be red and blue $\Phi_{\sigma}-$waves with normalized energyb
$\mathbf{E}(F^{\sigma})=\mathbf{E}(G^{\sigma})=1$. For any $Q=Q_{R}$, by using
Proposition 4.3, we have for all $\mathbf{z}_{0}$

$\bigl{\|}F^{\sigma}G^{\sigma}\bigr{\|}_{Z\bigl{(}\mathcal{X}^{\varpi,k,r}_{\Phi_{\sigma},\mathbf{z}_{0}}(Q)\bigr{)}}\\\
\leq(1+C\varpi)\Bigl{\|}[\mathcal{F}^{\sigma}]_{k}[\mathcal{G}^{\sigma}]_{C_{0}}\Bigr{\|}_{Z(\mathcal{X}^{\varpi,k,r}_{\Phi_{\sigma},\mathbf{z}_{0}}(Q^{*})}+\varpi^{-C}\Bigl{(}1+\frac{R}{2^{k}r^{\prime}}\Bigr{)}^{-\frac{1}{N}}2^{k\gamma(q)}.$

We are reduced to showing

$\Bigl{\|}[\mathcal{F}^{\sigma}]_{k}[\mathcal{G}^{\sigma}]_{C_{0}}\Bigr{\|}_{Z(\mathcal{X}^{\varpi,k,r}_{\Phi_{\sigma},\mathbf{z}_{0}}(Q^{*}))}\\\
\leq(1+C\varpi)\mathscr{A}^{\Phi_{\sigma}}(R/C_{0},r^{\\#},r^{\prime})\;\mathbf{E}_{r,C_{0}Q}(F^{\sigma},G^{\sigma})^{1/q^{\prime}}+\varpi^{-C}R^{-N/2}2^{k\gamma(q)}.$
(5.5)

Using the definition of $\mathscr{A}^{\Phi_{\sigma}}(R,r,r^{\prime})$, we have
for all $\varDelta$

$\bigl{\|}\mathcal{F}^{\sigma,\varDelta}\mathcal{G}^{\sigma,\varDelta}\bigr{\|}_{Z(\mathcal{X}^{\varpi,k,r}_{\Phi_{\sigma},\mathbf{z}_{0}}(\varDelta))}\\\
\leq\mathscr{A}^{\Phi_{\sigma}}(R/C_{0},r^{\\#},r^{\prime})\,\mathbf{E}_{r^{\\#},C_{0}\varDelta}(\mathcal{F}^{\sigma,\varDelta},\mathcal{G}^{\sigma,\varDelta})^{1/q^{\prime}}(\mathbf{E}(\mathcal{F}^{\sigma,\varDelta})\mathbf{E}(\mathcal{G}^{\sigma,\varDelta}))^{1/(2q)}.$

By Lemma 5.4, Proposition 4.5 with $2^{-C_{0}}CR\ll C_{0}^{-1}R$ so that
$C_{0}\varDelta\subset 5Q$, we obtain by Cauchy-Schwarz’s inequality

$\Bigl{\|}[\mathcal{F}^{\sigma}]_{k}[\mathcal{G}^{\sigma}]_{C_{0}}\Bigr{\|}^{q}_{Z(\mathcal{X}^{\varpi,k,r}_{\Phi_{\sigma},\mathbf{z}_{0}}(Q^{*}))}\leq\sum_{\varDelta\in\mathcal{Q}_{C_{0}}(Q^{*})}\bigl{\|}\mathcal{F}^{\sigma,\varDelta}\mathcal{G}^{\sigma,\varDelta}\bigr{\|}^{q}_{Z(\mathcal{X}_{\Phi_{\sigma},\mathbf{z}_{0}}^{\varpi,k,r}(Q^{*}))}\\\
\leq(1+C\varpi)\mathscr{A}^{\Phi_{\sigma}}(R/C_{0},r^{\\#},r^{\prime})^{q}\,\,\mathbf{E}_{r,C_{0}Q}(F^{\sigma},G^{\sigma})^{q/q^{\prime}}+\varpi^{-C}R^{-qN/2}2^{k\gamma(q)},$
(5.6)

and (5.5) follows by adjusting the constant $C$. ∎

####  5.1.3. Step 3. Proof of Proposition 5.3

With Proposition 5.5 and 5.6, we may complete the proof of Proposition 5.3 by
using the standard iteration argument. Please see [15, P.225] for details. ∎

