M\end{subarray}}\Big{\|}\big{(}\sum_{\theta\in\Theta(R^{-1/2})}|S_{\theta,\gamma}^{\lambda}f_{\theta}|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(B_{2R})}\lesssim\Big{\|}\big{(}\sum_{\theta\in\Theta(R^{-1/2})}|S_{\theta}^{\lambda}f_{\theta}|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(B_{2R})};$

here $B_{2R}$ is the $2R$-ball concentric to $B_{R}$. We now apply the wave
packet decomposition to $f$. By our scaling and combining (9.5) and Lemma 9.2,
we may assume that

$|S_{\theta}^{\lambda}f_{T}(\mathbf{x}_{B}+\mathbf{x})|\lesssim_{N,\varepsilon}R^{-\frac{n-1}{2}}a_{T}\chi_{T}(\mathbf{x})+R^{-N}\|f\|_{L^{\infty}(\mathbb{R}^{n-1})},\qquad\mathbf{x}\in
B(0,2R),T\in\mathbf{T}(R),$

where the coefficients $a_{T}\geq 0$ satisfy

$\sum_{T\in\mathbf{T}_{\theta}(R)}|a_{T}|^{2}=1\qquad\textrm{for all
$\theta\in\Theta(R^{-1/2})$.}$

Thus, by translating the variables,

$\Big{\|}\big{(}\sum_{\theta\in\Theta(R^{-1/2})}|S_{\theta}^{\lambda}f_{\theta}|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(B_{2R})}\lesssim
R^{-\frac{n-1}{2}}\Big{\|}\sum_{T\in\mathbf{T}(R)}|a_{T}|^{2}\chi_{T}\Big{\|}_{L^{s/2}(\mathbb{R}^{n})}^{1/2}$

holds up to a negligible error term. Combining this with the square function
estimate from Proposition 9.1, we deduce that

$\|S^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}\lesssim_{N,\varepsilon}R^{\frac{n-1}{4}(\frac{1}{2}-\frac{1}{s})-\frac{n-1}{2}+\frac{\varepsilon}{2}}\Big{\|}\sum_{T\in\mathbf{T}(R)}|a_{T}|^{2}\chi_{T}\Big{\|}_{L^{s/2}(\mathbb{R}^{n})}^{1/2}.$
(9.9)

The expression appearing in the norm on the right-hand side of (9.9) looks
similar to the multiplicity function appearing in the dual formulation of the
Kakeya maximal bound, except that it involves a sum over many parallel tubes.
To deal with this, we use an elementary probability trick that the authors
learned from the course notes of Tao [37]. For each
$\theta\in\Theta(R^{-1/2})$, we choose a tube
$T_{\theta}\in\mathbf{T}_{\theta}(R)$ independently at random, where each tube
$T\in\mathbf{T}_{\theta}(R)$ has probability $|a_{T}|^{2}$ of being selected.
It follows that

$\mathbb{E}\big{[}\chi_{T_{\theta}}(\mathbf{x})\big{]}=\sum_{T\in\mathbf{T}_{\theta}(R)}|a_{T}|^{2}\chi_{T}(\mathbf{x})$

and, by linearity of expectation,

$\mathbb{E}\Big{[}\sum_{\theta\in\Theta(R^{-1/2})}\chi_{T_{\theta}}(\mathbf{x})\Big{]}=\sum_{T\in\mathbf{T}(R)}|a_{T}|^{2}\chi_{T}(\mathbf{x})$
(9.10)

By combining (9.10) and Minkowski’s inequality,

$\Big{\|}\sum_{T\in\mathbf{T}(R)}|a_{T}|^{2}\chi_{T}\Big{\|}_{L^{s/2}(\mathbb{R}^{n})}\leq\mathbb{E}\Big{\|}\sum_{\theta\in\Theta(R^{-1/2})}\chi_{T_{\theta}}\Big{\|}_{L^{s/2}(\mathbb{R}^{n})}.$

