha-(m-1)}}\delta^{-\varepsilon}|\mathcal{T}_{j}(y)|\lesssim
R_{j}^{\beta d}\delta^{\varepsilon}.$

By choosing $\beta$ sufficiently small depending on $\varepsilon$, one has
that the above is $\lesssim\delta^{1000\beta d}\lesssim R_{j}^{-200\beta d}$.
The proof is complete by integrating this estimate in $y$.

∎

###  4.3. $L^{2}$ bound of the good part via weighted decoupling estimates

In this section, we prove Proposition 4.3, applying weighted refined
decoupling estimates. More precisely, we will apply Theorem 1.2(c). The
argument below proceeds very similarly as in [16, 8], except that we need to
reduce to the situation where the function has concentrated frequencies in
order to apply Theorem 1.2.

For $R>0$, let $\sigma_{R}$ denote the normalized surface measure on the
sphere of radius $R$ in $\mathbb{R}^{d}$. Our main estimate of the section is
the following.

######  Proposition 4.5.

Let $\alpha>0$ and $R>10R_{0}$, and let $\mu_{1,g}$ be as defined in the
above. Then, for all $\epsilon>0$, and $\beta,\varepsilon$ sufficiently small
depending on $\alpha,\epsilon$, there holds

(4.3) $\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lesssim
R^{-\Gamma(d,\alpha)+\epsilon}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi+{\rm
RapDec}(R),$

where

$\Gamma(d,\alpha)=\begin{cases}\frac{d\alpha}{d+1},&d\geq 4,\\\
\alpha-\frac{\alpha^{2}}{6},&d=3,\end{cases}$

and $\psi_{R}$ is a weight function which is $\sim 1$ on the annulus
$R-1\leq|\xi|\leq R+1$ and decays off of it. To be precise, we could take

$\psi_{R}(\xi)=\left(1+|R-|\xi||\right)^{-100}.$

It is a routine argument to check that Proposition 4.5 implies Proposition
4.3, for instance see [8, Proof of Proposition 2.2]. We include it here for
the sake of completeness.

Observe first that

$d^{x}_{*}(\mu_{1,g})(t)=t^{d-1}\mu_{1,g}*\sigma_{t}(x)\,.$

Since $\mu_{1,g}$ is essentially supported in the
$R_{0}^{-1/2+\beta}$-neighborhood of $E_{1}$, for $x\in E_{2}$, we only need
to consider $t\sim 1$. Hence, up to a loss of $\text{RapDec}(R_{0})$ which is
negligible in our argument, we have

$\begin{split}\int_{E_{2}}\|d_{*}^{x}(\mu_{1,g})\|_{L^{2}}^{2}\,d\mu_{2}(x)\lesssim&\int_{0}^{\infty}\int_{E_{2}}|\mu_{1,g}*\sigma_{t}(x)|^{2}\,d\mu_{2}(x)t^{d-1}\,dt\\\
\sim&\int_{0}^{\infty}\int_{E_{2}}|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)R^{d-1}\,dR,\end{split}$

where in the second step, we have used a limiting process and an
$L^{2}$-identity proved by Liu [17, Theorem 1.9].

For $R\leq 10R_{0}$, we use a simple estimate following from orthogonality
(for example, see [10, Lemma 5.2]):

$\int_{E_{2}}|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lesssim(R+1)^{d-1}R^{-(d-1)}\,.$

So the small $R$ contribution to
$\int\|d_{*}^{x}(\mu_{1,g})\|^{2}_{L^{2}}d\mu_{2}(x)$ is

$\displaystyle\int_{0}^{10R_{0}}(R+1)^{d-1}\,dR\lesssim R_{0}^{d}.$

Applying Proposition 4.5 for each $R>10R_{0}$ and dropping the rapidly
decaying tail as we may, one can bound the large $R$ contribution to
$\int\|d_{*}^{x}(\mu_{1,g})\|^{2}_{L^{2}}d\mu_{2}(x)$ by

