$\left(\int_{0}^{\infty}\|S_{L_{1}}(t)\bm{g}\|_{H^{2}_{x}L^{2}_{k}}^{2}dt\right)^{\frac{1}{2}}\lesssim\left(\int_{0}^{\infty}\|S_{\mathcal{B}}(t)\bm{g}\|_{H^{2}_{x}L^{2}_{k}}^{2}dt\right)^{\frac{1}{2}}\\\
+\left(\int_{0}^{\infty}\Big{\|}\int_{0}^{t}S_{L_{1}}(s)K_{2}S_{\mathcal{B}}(t-s)\bm{g}\,ds\Big{\|}_{H^{2}_{x}L^{2}_{k}}^{2}\,dt\right)^{\frac{1}{2}}.$
(4.38)

For the first term, we can use (4.24). For the second term, by Minkowski’s
inequality, (4), (4.32) and (4), we compute

$\displaystyle\left(\int_{0}^{\infty}\Big{\|}\int_{0}^{t}S_{L_{1}}(t-s)K_{2}S_{\mathcal{B}}(s)\bm{g}ds\Big{\|}_{H^{2}_{x}L^{2}_{k}}^{2}dt\right)^{\frac{1}{2}}\lesssim$
$\displaystyle\int_{0}^{\infty}\left(\int_{s}^{\infty}\|S_{L_{1}}(t-s)K_{2}S_{\mathcal{B}}(s)\bm{g}\|_{H^{2}_{x}L^{2}_{k}}^{2}dt\right)^{\frac{1}{2}}\,ds$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{\infty}\left(\int_{0}^{\infty}\|S_{L_{1}}(t)K_{2}S_{\mathcal{B}}(s)\bm{g}\|_{H^{3}_{x}L^{2}_{k}}^{2}dt\right)^{\frac{1}{2}}\,ds$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{\infty}\|K_{2}S_{\mathcal{B}}(t)\bm{g}\|_{H^{3}_{x}L^{2}_{v}(\mu^{-1/2})}\,dt$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{\infty}\|S_{\mathcal{B}}(t)\bm{g}\|_{H^{3}_{x}L^{2}_{4}}\,dt.$
(4.39)

Therefore, using (4) or (4.25), we continue (4.38) to deduce

$\displaystyle\left(\int_{0}^{\infty}\|S_{L_{1}}(t)\bm{g}\|_{H^{2}_{x}L^{2}_{k}}^{2}dt\right)^{\frac{1}{2}}\lesssim\min\Big{\\{}\|\bm{g}\|_{H^{3}_{x}L^{2}_{k+10}},\,\|\bm{g}\|_{H^{3}_{x}H^{-s}_{k-\gamma/2}}\Big{\\}}.$

This completes the proof of Lemma 4.1. ∎

##  5\. Proof of the main theorem

In this Section, we come to the proof of the main stability theorem. We first
define the time-decreasing weight function

$w(\alpha,\beta)=\langle
v\rangle^{k(1+(1+t)^{-\frac{1}{2}})-a|\alpha|-b|\beta|}.$ (5.1)

Then we have

$\displaystyle\frac{d}{dt}w^{2}(t,\alpha,\beta)$
$\displaystyle=\frac{d}{dt}e^{2(k(1+(1+t)^{-\frac{1}{2}})-a|\alpha|-b|\beta|+c)}\log\langle
v\rangle$ (5.2) $\displaystyle=-\frac{k}{(1+t)^{\frac{3}{2}}}\log\langle
v\rangle w^{2}(t,\alpha,\beta)\leq-k\log
2(1+t)^{-\frac{3}{2}}w^{2}(t,\alpha,\beta)$

Now we are ready to prove our main theorem.

###### Proof of Theorem 1.2.

We define the lifespan time $T^{*}$ by

$T^{*}:=\sup_{t\geq
0}\big{\\{}E_{6,k}(t)\leq\epsilon,E_{2,k}(t)\leq\epsilon(1+t)^{-3},\,\forall
t\in[0,T^{*})\big{\\}}.$ (5.3)

The fact that $T^{*}>0$ follows directly from local existence Theorem 2.17. We
next prove that $T^{*}=\infty$. Notice from (1.6) that

$\partial_{t}\partial^{\alpha}_{\beta}f_{\pm}+\partial^{\alpha}_{\beta}(v\cdot\nabla_{x}f_{\pm})\pm\partial^{\alpha}_{\beta}((E+v\times
B)\cdot\nabla_{v}f_{\pm})\mp\partial^{\alpha}_{\beta}(E\cdot
v)\mu=\partial^{\alpha}_{\beta}L_{\pm}f+\partial^{\alpha}_{\beta}\Gamma_{\pm}(f,f).$
(5.4)

we multiply (5.4) by $w(t,\alpha,\beta)$ and do the standard $L^{2}$ energy
estimate. Remind (5.2), we have