However, by applying the dual form of the Kakeya inequality and scaling, we
have

$\Big{\|}\sum_{\theta\in\Theta(R^{-1/2})}\chi_{T_{\theta}}\Big{\|}_{L^{s/2}(\mathbb{R}^{n})}\lesssim
R^{\frac{2n}{s}+\varepsilon}\|\mathcal{K}^{R^{-1/2}}\|_{L^{r}(\mathbb{R}^{n})\to
L^{1}(\mathbb{B}^{n-1})}.$ (9.11)

Here we assume that the amplitude $a$ in the operator is sufficiently
localised, so that the resulting rescaled tubes can be parametrised by
$(y,\omega)\in Y_{\phi}\times\Omega_{\phi}$ as in §1.2. Thus, (9.9) and (9.11)
give

$\|S^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}\lesssim_{\varepsilon}R^{\frac{n-1}{4}(\frac{1}{2}-\frac{1}{s})-\frac{n-1}{2}+\frac{n}{s}+\varepsilon}\|\mathcal{K}^{R^{-1/2}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{1}(\mathbb{B}^{n-1})}^{1/2}.$ (9.12)

Unpacking the statement $\mathbf{K}_{p\to
1}\big{(}\phi;\beta(n,p)-\eta\big{)}$, we have

$\|\mathcal{K}^{R^{-1/2}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{1}(\mathbb{B}^{n-1})}^{1/2}\lesssim_{\phi,a}R^{\beta(n,p)/4-\eta/4}=R^{\frac{n-1}{4}(\frac{1}{2}-\frac{1}{s})-\frac{\eta}{4}}.$

Combining this with (9.12), simplifying the resulting $R$ exponent using (9.8)
and recalling our choice of normalisation for $f$, we conclude that

$\|S^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}\lesssim_{\varepsilon}R^{\alpha_{\mathrm{H}}(n,s)-\frac{\eta}{4}+\varepsilon}\|f\|_{L^{\infty}(\mathbb{B}^{n-1})}.$

The desired result now follows from the definition of $\mathbf{H}_{\infty\to
s}(\phi;\alpha_{\mathrm{H}}(n,s)-\eta/4+\varepsilon)$.

II) Now assume $\mathbf{N}_{p\to p}\big{(}\phi;\beta(n,p)-\eta\big{)}$. We fix
$R:=\lambda$, write
$U_{\theta,\gamma}^{\lambda}:=U^{\lambda}[\phi;a_{\theta,\gamma}]$ and let
$P_{\theta}$ denote the Littlewood–Paley projector with multiplier
$\psi_{\theta}$ for each $\theta\in\Theta(\lambda^{-1/2})$. Fixing
$f\in\mathcal{S}(\mathbb{R}^{n-1})$, we may apply the square function estimate
to deduce that

$\|U^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}\lesssim_{N,\varepsilon}\lambda^{\frac{n-1}{4}(\frac{1}{2}-\frac{1}{s})+\frac{\varepsilon}{2}}\sum_{\begin{subarray}{c}\gamma\in\mathbb{N}_{0}^{n}\\\
|\gamma|\leq
M\end{subarray}}\Big{\|}\big{(}\sum_{\theta\in\Theta(\lambda^{-1/2})}|U_{\theta,\gamma}^{\lambda}\circ
P_{\theta}f|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(\mathbb{R}^{n})}$ (9.13)

holds up to the inclusion of a rapidly decaying error term.

As in the proof of part I), we apply pigeonholing to find some choice of
$\gamma_{\circ}\in\mathbb{N}^{n}_{0}$, depending on $f$, so that
$U_{\theta}^{\lambda}:=U_{\theta,\gamma_{\circ}}^{\lambda}$ satisfies

$\sum_{\begin{subarray}{c}\gamma\in\mathbb{N}_{0}^{n}\\\ |\gamma|\leq
M\end{subarray}}\Big{\|}\big{(}\sum_{\theta\in\Theta(R^{-1/2})}|U_{\theta,\gamma}^{\lambda}\circ
P_{\theta}f|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(\mathbb{R}^{n})}\lesssim\Big{\|}\big{(}\sum_{\theta\in\Theta(R^{-1/2})}|U_{\theta}^{\lambda}\circ
P_{\theta}f|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(\mathbb{R}^{n})}.$ (9.14)