###  5.2. Essential concentration along conic regions

Fix a pair of red and blue waves
$(\mathscr{F}^{\sigma},\mathscr{G}^{\sigma})\in\mathfrak{R}^{\sigma}_{R}\times\mathfrak{B}^{\sigma,2^{k}}_{R}$
with $\mathbf{E}(\mathscr{F}^{\sigma})=\mathbf{E}(\mathscr{G}^{\sigma})=1$.
The complete proof relies crucially on the Kakeya compression property below.

Before stating this result, let us start with a non-endpoint bilinear
estimate. For $0<r_{1}<r_{2}<+\infty$, we define the cubical annulus as

$Q^{\mathsf{ann}}(x_{Q},t_{Q};r_{1},r_{2})=Q(x_{Q},t_{Q};r_{2})\setminus
Q(x_{Q},t_{Q};r_{1}).$

The following non-endpoint bilinear estimate holds for localized blue or red
waves, on a dyadic annulus corresponds to Lemma 11.1 of Tao [15], and we omit
the proof since it is rather standard using the wave-packets and a
multiplicity estimate on the overlappedness of tubes. See also [23] for more
details.

######  Lemma 5.7.

Let $R\geq 2^{C_{0}k}$, $C_{0}2^{k}\leq r\leq R^{\frac{1}{2}+\frac{4}{N}}$,
and $D=D(\mathbf{z}_{D},\,C_{0}^{1/2}r)$ with $\mathbf{z}_{D}=(x_{D},t_{D})$.
Then, there exists $b>0$, depending only on $\sigma_{0},\,Z$ and $n$, such
that for any red and blue $\Phi_{\sigma}-$waves $F^{\sigma},G^{\sigma}$ of
frequency $1$ and $2^{k}$ respectively, with
$\mathbf{E}(F^{\sigma})=\mathbf{E}(G^{\sigma})=1$, we have

$\|(P_{D}^{\Phi_{\sigma}}F^{\sigma})G^{\sigma}\|_{Z(Q^{\mathsf{ann}}(\mathbf{z}_{D};R,2R))},\|F^{\sigma}\,(P_{D}^{\Phi_{\sigma}}G^{\sigma})\|_{Z(Q^{\mathsf{ann}}(\mathbf{z}_{D};R,2R))}\lesssim
R^{-b},$ (5.7)

where the implicit constants are uniform with respect to
$\sigma\in(0,\sigma_{0}]$

The main result of this subsection reads

######  Proposition 5.8.

Let $\Phi_{\sigma}\in\mathscr{E}_{\sigma}$ and
$(\mathscr{F}^{\sigma},\mathscr{G}^{\sigma})\in\mathfrak{R}^{\Phi_{\sigma},1}_{R}\times\mathfrak{B}^{\Phi_{\sigma},2^{k}}_{R}$
be the pair of red and blue waves fixed at the beginning of this section with
$\mathbf{E}(\mathscr{F}^{\sigma})=\mathbf{E}(\mathscr{G}^{\sigma})=1$. There
exists a constant $C>0$ depending only on $n,\varepsilon,\sigma_{0}$ and
$q,s,$ in the $Z-$norm such that for any $R\geq 2^{C_{0}k}$ and
$\delta\in(0,1/2)$, if $Q_{R}$ satisfies

$\|\mathscr{F}^{\sigma}\,\mathscr{G}^{\sigma}\|_{Z(Q_{R})}\geq\frac{1}{2}\,\overline{A}^{\sigma}(R),$
(5.8)

and we let $r_{\delta}$ be the supremum of all radii $r\geq{C_{0}}2^{k}$ such
that