$\frac{1}{2}\partial_{t}\|\partial^{\alpha}_{\beta}f_{\pm}w(\alpha,\beta)\|_{L^{2}_{x,v}}^{2}+k\log
2\|\partial^{\alpha}_{\beta}f_{\pm}w(\alpha,\beta)\|_{L^{2}_{x,v}}^{2}-(\partial^{\alpha}_{\beta}f_{\pm},\partial^{\alpha}_{\beta}f_{\pm}\partial_{t}w^{2}(\alpha,\beta))_{L^{2}_{x,v}}+(\partial^{\alpha}_{\beta}(v\cdot\nabla_{x}f_{\pm}),f_{\pm}w^{2}(\alpha,\beta))_{L^{2}_{x,v}}\\\
\pm(\partial^{\alpha}_{\beta}((E+v\times
B)\cdot\nabla_{v}f_{\pm}),f_{\pm}w^{2}(\alpha,\beta))_{L^{2}_{x,v}}\mp(\partial^{\alpha}_{\beta}(E\cdot
v)\mu,f_{\pm}w^{2}(\alpha,\beta))_{L^{2}_{x,v}}\\\
=+(L_{\pm}\partial^{\alpha}_{\beta}f_{\pm},f_{\pm}w^{2}(\alpha,\beta))_{L^{2}_{x,v}}+(\partial^{\alpha}_{\beta}\Gamma_{\pm}(f,f),f_{\pm}w^{2}(\alpha,\beta))_{L^{2}_{x,v}}.$

we make $\partial^{\alpha}$ for (LABEL:VMB2) and sum up over $|\alpha|\leq 7$,
and after summing $\alpha,\beta$ with $|\alpha|+|\beta|\leq 7$, collecting the
Lemmas from 2.11 to 2.16, there exist constants $\lambda_{k},C_{k},C>0$, such
that

$\displaystyle\partial_{t}E_{7,k}(t)+k\log
2(1+t)^{-\frac{3}{2}}E^{f}_{7,k}(t)+\lambda_{k}D_{7,k}(t)\leq$
$\displaystyle\,C_{k}\|[E,B]\|_{H^{7}_{x}}\|f\|_{H^{7}_{x}}+\sqrt{E_{2,k}(t)}E^{f}_{7,k}(t)$
(5.5)
$\displaystyle+\sqrt{E_{6,k}(t)}\sqrt{E_{7,k}(t)}\sqrt{D_{7,k}(t)}+C\|f\|_{H^{7}_{x,v}}^{2},$

where the energy functionals are given in (1.12) and (1.13). It is obvious
that

$||f,E,B||^{2}_{H^{7}_{x,v}}\leq E_{7,k}(t)$ (5.6)

notice that on $[0,T^{*}]$, we have

$\sqrt{E_{2,k}(t)}\leq\epsilon(1+t)^{-\frac{3}{2}}\leq k\log
2(1+t)^{-\frac{3}{2}}$ (5.7)

and

$\sqrt{E_{6,k}(t)}\sqrt{E_{7,k}(t)}\sqrt{D_{7,k}(t)}\leq\lambda_{k}D_{7,k}(t)+\frac{1}{4\lambda_{k}}E_{6,k}(t)E_{7,k}(t)\leq\lambda_{k}D_{7,k}(t)+E_{7,k}(t)$
(5.8)

after combining (5.5) to (5.8) together, there exists a constant $C>0$, such
that $\partial_{t}E_{7,k}(f)\leq CE_{7,k}(f)$ for any $t\in[0,T^{*}]$. By the
_a priori_ assumption (5.3), for any $t\in[0,T^{*})$, we have
$E_{7,k}(f)(t)\leq e^{Ct}E_{7,k}(f_{0})$. It implies that, if
$\bm{g}(t)=(f_{+}(t),f_{-}(t),E(t),B(t))$ is the solution to (1.6) and (1.7),
then for any $|\alpha|\leq 6$, we have

$\displaystyle\int_{0}^{\infty}(S_{L_{1}}(\tau)L_{1}\partial^{\alpha}\bm{g}(t),S_{L_{1}}(\tau)\partial^{\alpha}\bm{g}(t))_{L^{2}_{x}L^{2}_{v}}d\tau$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}\frac{d}{d\tau}\|S_{L_{1}}(\tau)\partial^{\alpha}\bm{g}(t)\|_{L^{2}_{x}L^{2}_{v}}^{2}d\tau$
$\displaystyle=$
$\displaystyle\lim_{\tau\to\infty}\frac{1}{2}\|S_{L_{1}}(\tau)\partial^{\alpha}\bm{g}(t)\|_{L^{2}_{x}L^{2}_{v}}^{2}-\frac{1}{2}\|\partial^{\alpha}\bm{g}(t)\|_{L^{2}_{x}L^{2}_{v}}^{2}$
$\displaystyle\leq$
$\displaystyle\lim_{\tau\to\infty}C(1+\tau)^{-1}\|\bm{g}(t)\|_{H^{7}_{x}L^{2}_{4}}^{2}-\frac{1}{2}\|\partial^{\alpha}\bm{g}(t)\|_{L^{2}_{x}L^{2}_{v}}^{2}.$
$\displaystyle\leq$
$\displaystyle\lim_{\tau\to\infty}C(1+\tau)^{-1}e^{Ct}\|\bm{g}(0)\|_{H^{7}_{x}L^{2}_{4}}^{2}-\frac{1}{2}\|\partial^{\alpha}\bm{g}(t)\|_{L^{2}_{x}L^{2}_{v}}^{2}=-\frac{1}{2}\|\partial^{\alpha}\bm{g}(t)\|_{L^{2}_{x}L^{2}_{v}}^{2},$
(5.9)