Let $K^{\lambda}_{\theta}$ denote the kernel of $U_{\theta}^{\lambda}$ for
each $\theta\in\Theta(R^{-1/2})$. By Lemma 9.3, we have
$\|K^{\lambda}_{\theta}(\mathbf{x};\,\cdot\,)\|_{L^{1}(\mathbb{R}^{n-1})}\lesssim
1$ and so, by Cauchy–Schwarz,

$|U_{\theta}^{\lambda}\circ
P_{\theta}f(\mathbf{x})|^{2}\lesssim\int_{\mathbb{R}^{n-1}}|K^{\lambda}_{\theta}(\mathbf{x};\omega)||P_{\theta}f(\omega)|^{2}\,\mathrm{d}\omega.$

For each $N\in\mathbb{N}$ we define a maximal function

$\mathcal{N}_{N}^{\lambda^{-1/2}}g(\omega):=\max_{\begin{subarray}{c}\theta\in\Theta(\lambda^{-1/2})\\\
y_{\theta}\in
Y_{\phi}\end{subarray}}\lambda^{-\frac{n+1}{2}}\int_{B(0,\rho\lambda)}\big{(}1+\lambda^{-1/2}|\partial_{y}\phi^{\lambda}(\mathbf{x};y_{\theta})-\omega|\big{)}^{-N}|g(\mathbf{x})|\,\mathrm{d}\mathbf{x}$

for $g\in L^{p}(\mathbb{R}^{n})$. Here $Y_{\phi}$ is the set appearing in §1.2
and $0<\rho<1$ depends on the size of the support of the underlying amplitude
$a$. By combining (9.13) and (9.14) with a standard duality argument and Lemma
9.3, we obtain

$\|U^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}^{2}\lesssim_{N,\varepsilon}\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{s})+1+\varepsilon}\int_{\mathbb{R}^{n-1}}\sum_{\theta\in\Theta(\lambda^{-1/2})}|P_{\theta}f(\omega)|^{2}\mathcal{N}_{N}^{\lambda^{-1/2}}g(\omega)\,\mathrm{d}\omega$
(9.15)

for some choice of $g\in L^{p}(\mathbb{R}^{n})$ satisfying
$\|g\|_{L^{p}(\mathbb{R}^{n})}\leq 1$. Note that we pick up an extra factor of
$\lambda$ owing to the normalisation in the kernel estimate in Lemma 9.3 In
addition, here (as in the proof of I) we assume that the amplitude $a$ in the
operator is sufficiently localised.

It is well-known (see, for instance, [33]) that

$\Big{\|}\big{(}\sum_{\theta\in\Theta(\lambda^{-1/2})}|P_{\theta}f|^{2}\big{)}^{1/2}\Big{\|}_{L^{s}(\mathbb{R}^{n-1})}\lesssim\|f\|_{L^{s}(\mathbb{R}^{n-1})}.$
(9.16)

Thus, matters reduce to showing

$\|\mathcal{N}_{N}^{\lambda^{-1/2}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{p}(B^{n-1}(0,\lambda))}\lesssim_{N,\varepsilon}\lambda^{\frac{2}{s}-1+\varepsilon}\|\mathcal{N}^{\lambda^{-1/2}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{p}(\mathbb{B}^{n-1})}$ (9.17)

for $N$ sufficiently large. Indeed, once we have (9.17), we may apply Hölder’s
inequality, (9.16) and (9.17) to estimate the right-hand side of (9.15),
giving

$\|U^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}\lesssim\lambda^{\frac{n-1}{4}(\frac{1}{2}-\frac{1}{s})+\frac{1}{s}+\varepsilon}\|\mathcal{N}^{\lambda^{-1/2}}\|_{L^{p}(\mathbb{R}^{n})\to
L^{p}(\mathbb{B}^{n-1})}^{1/2}\|f\|_{L^{s}(\mathbb{R}^{n-1})}.$