$\mathbf{E}_{r,C_{0}Q_{R}}(\mathscr{F}^{\sigma},\mathscr{G}^{\sigma})\leq
1-\delta$ (5.9)

holds and let $r_{\delta}={C_{0}}2^{k}$ if no such radius exists, then there
exists a cube $\widetilde{Q}_{\overline{R}_{\delta}}$ of size
$\overline{R}_{\delta}\in[2^{C_{0}k},R]$ and
$\mathbf{z}_{\delta}\in\mathbb{R}^{n+1}$ such that
$\overline{R}_{\delta}^{\frac{1}{2}+\frac{4}{N}}\leq r_{\delta}$ when
$r_{\delta}\geq 2^{C_{0}k}$, for which we have

$\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}\leq(1-C(\delta+C_{0}^{-C})^{q})^{-2/q}\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(\Omega_{\delta}^{\Phi_{\sigma}})}+\;2^{CC_{0}}2^{k\gamma(p,\varepsilon)}\,,$
(5.10)

where
$\Omega_{\delta}^{\Phi_{\sigma}}:=\widetilde{Q}_{\overline{R}_{\delta}}\cap\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta}$
with
$\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta}=\mathbf{\Lambda}^{\Phi_{\sigma}}_{\mathsf{purple}}(\mathbf{z}_{\delta},\,C_{0}(r_{\delta}+1))$.

####  5.2.1.  The medium or low concentration case: $r_{\delta}\geq
R^{1/2+4/N}$

In this case, we show there is a constant $C$ such that for some
$\mathbf{z}_{\delta}\in\mathbb{R}^{n+1}$, we have

$\bigl{\|}\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\bigr{\|}_{Z(Q_{R})}\leq\bigl{(}1-C(\delta+C_{0}^{-C})^{q}\bigr{)}^{-1/q}\bigl{\|}\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\bigr{\|}_{Z(Q_{R}\cap\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta})}$
(5.11)

holds with $C$ independent of $Q_{R}$ and $\mathbf{z}_{\delta}$. In this case,
$\overline{R}_{\delta}=R$ and $\widetilde{Q}_{\overline{R}_{\delta}}=Q_{R}$.

By definition, there is $D_{\delta}=D(\mathbf{z}_{\delta},r_{\delta})$ with
$\mathbf{z}_{\delta}=(x_{0},t_{0})$ and $t_{0}\in\mathsf{LS}(C_{0}Q_{R})$,
such that we have

$\min\Bigl{(}\|\mathscr{F}^{\sigma}\|_{L^{2}(D_{\delta})}^{2},\|\mathscr{G}^{\sigma}\|_{L^{2}(D_{\delta})}^{2}\Bigr{)}\geq
1-2\delta\,.$ (5.12)

Let
$D^{\natural}=C_{0}^{1/2}D_{\delta}=D(\mathbf{z}_{\delta},\,C_{0}^{1/2}r_{\delta})$
and write

$\mathscr{F}^{\sigma}=P^{\Phi_{\sigma}}_{D^{\natural}}\mathscr{F}^{\sigma}+(1-P_{D^{\natural}}^{\Phi_{\sigma}})\mathscr{F}^{\sigma},\;\;\mathscr{G}^{\sigma}=P_{D^{\natural}}^{\Phi_{\sigma}}\mathscr{G}^{\sigma}+(1-P_{D^{\natural}}^{\Phi_{\sigma}})\mathscr{G}^{\sigma}.$

By Lemma 5.4 and the condition
$2^{k\gamma(q,\varepsilon)}\leq\overline{A}^{\sigma}(R)\leq
2\,\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}$, it suffices to
show for some universal constant $C>0$, we have

$\|(P_{D^{\natural}}^{\Phi_{\sigma}}\mathscr{F}^{\sigma})\,\mathscr{G}^{\sigma}\|_{Z(Q_{R}\setminus\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta})}\lesssim
C_{0}^{-C},$ (5.13)
$\|(1-P_{D^{\natural}}^{\Phi_{\sigma}})\mathscr{F}^{\sigma}\,P_{D^{\natural}}^{\Phi_{\sigma}}\mathscr{G}^{\sigma}\|_{Z(Q_{R}\setminus\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta})}\lesssim
C_{0}^{-C},$ (5.14)

and

$\|(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{F}^{\sigma}\,(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{G}^{\sigma}\|_{Z(Q_{R})}\lesssim(\delta+C_{0}^{-C})\;\;\overline{A}^{\sigma}(R).$
(5.15)