$\begin{split}&\lesssim\int_{10R_{0}}^{\infty}\int_{\mathbb{R}^{d}}R^{-\Gamma(d,\alpha)+\epsilon}\psi_{R}(\xi)|\hat{\mu}_{1}(\xi)|^{2}\,d\xi
dR\\\
&\lesssim\int_{\mathbb{R}^{d}}|\xi|^{-\Gamma(d,\alpha)+\epsilon}|\hat{\mu}_{1}(\xi)|^{2}\,d\xi\sim
I_{\beta}(\mu_{1}),\end{split}$

where $\beta=d-\Gamma(d,\alpha)+\epsilon$, by a Fourier representation for
$I_{\beta}(\mu_{1})$, the $\beta$-dimensional energy of $\mu_{1}$. One thus
has $I_{\beta}(\mu_{1})<\infty$ if $\beta<\alpha$, which is equivalent to

$\alpha>\begin{cases}\frac{d}{2}+\frac{1}{4}-\frac{1}{4(2d+1)},&d\geq 4,\\\
\frac{3}{2}+\frac{1}{4}+\frac{17-12\sqrt{2}}{4},&d=3.\end{cases}$

The proof of Proposition 4.3 is thus complete.

We are now ready to prove Proposition 4.5. First, let’s recall the following
[19, Lemma 7.5], which controls the total number of essentially distinct
concentrated plates from a fixed scale.

######  Lemma 4.6.

Let $m-1<\alpha\leq m$. There is $N=N(\alpha,m)$ such that the following
holds: let $\nu$ be an $\alpha$-dimensional measure with constant $C_{\nu}\geq
1$ and let $\mathcal{H}=\\{H\in\mathcal{E}_{r,m}:\nu(H)\geq a\\}$. Then
$|\mathcal{H}|\lesssim(\frac{C_{\nu}}{a})^{N}$.

###### Proof of Proposition 4.5.

Fix $\epsilon>0$ and $R>10R_{0}$. By definition,

$\mu_{1,g}*\hat{\sigma}_{R}=\sum_{R_{j}\sim
R}\sum_{\tau}\sum_{T\in\mathbb{T}_{j,\tau}:\,T\textrm{
good}}M_{T}\mu_{1}*\hat{\sigma}_{R}+{\rm RapDec}(R).$

Since the contribution of ${\rm RapDec}(R)$ is already taken into account in
the statement of Proposition 4.5, we may ignore the tail ${\rm RapDec}(R)$ in
the argument below. The following reduction is the same as in [8].

Let $\eta_{1}$ be a bump function adapted to the unit ball and define

$f_{T}=\eta_{1}\left(M_{T}\mu_{1}*\hat{\sigma}_{R}\right).$

It is easy to see that $f_{T}$ is microlocalized to $(T,\theta(T))$ (similarly
as defined in Section 2, with $f$ essentially supported in $2T$ and $\hat{f}$
essentially supported in $2\theta(T)$, the block of dimension
$R^{1/2}\times\cdots\times R^{1/2}\times 1$ in the partition of the
$1$-neighborhood of $RS^{d-1}$ corresponding to the long direction of $T$).

We now apply a series of dyadic pigeonholing to the integral to be estimated.
First, there exists $\lambda>0$ such that

$\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lesssim\log
R\int|f_{\lambda}(x)|^{2}d\mu_{2}(x),$

where

$f_{\lambda}=\sum_{T\in\mathbb{W}_{\lambda}}f_{T},\quad\mathbb{W}_{\lambda}:=\bigcup_{R_{j}\sim
R}\bigcup_{\tau}\Big{\\{}T\in\mathbb{T}_{j,\tau}:T\text{ good
},\|f_{T}\|_{L^{p}}\sim\lambda\Big{\\}}.$

For each $R$, since there are only finitely many $R_{j}\sim R$, for the sake
of brevity we will drop the first union in $\mathbb{W}_{\lambda}$ and assume
without loss of generality that $R_{j}=R$ for all $T\in\mathbb{W}_{\lambda}$
from this point on.

In addition, considering a sequence of dyadic scales
$r\in[2C_{0}R^{-\frac{1}{2}+\beta},1]$, by dyadic pigeonholing, one can reduce
to a subcollection $\mathbb{W}_{\lambda,r}$ for a fixed $r$, where

$\mathbb{W}_{\lambda,r}:=\\{T\in\mathbb{W}_{\lambda}:\,r(T)=r\\}.$

Here, recall that

$r(T):=\min\\{{\rm dyadic}\,\,r\in[2C_{0}R^{-\frac{1}{2}+\beta},1]:\,\exists
H\in\mathcal{H}_{r}\,\,{\rm s.t.}\,2T\subset H\\},$

where $\mathcal{H}_{r}$ denotes a collection of
$(2R^{-\frac{1}{2}+\beta})^{\eta}$-concentrated $(r,m)$-plates of
$\mathcal{E}_{r,m}$.