Unpacking the definition of $\mathbf{N}_{p\to
p}\big{(}\phi;\beta(n,p)-\eta\big{)}$, we deduce that

$\|U^{\lambda}f\|_{L^{s}(\mathbb{R}^{n})}\lesssim\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{s})+\frac{1}{s}-\frac{\eta}{4}+\varepsilon}\|f\|_{L^{s}(\mathbb{R}^{n-1})}.$

The desired result now follows from the definition of
$\mathbf{LS}_{s}(\phi;\alpha_{\mathrm{LS}}(n,s)-\eta/4+\varepsilon)$.

Turning to the proof of (9.17), by choosing $N$ large enough we can ensure

$\mathcal{N}_{N}^{\lambda^{-1/2}}g(\omega)\lesssim\sup_{y\in\mathbb{B}^{n-1}}\fint_{T_{y,\omega}}|g|+\lambda^{-10n}\|g\|_{L^{r}(\mathbb{R}^{n})}$

where $T_{y,\omega}$ is the $\lambda$-tube with error $\varepsilon$ as defined
in (9.4) and $\omega\in B(0,\lambda)$. Note that any $\lambda$-tube with error
$\varepsilon$ can be covered by $O(\lambda^{\varepsilon/100})$ parallel
$\lambda$-tubes with error $0$. Combining this observation with a simple
rescaling, and recalling that $-1/p=2/s-1$, we obtain the desired estimate
(9.17). ∎

###  9.6. The local-to-global principle

The final ingredient of the proof of Proposition 2.10 is the following ‘local-
to-global’ lemma.

######  Lemma 9.5 (Bourgain [8]).

Let $2\leq s\leq q(n)$, $\alpha s<\tfrac{n+1}{2}$ and

$q(n,s,\alpha):=2+\frac{2s}{n+1-2\alpha s}.$

For $q>q(n,s,\alpha)$ and $\phi\colon\mathbb{D}^{n}\to\mathbb{R}$ a Hörmander-
type phase,

$\mathbf{H}_{\infty\to s}(\phi;\alpha)\Rightarrow\mathbf{H}_{\infty\to
q}(\phi).$

The $n=3$ case of Lemma 9.5 is implicit in [8]. For completeness, we present a
full proof, following the general scheme of [8].

###### Proof (of Lemma 9.5).

Fix $q>q(n,s,\alpha)$ and let $f\in L^{\infty}(\mathbb{B}^{n-1})$. Let
$\mathfrak{R}z$ denote the real part of $z\in\mathbb{C}$. For $\sigma>0$,
define the superlevel set

$F_{\sigma}:=\big{\\{}\mathbf{x}\in\mathbb{R}^{n}:\mathfrak{R}(S^{\lambda}f(\mathbf{x}))>\sigma\big{\\}}.$

By Chebyshev’s inequality,

$\sigma|F_{\sigma}|\leq\mathfrak{R}\int_{F_{\sigma}}S^{\lambda}f(\mathbf{x})\,\mathrm{d}\mathbf{x}=\mathfrak{R}\langle
S^{\lambda}f,\chi_{F_{\sigma}}\rangle.$

Thus, by duality and the Cauchy–Schwarz inequality, we have

$\sigma|F_{\sigma}|\lesssim\|S^{\lambda,*}\chi_{F_{\sigma}}\|_{L^{2}(\mathbb{B}^{n-1})}\|f\|_{L^{2}(\mathbb{B}^{n-1})},$
(9.18)

where $S^{\lambda,*}$ is the adjoint operator

$S^{\lambda,*}g(y):=\int_{B^{n}}e^{-i\phi^{\lambda}(\mathbf{x};y)}\overline{a^{\lambda}(\mathbf{x};y)}g(\mathbf{x})\,\mathrm{d}\mathbf{x}.$