To see (5.13) and (5.14), by using energy estimates, we are reduced to

$\|P_{D^{\natural}}^{\Phi_{\sigma}}\mathscr{F}^{\sigma}\|_{L^{\infty}(Q_{R}\setminus\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta})}\lesssim_{N}R^{-N/2},\,\|P_{D^{\natural}}^{\Phi_{\sigma}}\mathscr{G}^{\sigma}\|_{L^{\infty}(Q_{R}\setminus\mathbf{\Lambda}^{\Phi_{\sigma}}_{\delta})}\lesssim_{N}R^{-N/2},$
(5.16)

which is obvious in view of Lemma 3.9 and $r_{\delta}\geq R^{1/2+4/N}$.

To show (5.15), by the induction argument and (5.9), (3.26), (3.27) as well as
the assumption on $r_{\delta}$, we have

$\mathbf{E}\bigl{(}(1-P_{D^{\natural}}^{\Phi_{\sigma}})\mathscr{F}^{\sigma}\bigr{)}\lesssim\delta+R^{-N/2},\;\;\mathbf{E}\bigl{(}(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{G}^{\sigma}\bigr{)}\lesssim\delta+R^{-N/2}.$

It is clear that

$(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{F}^{\sigma}\in\mathfrak{R}^{\Phi_{\sigma},1}_{R^{\prime}},\quad(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{G}^{\sigma}\in\mathfrak{B}^{\Phi_{\sigma},2^{k}}_{R^{\prime}},$

with $R^{\prime}=\frac{R}{(1+o(1))}$. This yields (5.15) by finitely
partitioning $Q_{R}$ and using the definition of $A^{\sigma}(R^{\prime})$ and
the monotonicity of $\overline{A}^{\sigma}(R)$. The proof is complete.

####  5.2.2. The high concentration case: $r_{\delta}\leq R^{1/2+4/N}$

We turn to the case where the blue and red waves are highly concentrated.
Define

$\overline{R}_{\delta}=\max\Bigl{(}2^{C_{0}k},\;r_{\delta}^{1/(1/2+4/N)}\Bigr{)}.$

Consider the case $\overline{R}_{\delta}>2^{C_{0}k}$. In this case, we
necessarily have $r_{\delta}>2^{C_{0}k/2}$ and there is $\mathbf{z}_{\delta}$
such that we have (5.12). Let
$\widetilde{Q}=Q^{\mathbf{z}_{\delta}}_{\overline{R}_{\delta}}$ be the cube of
size $\overline{R}_{\delta}$ centered at $\mathbf{z}_{\delta}$. By splitting
$Q_{R}=\bigl{(}Q_{R}\cap\widetilde{Q}\bigr{)}\cup\bigl{(}Q_{R}\setminus\widetilde{Q}\bigr{)}$
and using Lemma 5.4, we have

$\bigl{\|}\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\bigr{\|}_{Z(Q_{R})}^{q}\leq\bigl{\|}\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\bigr{\|}_{Z(\widetilde{Q})}^{q}+\bigl{\|}\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\bigr{\|}_{Z(Q_{R}\setminus\widetilde{Q})}^{q}.$

For the first term on $\widetilde{Q}$, the argument as in the medium or low
concentration case leads to an estimate of the form (5.11) with $Q_{R}$ there
replaced by $\widetilde{Q}$.

For the second term, write

$\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}=\underbrace{\bigl{(}(P^{\Phi_{\sigma}}_{D^{\natural}}\mathscr{F}^{\sigma})\mathscr{G}^{\sigma}\bigr{)}}_{:=I}+\underbrace{\bigl{(}(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{F}^{\sigma}\;P^{\Phi_{\sigma}}_{D^{\natural}}\mathscr{G}^{\sigma}\bigr{)}}_{:=II}+\underbrace{\bigl{(}(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{F}^{\sigma}\,(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{G}^{\sigma}\bigr{)}}_{:=III}$

For $I$ and $II$, dyadic decomposing $Q_{R}\setminus\widetilde{Q}$ into annuli
around $\mathbf{z}_{\delta}$ of the form
$Q^{\mathsf{ann}}(\mathbf{z}_{\delta};2^{j},2^{j+1})$ with
$2^{j}\gtrsim\overline{R}_{\delta}$. Taking $C_{0}$ large and applying Lemma
5.7 then summing over dyadic $2^{-jb}$, we are done.