Fixing such an $r$ from now on, one can further conclude from Lemma 4.6 that
there are at most $\sim R^{\frac{N\eta}{2}}$ different $H\in\mathcal{H}_{r}$.
Therefore, choosing $\eta\ll\frac{1}{N}$ (depending on $\varepsilon$), one can
assume that there is a fixed $(r,m)$-plate $H\in\mathcal{H}_{r}$ such that

$\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lessapprox\int|f_{\lambda,r,H}(x)|^{2}d\mu_{2}(x),$

where

$f_{\lambda,r,H}=\sum_{T\in\mathbb{W}_{\lambda,r,H}}f_{T},\quad\mathbb{W}_{\lambda,r,H}:=\\{T\in\mathbb{W}_{\lambda,r}:2T\subset
H\\}.$

By dyadic pigeonholing again, one is able to find $M>0$ such that the right
hand side of the inequality above is further dominated by

$\lessapprox\int_{Y_{M}}|f_{\lambda,r,H}(x)|^{2}d\mu_{2}(x),$

where the region $Y_{M}=\bigcup_{q\in\mathcal{Q}_{M}}q$ is contained in the
unit ball and

$\mathcal{Q}_{M}:=\\{R^{-1/2}\textrm{-cube }q:q\textrm{ intersects }\sim
M\textrm{ tubes }T\in\mathbb{W}_{\lambda,r,H}\\}.$

Fix $p\in[p_{d},p_{m}]$ and write $f=f_{\lambda,r,H}$, $Y=Y_{M}$, and
$\mathbb{W}=\mathbb{W}_{\lambda,r,H}$ for the sake of brevity. We first claim
that there holds the following $L^{p}$ estimate:

(4.4) $\begin{split}&\|f\|_{L^{p}(Y;\,\mu_{2})}\\\
\lessapprox&r^{\left(\frac{1}{p}-\frac{1}{p_{m}}\right)(d-\alpha)}(r^{2}R)^{\frac{d-1}{4}-\frac{d+1}{2p}}R^{\frac{d-\alpha}{p}}\left(\frac{M}{|\mathbb{W}|}\right)^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|^{2}_{L^{p}}\right)^{1/2}.\end{split}$

This is essentially a rescaled version of our new decoupling estimate, more
precisely, Theorem 1.2(c). We will leave the justification of it to the end of
the proof and proceed now assuming it holds true.

Since $f$ only involves good wave packets, by considering the quantity

$\sum_{q\in\mathcal{Q}_{M}}\sum_{T\in\mathbb{W}:T\cap
q\neq\emptyset}\mu_{2}(q),$

one obtains that

(4.5)
$M\mu_{2}(Y)\lesssim|\mathbb{W}|\frac{R^{(-\frac{1}{2}+\beta)(\alpha-2\varepsilon)}}{r^{\alpha-(m-1)}}.$

The quantity on the right hand side above follows from the threshold defining
good wave packets.

Hence, combining (4.4) and Hölder’s inequality, one has that

$\begin{split}&\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lessapprox\int_{Y}|f(x)|^{2}\,d\mu_{2}(x)\\\
\leq&\left(\int_{Y}|f(x)|^{p}\,d\mu_{2}(x)\right)^{\frac{2}{p}}\mu_{2}(Y)^{1-\frac{2}{p}}\\\
\lessapprox&r^{2\left(\frac{1}{p}-\frac{1}{p_{m}}\right)(d-\alpha)}(r^{2}R)^{\frac{d-1}{2}-\frac{d+1}{p}}R^{\frac{2(d-\alpha)}{p}}\left(\frac{R^{(-\frac{1}{2}+\beta)(\alpha-2\varepsilon)}}{r^{\alpha-(m-1)}}\right)^{1-\frac{2}{p}}\sum_{T\in\mathbb{W}}\|f_{T}\|^{2}_{L^{p}}\\\
=&r^{2\left(\frac{1}{p}-\frac{1}{p_{m}}\right)(d-\alpha)+d-1-\frac{2(d+1)}{p}-\left(1-\frac{2}{p}\right)(\alpha-m+1)}R^{(d-1-\alpha)(\frac{1}{2}+\frac{1}{p})+O_{\alpha}(\beta)+\varepsilon}\sum_{T\in\mathbb{W}}\|f_{T}\|^{2}_{L^{p}}.\end{split}$