By expanding out the square of the $L^{2}$-norm in (9.18), we have

$\|S^{\lambda,*}\chi_{F_{\sigma}}\|_{L^{2}(\mathbb{B}^{n-1})}^{2}=\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}},\chi_{F_{\sigma}}\rangle$

where

$\mathcal{S}^{\lambda}g(\mathbf{x}):=S^{\lambda}S^{\lambda,*}g(\mathbf{x})=\int_{\mathbb{R}^{n}}K^{\lambda}(\mathbf{x},\mathbf{z})g(\mathbf{z})\,\mathrm{d}\mathbf{z}$

for the kernel

$K^{\lambda}(\mathbf{x},\mathbf{z}):=\int_{\mathbb{R}^{n-1}}e^{i(\phi^{\lambda}(\mathbf{x};y)-\phi^{\lambda}(\mathbf{z};y))}a^{\lambda}(\mathbf{x};y)\overline{a^{\lambda}(\mathbf{z};y)}\,\mathrm{d}y.$

The hypotheses H1) and H2), together with a stationary phase argument, imply
that

$|K^{\lambda}(\mathbf{x},\mathbf{z})|\lesssim(1+|\mathbf{x}-\mathbf{z}|)^{-(n-1)/2}.$
(9.19)

We now perform a Whitney decomposition in $\mathbb{R}^{n}\times\mathbb{R}^{n}$
with respect to the diagonal
$\\{(\mathbf{x},\mathbf{x}):\mathbf{x}\in\mathbb{R}^{n}\\}$. In particular,
for each $k\in\mathbb{Z}$ we let $\mathcal{Q}_{k}$ denote the set of dyadic
cubes in $\mathbb{R}^{n}$ of side-length $2^{k}$. Given
$Q_{1},Q_{2}\in\mathcal{Q}_{k}$, we write $Q_{1}\sim Q_{2}$ if $Q_{1}$ and
$Q_{2}$ are disjoint, but their parent intervals are not disjoint. It then
follows that

$1=\sum_{k\in\mathbb{Z}}\sum_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathcal{Q}_{k}\\\
Q_{1}\sim
Q_{2}\end{subarray}}\chi_{Q_{1}}(\mathbf{x})\chi_{Q_{2}}(\mathbf{z})$

and so we may decompose

$\|S^{\lambda,*}\chi_{F_{\sigma}}\|_{L^{2}(\mathbb{B}^{n-1})}^{2}=\sum_{k\in\mathbb{Z}}\sum_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathcal{Q}_{k}\\\
Q_{1}\sim Q_{2}\end{subarray}}\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap
Q_{1}},\chi_{F_{\sigma}\cap Q_{2}}\rangle.$ (9.20)

Fix $k\in\mathbb{Z}$ and $Q_{1},Q_{2}\in\mathcal{Q}_{k}$ satisfying $Q_{1}\sim
Q_{2}$. Using (9.19), we may estimate

$\displaystyle|\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap
Q_{1}},\chi_{F_{\sigma}\cap Q_{2}}\rangle|$
$\displaystyle\leq\sup_{\begin{subarray}{c}\mathbf{x}\in Q_{2}\\\
\mathbf{z}\in
Q_{1}\end{subarray}}|K^{\lambda}(\mathbf{x},\mathbf{z})||F_{\sigma}\cap
Q_{1}||F_{\sigma}\cap Q_{2}|$ $\displaystyle\lesssim
2^{-k(n-1)/2}|F_{\sigma}\cap Q_{1}||F_{\sigma}\cap Q_{2}|,$

since the hypothesis $Q_{1},Q_{2}\in\mathcal{Q}_{k}$ implies
$\mathrm{dist}(Q_{1},Q_{2})\gtrsim 2^{k}$. Summing this bound therefore yields

$\sum_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathcal{Q}_{k}\\\ Q_{1}\sim
Q_{2}\end{subarray}}|\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap
Q_{1}},\chi_{F_{\sigma}\cap Q_{2}}\rangle|\lesssim
2^{-k(n-1)/2}M|F_{\sigma}|\max_{Q\in\mathcal{Q}_{k}}|F_{\sigma}\cap Q|,$
(9.21)

where
$M:=\max_{Q_{1}\in\mathcal{Q}_{k}}\\#\\{Q_{2}\in\mathcal{Q}_{k}:Q_{1}\sim
Q_{2}\\}$ satisfies $M\lesssim 1$.