It remains to handle the $III-$term. Denote

$\mathring{\mathscr{F}}^{\sigma}=(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{F}^{\sigma},\quad\mathring{\mathscr{G}}^{\sigma}=(1-P^{\Phi_{\sigma}}_{D^{\natural}})\mathscr{G}^{\sigma}\,.$

Note that $\mathring{\mathscr{F}}^{\sigma},\mathring{\mathscr{G}}^{\sigma}$
are red and blue waves without the relaxed margin conditions required in
$\mathfrak{R}^{\Phi_{\sigma}}_{R}$ and $\mathfrak{B}^{\Phi_{\sigma}}_{R}$.
Thus, we can not apply the inductive argument as in the case when
$r_{\delta}\geq R^{1/2+4/N}$. To over come this obstacle, we may use the same
method in [23, Appendix B], to which we refer for details.

Switching back and affording a fixed universal constant, we have

$\|III\|_{Z(Q_{R})}\lesssim\;\delta\overline{A}^{\sigma}(R)+2^{O_{\varepsilon}(C_{0})}2^{k\gamma(q,\varepsilon)}$
(5.17)

Thus, by using (5.8)

$\|III\|_{Z(Q_{R})}\lesssim\;\delta\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}+2^{O_{\varepsilon}(C_{0})}2^{k\gamma(q,\varepsilon)},$

Plugging this back we are done.

It remains to consider the case $\overline{R}_{\delta}=2^{2C_{0}k}$. In this
case, the energy is concentrated in a scale $\leq 2^{C_{0}k}$, we use the
small scale estimate at the very beginning of the section to see thatr
$\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(\widetilde{Q})}\lesssim
2^{k\gamma(q,\varepsilon)}$ with $\varepsilon$ small.

Collecting all these estimates, we obtain (5.10) and the proof of Proposition
5.8 is complete. ∎

###  5.3. End of the proof

We are ready to show Theorem 1.2. Let $C_{1}$ and $C_{2}$ be the structural
constants given by Proposition 5.8 and Proposition 5.3 respectively. We may
take $C_{1}$ large and take $\delta=C_{0}^{-C_{1}/100}$.

Let
$(\mathscr{F}^{\sigma},\mathscr{G}^{\sigma})\in\mathfrak{R}^{\sigma}_{R}\times\mathfrak{B}^{\sigma,2^{k}}_{R}$
such that there exists $\Phi_{\sigma}\in\mathscr{E}_{\sigma}$ for which we
have $\mathscr{F}^{\sigma}\in\mathfrak{R}^{\Phi_{\sigma}}_{R}$ and
$\mathscr{G}^{\sigma}\in\mathfrak{B}^{\Phi_{\sigma},2^{k}}_{R}$. For any
spacetime cube $Q_{R}$, if it satisfies the condition (5.8), we let
$\mathbf{z}_{\delta},r_{\delta},D_{\delta}$ be given by Proposition 5.8. If
$\overline{R}_{\delta}>2^{C_{0}k}$, then by definition of
$\mathscr{A}^{\Phi_{\sigma}}(\overline{R}_{\delta},r_{\delta},C_{0}r_{\delta})$,
(5.10), Proposition 5.3, (5.9), we get

$\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}\\\
\leq\bigl{(}1-C_{1}(\delta+C_{0}^{-C_{1}})^{q}\bigr{)}^{-2/q}(1-\delta)^{1/q^{\prime}}\Bigl{(}(1+C_{2}2^{-C_{0}})\overline{A}^{\sigma}(R)+2^{C_{2}C_{0}}2^{k\gamma(q,\varepsilon)}\Bigr{)}\\\
+2^{O(C_{0})}2^{k\gamma(q,\varepsilon)}.$

Using $q>1$, and taking $C_{0}$ large if necessary (depending only on
$q,C_{1}$), one has $\exists\;\delta_{\mathbf{o}}\in(0,1/10)$ and
$0<C_{\mathbf{o}}<\infty$, depending only on $C_{0},C_{1},C_{2}$ and $q$, such
that

$\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}\leq(1-\delta_{\mathbf{o}})\,\overline{A}^{\sigma}(R)+C_{\mathbf{o}}\,2^{k\gamma(k,\varepsilon)}.$

If $\overline{R}_{\delta}=2^{C_{0}k}$, then we have the trivial estimate
$\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z({Q}_{R})}\leq\,C_{\mathbf{o}}2^{k\gamma(q,\varepsilon)}$
by the small scale estimate, in view of the proof in the last subsection.