From standard computation (see for instance [8, Proof of Lemma 4.1]), one has
the simple bound

$\begin{split}\|f_{T}\|_{L^{p}}\lesssim&\|f_{T}\|_{L^{\infty}}|T|^{1/p}\lesssim\sigma_{R}(\theta(T))^{1/2}|T|^{1/p}\|\widehat{M_{T}\mu_{1}}\|_{L^{2}(d\sigma_{R})}\\\
=&R^{-(\frac{1}{2p}+\frac{1}{4})(d-1)+O_{\alpha}(\beta)}\|\widehat{M_{T}\mu_{1}}\|_{L^{2}(d\sigma_{R})}.\end{split}$

Plugging this back into the above and applying orthogonality, one has that

$\begin{split}&\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\\\
\lessapprox&r^{2\left(\frac{1}{p}-\frac{1}{p_{m}}\right)(d-\alpha)+d-1-\frac{2(d+1)}{p}-\left(1-\frac{2}{p}\right)(\alpha-m+1)}\cdot\\\
&\qquad\qquad\qquad
R^{-\alpha(\frac{1}{2}+\frac{1}{p})+O_{\alpha}(\beta)+\varepsilon}\sum_{T\in\mathbb{W}}\|\widehat{M_{T}\mu_{1}}\|^{2}_{L^{2}(d\sigma_{R})}\\\
\lesssim&r^{\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}}R^{-\alpha(\frac{1}{2}+\frac{1}{p})+O_{\alpha}(\beta)+\varepsilon}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.\end{split}$

Thus, for sufficiently small $\beta,\varepsilon$ depending on $\alpha$ and
$\epsilon$,

$\begin{split}&\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\\\
\lesssim&r^{\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}}R^{-\alpha(\frac{1}{2}+\frac{1}{p})+\epsilon}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.\end{split}$

Since $r$ can be anything ranging between $R^{-\frac{1}{2}+\beta}$ and $1$,
one needs to maximize the right hand side of the above estimate over all
possible values of $r$. To do this, we first rewrite $r=R^{-\gamma}$, where
$0\leq\gamma\leq\frac{1}{2}$. Then, the estimate above becomes

(4.6) $\begin{split}&\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\\\
\lessapprox&R^{-\gamma\left(\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}\right)-\alpha(\frac{1}{2}+\frac{1}{p})+\epsilon}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi\\\
=&R^{-\gamma\left(\frac{2(d-\alpha)}{m+1}+m-2\right)-\frac{\alpha}{2}+\epsilon}R^{\frac{2m\gamma-\alpha}{p}}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.\end{split}$

First, we look at the case $d\geq 4$. Note that one can assume without loss of
generality that $\frac{d}{2}<\alpha<\frac{d}{2}+\frac{1}{3}$, hence
$m=\frac{d+1}{2}$ if $d$ is odd, and $m=\frac{d}{2}+1$ if $d$ is even. A quick
calculation then shows that $\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}>0$ for
all $p\in[p_{d},p_{m}]$. Therefore, the estimate is maximized at $\gamma=0$,
becoming

$\begin{split}&\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\\\
\lessapprox&R^{-\alpha(\frac{1}{2}+\frac{1}{p})+\epsilon}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.\end{split}$

This bound can then be minimized at $p=p_{d}$, hence one concludes that

(4.7) $\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lessapprox
R^{-\frac{d\alpha}{d+1}+\epsilon}R^{-(d-1)}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.$

The computation in the case $d=3$ ($m=2$) is a bit more complicated. Observing
(4.6), one sees that in the case $\frac{\alpha}{4}\leq\gamma\leq\frac{1}{2}$,
the optimal choice of $p$ is $p=p_{m}=6$, which gives