On the other hand, using Cauchy–Schwarz, we may bound

$|\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap Q_{1}},\chi_{F_{\sigma}\cap
Q_{2}}\rangle|\leq\|S^{\lambda,*}\chi_{F_{\sigma}\cap
Q_{1}}\|_{L^{2}(\mathbb{B}^{n-1})}\|S^{\lambda,*}\chi_{F_{\sigma}\cap
Q_{2}}\|_{L^{2}(\mathbb{B}^{n-1})}.$

We now apply Hörmander’s $L^{2}$ estimate to deduce that

$|\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap Q_{1}},\chi_{F_{\sigma}\cap
Q_{2}}\rangle|\lesssim 2^{k}|F_{\sigma}\cap Q_{1}|^{1/2}|F_{\sigma}\cap
Q_{2}|^{1/2}$

Summing this bound and multiple applications of Cauchy–Schwarz therefore
yields

$\sum_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathcal{Q}_{k}\\\ Q_{1}\sim
Q_{2}\end{subarray}}|\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap
Q_{1}},\chi_{F_{\sigma}\cap Q_{2}}\rangle|\lesssim 2^{k}M|F_{\sigma}|.$ (9.22)

Assume $\mathbf{H}_{\infty\to s}(\phi;\alpha)$, where $\alpha$ and $s$ satisfy
the hypotheses of the lemma. Let $E\subseteq\mathbb{B}^{n-1}$ be measurable
and take $f:=i^{m}\chi_{E}$ in the general framework, where $i$ is the
imaginary unit and $0\leq m\leq 3$. We estimate the local quantity
$|F_{\sigma}\cap Q|$ appearing in (9.21) using the hypothesised local
estimate. In particular, by unpacking the definition of $\mathbf{H}_{\infty\to
s}(\phi;\alpha)$ and Chebyshev’s inequality,

$\sigma^{s}|F_{\sigma}\cap
Q|\leq\|S^{\lambda}\chi_{E}\|_{L^{s}(Q)}^{s}\lesssim
2^{ks\alpha}\qquad\textrm{for all $Q\in\mathcal{Q}_{k}$.}$

Combining this with (9.21), we obtain

$\sum_{\begin{subarray}{c}Q_{1},Q_{2}\in\mathcal{Q}_{k}\\\ Q_{1}\sim
Q_{2}\end{subarray}}|\langle\mathcal{S}^{\lambda}\chi_{F_{\sigma}\cap
Q_{1}},\chi_{F_{\sigma}\cap Q_{2}}\rangle|\lesssim
2^{-k(\frac{n-1}{2}-s\alpha)}\sigma^{-s}M|F_{\sigma}|.$ (9.23)

Together, (9.20), (9.23) and (9.22), and the fact that $M\lesssim 1$, imply
that

$\|S^{\lambda,*}\chi_{F_{\sigma}}\|_{L^{2}(\mathbb{B}^{n-1})}^{2}\lesssim\sum_{k\in\mathbb{Z}}\min\\{2^{-k(\frac{n-1}{2}-s\alpha)}\sigma^{-s},2^{k}\\}|F_{\sigma}|.$
(9.24)

Let $k_{0}\in\mathbb{Z}$ be such that, for $k=k_{0}$, the two terms in the
minimum are comparable; this is equivalent to choosing

$2^{k_{0}}\sim\sigma^{-q+2+\eta}\qquad\textrm{where}\quad\eta:=q-q(n,s,\alpha)>0.$

By partitioning the range of summation on the right-hand side of (9.24)
according to whether $k>k_{0}$ or $k\leq k_{0}$, we deduce that

$\|S^{\lambda,*}\chi_{F_{\sigma}}\|_{L^{2}(\mathbb{B}^{n-1})}^{2}\lesssim\sigma^{-q+2+\eta}|F_{\sigma}|.$