Thus, we have

$\displaystyle\max_{Q_{R}}\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}$
$\displaystyle\leq$
$\displaystyle\max\Bigl{\\{}\max_{Q_{R}:\eqref{eq:crit}\text{holds}}\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})},\max_{Q_{R}:\eqref{eq:crit}\text{
fails}}\|\mathscr{F}^{\sigma}\mathscr{G}^{\sigma}\|_{Z(Q_{R})}\Bigr{\\}}+C_{\mathbf{o}}\,2^{k\gamma(q,\varepsilon)}$
$\displaystyle\leq$
$\displaystyle\max\Bigl{\\{}(1-\delta_{\mathbf{o}})\overline{A}^{\sigma}(R)+{C}_{\mathbf{o}}2^{k\gamma(k,\varepsilon)},\;\frac{1}{2}\,\overline{A}^{\sigma}(R)\Bigr{\\}}+C_{\mathbf{o}}2^{k\gamma(q,\varepsilon)}$
$\displaystyle\leq$
$\displaystyle(1-\delta_{\mathbf{o}})\overline{A}^{\sigma}(R)+2{C}_{\mathbf{o}}\,2^{k\gamma(q,\varepsilon)}\,.$

Noting the right side is independent of
$(\mathscr{F}^{\sigma},\mathscr{G}^{\sigma})\in\mathfrak{R}^{\sigma}_{R}\times\mathfrak{B}^{\sigma,2^{k}}_{R}$,
we get

$A^{\sigma}(R)\leq(1-\delta_{\mathbf{o}})\overline{A}^{\sigma}(R)+2{C}_{\mathbf{o}}2^{k\gamma(q,\varepsilon)}.$

Note that $\overline{A}^{\sigma}(R)$ is finite for each $\sigma,R$. Taking
suprema, we obtain

$\overline{A}^{\sigma}(R)\leq\frac{2{C}_{\mathbf{o}}}{\delta_{\mathbf{o}}}\,2^{k\gamma(q,\varepsilon)}.$

Since the right side is uniform with respect to $\sigma\leq\sigma_{0}$, the
proof is thus complete by taking suprema over $\sigma\in(0,\sigma_{0}]$.∎

######  Remark 5.9.

To get Proposition 2.8, the above same proof works equally well based on the
the wave tables in [15], with modifications in the interpolation argument to
meet the mixed-norms.

##  6\. Endpoint bilinear restriction estimates on the sphere concerning a
conjecuture by Foschi and Klainerman

Let $n\geq 2$ and
$\mathbb{S}=\mathbb{S}^{n-1}=\\{\xi\in\mathbb{R}^{n}:|\xi|=1\\}$ be the unit
sphere in $\mathbb{R}^{n}$. Define the Stein operator to be the adjoint of the
Fourier restriction operator

$Sf(x)=\int_{\mathbb{S}}e^{ix\cdot\xi}f(\xi)\,d\sigma(\xi),$

with $d\sigma$ being the surface measure. It is conjecured in [6] that
555Conjecture 17.2 if we let $\Omega_{1},\Omega_{2}$ be two disjoint compact
subsets of $\mathbb{S}$ such that

$\mathsf{dist}(\Omega_{1},\Omega_{2})>0,\;\mathsf{dist}(\Omega_{1},-\Omega_{2})>0\;,$

then for every $p\geq\frac{n+2}{n}$, there is a finite constant
$C=C_{\Omega_{1},\Omega_{2},p}>0$ such that

$\displaystyle\bigl{\|}Sf\cdot Sg\bigr{\|}_{L^{p}(\mathbb{R}^{n})}\leq
C\|f\|_{L^{2}(\Omega_{1})}\|g\|_{L^{2}(\Omega_{2})}$ (6.1)

holds for all $f$ supported in $\Omega_{1}$, and $g$ supported in
$\Omega_{2}$.