$\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lessapprox
R^{-\gamma(\frac{4}{3}-\frac{2\alpha}{3})-\frac{2\alpha}{3}+\epsilon}R^{-2}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.$

This bound is the worst when $\gamma=\frac{\alpha}{4}$, at which it becomes

(4.8) $\int|\mu_{1,g}*\hat{\sigma}_{R}(x)|^{2}\,d\mu_{2}(x)\lessapprox
R^{\frac{\alpha^{2}}{6}-\alpha+\epsilon}R^{-2}\int|\hat{\mu}_{1}|^{2}\psi_{R}\,d\xi.$

In the case $0\leq\gamma\leq\frac{\alpha}{4}$, (4.6) is optimized at
$p=p_{d}=4$, which is then maximized at $\gamma=\frac{\alpha}{4}$, producing
the same bound as (4.8).

In sum, one concludes that the desired estimate (4.3) holds true in all
dimensions $d\geq 3$.

We are left with the justification of estimate (4.4).

First, note that $f$ has Fourier support in the $1$-neighborhood of the sphere
of radius $R$, one has that

$\|f\|_{L^{p}(Y;\,\mu_{2})}^{p}\lesssim\|f\|_{L^{p}(Y;\,\mu_{2}\ast\eta_{\frac{1}{R}})}^{p}=R^{-d}\int_{RY}|f(R^{-1}y)|^{p}\mu_{2}\ast\eta_{\frac{1}{R}}(R^{-1}y)\,dy,$

where $\eta_{\frac{1}{R}}$ is a bump function of integral $1$ that is
essentially supported on the ball of radius $\frac{1}{R}$.

The above quantity can be rewritten as

$\sim R^{d-\alpha}R^{-d}\|F\|^{p}_{L^{p}(RY;\,Hdy)}\,,$

where $F(y):=f(R^{-1}y)$ and weight function
$H(y):=c_{1}R^{\alpha-d}\mu_{2}\ast\eta_{\frac{1}{R}}(R^{-1}y)$. It is easy to
see that Theorem 1.2(c) applies. Indeed, $F$ obviously satisfies all the
required conditions. From the observation that
$\|\mu_{2}\ast\eta_{\frac{1}{R}}\|_{L^{\infty}}\lesssim R^{d-\alpha}$, one can
choose a constant $c_{1}$ such that $H\leq 1$. Moreover, for any ball
$Q^{\prime}$ of radius $\frac{1}{r}$, the $\alpha$-dimensional condition of
$\mu_{2}$ and the fact that $\int\eta_{\frac{1}{R}}=1$ imply that
$\int_{Q^{\prime}}H(y)\,dy\lesssim r^{-\alpha}$, (in fact, we have this ball
condition for any radius $t>0$, but we only need this for $t=1/r$ in order to
apply Theorem 1.2(c)).

Therefore, Theorem 1.2(c) yields that

$\begin{split}&\|f\|_{L^{p}(Y;\mu_{2})}\\\
\lessapprox&R^{\frac{d-\alpha}{p}}R^{-\frac{d}{p}}r^{(d-\alpha)(\frac{1}{p}-\frac{1}{p_{m}})}(r^{2}R)^{\frac{d-1}{4}-\frac{d+1}{2p}}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|F_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}\\\
=&R^{\frac{d-\alpha}{p}}r^{(d-\alpha)(\frac{1}{p}-\frac{1}{p_{m}})}(r^{2}R)^{\frac{d-1}{4}-\frac{d+1}{2p}}M^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{p}\right)^{\frac{1}{p}},\end{split}$

where $F_{T}(y):=f_{T}(R^{-1}y)$. Then, recalling that we have pigeonholed at
the beginning to reduce to the case that all $\|f_{T}\|_{L^{p}}$ are roughly
constant, one thus concludes that the right hand side in the above, up to a
constant, coincides with the right hand side of the desired inequality (4.4).
The proof is complete.

∎

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Xiumin Du, Northwestern University<EMAIL_ADDRESS>

Yumeng Ou, University of Pennsylvania<EMAIL_ADDRESS>

Kevin Ren, Princeton University<EMAIL_ADDRESS>

Ruixiang Zhang, UC Berkeley<EMAIL_ADDRESS>