Finally, we combine the above inequality with (9.18) to obtain

$\sigma^{2}|F_{\sigma}|^{2}\lesssim\sigma^{-q+2+\eta}|F_{\sigma}|,$

which rearranges to give

$|F_{\sigma}|\lesssim\sigma^{-q+\eta}.$ (9.25)

Note that $\|S^{\lambda}\chi_{E}\|_{\infty}\lesssim_{a}1$, where $a$ denotes
the amplitude associated to $S^{\lambda}$. Thus, in (9.25) it suffices to
consider $0<\sigma\lesssim_{a}1$ only, since otherwise the left-hand side is
zero. Since $\eta>0$, the bound (9.25) therefore implies

$|\\{\mathbf{x}\in\mathbb{R}^{n}:|S^{\lambda}\chi_{E}(\mathbf{x})|>\sigma\\}|\lesssim_{\phi,a,q}\sigma^{-q}\qquad\textrm{for
all $\sigma>0$.}$

The desired result now follows by restricted weak-type interpolation. ∎

###  9.7. Concluding the proof of Proposition 2.10

It remains to combine our earlier lemmas to conclude the proof of the
geometric reduction.

###### Proof (of Proposition 2.10).

Fix $p(n)<p<\infty$ and $0\leq\beta<\beta(n;p)$, so that
$\eta:=\beta(n,p)-\beta>0$. Let $2<s<q(n)$ satisfy $p=(s/2)^{\prime}$.

I) Take $\alpha:=\alpha_{\mathrm{H}}(n,s)-\eta/4$, where
$\alpha_{\mathrm{H}}(n,s)$ is as in (9.8), so that

$n+1-2\alpha s=\frac{(n-1+\eta)s}{2}>0.$

It follows that the exponent $q(n,s,\alpha)$ as featured in Lemma 9.5
satisfies

$q(n,s,\alpha):=2+\frac{4}{n-1+\eta}<q(n)=2+\frac{4}{n-1}.$

Furthermore, $\alpha s<\frac{n+1}{2}$ for $n\geq 3$. Combining Lemma 9.4 and
Lemma 9.5, given any $q(n,s,\alpha)<q<q(n)$ we may choose $\varepsilon>0$
sufficiently small so that

$\mathbf{K}_{p\to 1}(\phi;\beta)\Rightarrow\mathbf{H}_{\infty\to
s}(\phi;\alpha_{\mathrm{H}}(n,s)-\eta/4+\varepsilon)\Rightarrow\mathbf{H}_{\infty\to
q}(\phi).$

This concludes the proof of I).

II) Taking $\alpha:=\alpha_{\mathrm{H}}(n,s)-\eta/8$, Lemma 9.4 already
ensures that

$\mathbf{N}_{p\to p}(\phi;\beta)\Rightarrow\mathbf{LS}_{s}(\phi;\alpha),$

which concludes the proof of II). ∎

##  Appendix A Tools from real algebraic geometry

For the reader’s convenience, here we recall the definitions and results from
real algebraic geometry that play a role in our arguments in §4.

######  Definition A.1.

A set $S\subset\mathbb{R}^{n}$ is _semialgebraic_ if there exists a finite
collection of polynomials $P_{i,j}$,
$Q_{i,j}\colon\mathbb{R}^{n}\to\mathbb{R}$ for $1\leq i\leq r$, $1\leq j\leq
s$ such that

$S=\bigcup_{i=1}^{r}\big{\\{}x\in\mathbb{R}^{n}:P_{i,1}(x)=\cdots=P_{i,s}(x)=0,\,Q_{i,1}(x)>0,\dots,Q_{i,s}(x)>0\big{\\}}.$
(A.1)

Given a semialgebraic set $S\subset\mathbb{R}^{n}$ the complexity of $S$ is

$\inf\Big{(}\sum_{i,j}\deg P_{i,j}+\deg Q_{i,j}\Big{)}$

where the infimum is taken over all possible representations of $S$ of the
form (A.1).