Nowadays, this question is usually referred as the bilinear restriction
theorems on elliptic type surfaces. In [16], the sharp result was established
by Tao for all $p>1+\frac{2}{n}$, except the endpoint. The purpose of this
section is to invest a possible way of combining the method of descent with
the argument for Theorem 1.2 to get the endpoint case. To illustrate the idea,
we assume $\Omega_{1},\Omega_{2}$ are contained in a small neighbourhood of
the north pole $e_{n}=(0,\ldots,0,1)$. By modulating the integrand in the
$L^{p}-$norm on the left side of (6.1), it is natural to consider the Monge
function $\Psi(\xi^{\prime})=\sqrt{1-|\xi^{\prime}|^{2}}-1$ and introducing
the $\lambda-$dependent operator

$S^{\lambda}_{\Omega}f(x,t)=\int_{-1}^{1}\int_{\Omega}e^{i(x^{\prime}\cdot\xi^{\prime}+ts+x_{n}\frac{\Psi(\xi^{\prime})}{\lambda+s})}f(\xi^{\prime},s)d\xi^{\prime}ds,$

where $t,s$ are auxiliary variables.

Taylor expand
$\Psi(\xi^{\prime})=\frac{|\xi^{\prime}|^{2}}{2}+\mathcal{E}(\xi^{\prime})$
for $|\xi^{\prime}|\ll 1$ and look at the surface

$\varSigma^{\lambda}:=\\{(\xi^{\prime},s,\,\Psi(\xi^{\prime})/(\lambda+s)):|\xi^{\prime}|\ll
1,|s|\leq 1\\}$

Ignoring the error term $\frac{\mathcal{E}(\xi^{\prime})}{\lambda+s}$, the
main term is a paraboloid as in [23], so that normal directions at each point
of $\varSigma^{\lambda}$ should be almost contained in a conic surface due to
the main term. To justify this rigorously, it remains to control the
perturbation term. To this end, one may introduce a similar class of functions
including $\frac{\Psi(\xi^{\prime})}{\lambda+s}$ as a special case such that
the stabiliy result holds as $\lambda\to+\infty$

Let $I=[-1,1]$ and $\mathscr{C}^{\lambda}(R)$ be the optimal constant $C$ for
which

$\|S^{\lambda}_{\Omega_{1}}f\cdot
S^{\lambda}_{\Omega_{2}}g\|_{L^{p}(Q^{\lambda}_{R})}\leq
C\lambda^{1/p}\|f\|_{L^{2}(\Omega_{1}\times I)}\|g\|_{L^{2}(\Omega_{2}\times
I)}$

for all $f,g$ and all $\lambda-$stretched $R$-cube
$Q^{\lambda}_{R}\subset\mathbb{R}^{n+1}_{x,t}$. Here, by stretch we mean
multiplying $\lambda$ to the side along $x_{n}-$direction. Construct the wave
tables and use bootstrap under the condition $R\leq\lambda$ ultimately to get
$\mathscr{C}^{\lambda}(R)\lesssim 1$ for all $R\leq\lambda$.

Let $\lambda=R$. Changing variables, and letting $R\to\infty$ using Lebesgue’s
dominated convergence theorem and then Fatou’s lemma, we obtain after
integrating $t$ out

######  Theorem 6.1.

Under the above conditions on $\Omega_{1},\Omega_{2}$, (6.1) holds for all
$p\geq\frac{n+2}{n}$.