######  Definition A.2.

Let $S\subseteq\mathbb{R}^{n}$ and $T\subseteq\mathbb{R}^{m}$ be semialgebraic
sets. We say $f\colon T\rightarrow S$ is a semialgebraic map if its graph is a
semialgebraic subset of $\mathbb{R}^{m+n}$. The dimension of $S$ is the
largest $d\in\mathbb{N}_{0}$ such that there exists an injective semialgebraic
map from $(0,1)^{d}$ to $S$.

We briefly recount three key results concerning semialgbraic sets, used in §4.

#### Wongkew’s theorem

We make considerable use of the following theorem of Wongkew [40], which
bounds the volume of neighbourhoods of algebraic varieties.

######  Theorem A.3 (Wongkew [40]).

Suppose $\mathbf{Z}$ is an $m$-dimensional variety in $\mathbb{R}^{n}$ with
$\deg\mathbf{Z}\leq d$. For any $0<\rho\leq\lambda$ and $\lambda$-ball
$B_{\lambda}$ the neighbourhood $N_{\\!\rho}(\mathbf{Z}\cap B_{\lambda})$ can
be covered by $O_{d}((\lambda/\rho)^{m})$ balls of radius $\rho$.

#### The Tarski–Seidenberg projection theorem

A fundamental result in the theory of semialgebraic sets is the
Tarski–Seidenberg projection theorem, which is also referred to as quantifier
elimination. A useful reference for this material is [1].

######  Theorem A.4 (Tarski–Seidenberg).

Let $\Pi$ be the orthogonal projection of $\mathbb{R}^{n}$ into its first
$n-1$ coordinates. Then for every $E\geq 1$, there is a constant $C(n,E)>0$ so
that, for every semialgebraic $S\subset\mathbb{R}^{n}$ of complexity at most
$E$, the projection $\Pi(S)$ has complexity at most $C(n,E)$.

We repeatedly use Theorem A.4 to form semialgebraic sections.

######  Corollary A.5.

Let $S\subset\mathbb{R}^{2n}$ be a compact semialgebraic set of complexity at
most $E$. Let $\Pi$ be the orthogonal projection into the final $n$
coordinates $(\mathbf{a},\mathbf{d})\mapsto\mathbf{d}.$ Then there is a
constant $C(n,E)>0$, depending only on $n$ and $E$, and a semialgebraic set
$Z$, of complexity at most $C(n,E)$, so that

$Z\subset S,\quad\quad\Pi(Z)=\Pi(S),$

and so that for each $\mathbf{d},$ there is at most one $\mathbf{a}$ with
$(\mathbf{a},\mathbf{d})\in Z.$

This is Lemma 2.2 from [28]. It is a direct consequence of Theorem A.4, as
discussed in [28].

#### Gromov’s algebraic lemma

The final key tool is the existence of useful parameterisations of
semialgebraic sets, as guaranteed by the following lemma.

######  Lemma A.6 (Gromov).

For all integers $E,n,r\geq 1$, there exists $M(E,n,r)<\infty$ with the
following properties. For any compact semialgebraic set $A\subset[0,1]^{n}$ of
dimension $m$ and complexity at most $E$, there exists an integer $N\leq
M(E,n,r)$ and $C^{r}$ maps
$\phi_{1},\dots,\phi_{N}:[0,1]^{m}\longrightarrow[0,1]^{n}$ such that

$\bigcup_{j=1}^{N}\phi_{j}([0,1]^{m})=A\quad\text{and}\quad\|\phi_{j}\|_{C^{r}}:=\max_{|\alpha|\leq
r}\|\partial^{\alpha}\phi_{j}\|_{\infty}\leq 1.$

This result was originally stated by Gromov. Detailed proofs were later given
by Pila and Wilkie [31] and Burguet [10].

### Acknowledgements

Shaoming Guo is partly supported by the Nankai Zhide Foundation, and partly
supported by NSF-2044828. James Wright is supported by a Leverhulme Research
Fellowship: project no. RF-2023-709$\backslash$9.

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