## References

  * [1] Antonić, N. and Ivec, I., _On the Hörmander-Mihlin theorem for mixed-norm Lebesgue spaces._ J. Math. Anal. Appl. 433 (2016), no. 1, 176–199. MR3388786.
  * [2] Bennett, J.; Carbery, A., and Tao, T., _On the multilinear restriction and Kakeya conjectures._ Acta. Math. 196 (2006), no. 2, 261–302. MR2275834.
  * [3] Bourgain, J. _Estimates for cone multipliers._ Geometric aspects of functional analysis (Israel, 1992-1994), 41–60, Oper. Theory Adv. Appl., 77, Birkhäuser, Basel, 1995. MR1353448.
  * [4] Carbery, A., and Valdimarsson, S. _The endpoint multilinear Kakeya theorem via the Borsuk-Ulam theorem._ J. Funct. Anal. 264 (2013), no. 7, 1643-1663. MR3019726.
  * [5] Demeter, C. _Fourier restriction, decoupling, and applications._ Cambridge Studies in Advanced Mathematics, 184. Cambridge University Press, Cambridge 2020. xvi+331 pp. ISBN:978-1-108-49970-5. MR3971577.
  * [6] Foschi, D. and Klainerman, S. _Bilinear space-time estimates for homogeneous wave equations._ Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 211–274. MR 1755116.
  * [7] Del Nin, Giacomo, and Merlo, Andrea, _Endpoint Fourier restriction and unrectifiability._ Proc. Amer. Math. Soc. 150(2022), no. 5, 2137-2144. MR 4392348.
  * [8] Guth, L. _The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture._ Acta Math. 205(2010), no. 2, 263-286. MR2746348.
  * [9] Haberman, B. _Uniqueness in Calderón’s problem for conductivities with unbounded gradient._ Comm. Math. Phys. 340 (2015), no. 2, 639-659. MR3397029.
  * [10] Ham, S.; Kown, Y., and Lee, S. _Uniqueness in the Calderón problem and bilinear restriction estimates._ J. Funct. Anal. 281 (2021), no. 8, Paper No.109119,58pp. MR4273826.
  * [11] Klainerman, S., and Tataru, D. _On the optimal local regularity for Yang-Mills equations in $\mathbb{R}^{4+1}$._ J. Amer. Math. Soc. 12 (1999), no. 1, 93–116. MR1626261.
  * [12] Lee, S., Rogers, K. M., and Vargas, A. _Sharp null form estimates for the wave equation in $\mathbb{R}^{3+1}$._ Int. Math. Res. Not., IMRN (2008), Art. ID rnn 096, 18,pp. MR2439536.
  * [13] Lee, S. and Vargas, A. _Sharp null form estimates for the wave equation._ Amer. J. Math 130 (2008), no. 5, 1279–1326. MR2450209.
  * [14] Ponce-Vanegas, F. _A bilinear strategy for Calderón’s problem._ Rev. Mat. Iberoam. 37 (2021), no. 6, 2119-2160. MR4310288.
  * [15] Tao, T. _Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates._ Math. Z. 238 (2001), no. 2, 215–268. MR1865417.
  * [16] Tao, T. _A sharp bilinear restrictions estimate for paraboloids._ Geom. Funct. Anal. 13 (2003) no. 6, 1359-1384. MR 2033842.
  * [17] Tao, T.; Vargas, A. and Vega, L. _A bilinear approach to the restriction and Kakeya conjectures._ J. Amer. Math. Soc. 11 (1998), no. 4, 967–1000. MR1625056.
  * [18] Tao, T. and Vargas, A. _A bilinear approach to cone multipliers I: Restriction estimates._ Geom. Funct. Anal. 10 (2000), no.1, 185–215. MR1748920.
  * [19] Tao, T. and Vargas, A. _A bilinear approach to cone multipliers II: Applications._ Geom. Funct. Anal. 10 (2000), no. 1, 216–258. MR1748921.
  * [20] Tataru, D., _Null form estimates for second order hyperbolic operators with rough coefficients._ Harmonic analysis at Mount Holyoke, (South Hadley, MA, 2001), 383-409, Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003 . MR1979953.
  * [21] Temur, F. _An endline bilinear cone restriction estimate for mixed norms._ Math. Z. 273 (2013), no. 3-4, 1197–1214. MR3030696.
  * [22] Wolff, T. _A sharp bilinear cone restriction estimate._ Ann. of Math. (2) 153 (2001) no. 3, 661-698. MR1836285.
  * [23] Yang, J. _An endline bilinear restriction estimate for paraboloids._ arXiv:math/2202.13905v2.