where we used the large-time behavior (4.4). For $2\leq K\leq 6$, under the
definition of $T^{*}$, we can deduce similarly as (5.5) that

$\displaystyle\partial_{t}E_{K,k}(t)+k\log
2(1+t)^{-\frac{3}{2}}E^{f}_{K,k}(t)+\lambda D_{K,k}(t)\leq$
$\displaystyle\,C_{k}\|[E,B]\|_{H^{K}_{x}}\|f\|_{H^{K}_{x}}+\sqrt{E_{2,k}(t)}E^{f}_{K,k}(t)$
(5.10) $\displaystyle+\sqrt{E_{K,k}(t)}{D_{K,k}(t)}+C\|f\|_{H^{K}_{x,v}}^{2},$

remind the _a priori_ assumption (5.3). Similarly as the argument for $K=7$,
we obtain that for any $K=2,3,4,5,6$,

$\partial_{t}E_{K,k}(f)+\lambda
D_{K,k}(f)\lesssim\|[E,B]\|_{H^{K}_{x}}\|f\|_{H^{K}_{x}}+\|f\|_{H^{K}_{x,v}}^{2}.$
(5.11)

morevoer, for the term $\|f\|_{H^{K}_{x,v}}^{2}$, notice that for any
$\alpha,\beta$ with $|\beta|\geq 1,|\alpha|+|\beta|\leq K$, and any $\eta>0$
small, by interpolation we have

$\|\partial^{\alpha}_{\beta}f\|_{L^{2}_{x,v}}^{2}\leq
C\|\partial^{\alpha}f\|_{L^{2}_{x}H^{\beta}_{v}}^{2}\leq\eta\|\partial^{\alpha}f\|_{L^{2}_{x}H^{\beta+s}_{v}}^{2}+C_{\eta}\|\partial^{\alpha}f\|_{L^{2}_{x,v}}^{2}\leq\eta
D_{K,k}(f)+C_{\eta}\|\partial^{\alpha}f\|_{L^{2}_{x,v}}^{2},$

which implies that

$C_{k}\|f\|_{H^{K}_{x,v}}^{2}\leq\frac{\lambda}{4}D_{K,k}(f)+C_{k}\|f\|_{H^{K}_{x}L^{2}_{v}}^{2}.$

Thus for $2\leq K\leq 6$ we have

$\displaystyle\partial_{t}E_{K,k}(f)+\lambda D_{K,k}(f)\lesssim$
$\displaystyle\|[E,B]\|_{H^{K}_{x}}^{2}+\|f\|_{H^{K}_{x}L^{2}_{v}}^{2}.$
(5.12)

Therefore, if $\bm{g}(t)=(f_{+}(t),f_{-}(t),E(t),B(t))$ is the solution to
(1.6) and (1.7), we define the norm
${{|\kern-1.07639pt|\kern-1.07639pt|}}\cdot{{|\kern-1.07639pt|\kern-1.07639pt|}}_{K,k}$
for $K\geq 6$ by

${{|\kern-1.07639pt|\kern-1.07639pt|}}g{{|\kern-1.07639pt|\kern-1.07639pt|}}_{K,k}^{2}:=E^{\bm{g}}_{K,k}(t)+\eta\int_{0}^{\infty}(S_{L_{1}}(\tau)\bm{g},S_{L_{1}}(\tau)\bm{g})_{H^{K}_{x}L^{2}_{v}}d\tau.$
(5.13)

From Lemma 4.1, we have

$\displaystyle\int_{0}^{\infty}(S_{L_{1}}(\tau)\bm{g}(t),S_{L_{1}}(\tau)\bm{g}(t))_{H^{K}_{x}L^{2}_{v}}d\tau
d\tau\lesssim\|\bm{g}(t)\|_{H^{K+1}_{x}L^{2}_{4}}^{2}\int_{0}^{\infty}C(1+\tau)^{-2}d\tau\lesssim\|\bm{g}(t)\|_{H^{K+1}_{x}L^{2}_{4}}^{2}$

Thus, the norm
${{|\kern-1.07639pt|\kern-1.07639pt|}}g{{|\kern-1.07639pt|\kern-1.07639pt|}}^{2}_{K,k}$
may not be equivalent to $E_{{K+1},k}(t)$ because of the loss of one-order
derivative, but it is larger than $E_{K,k}(t)$. Recall from (3.1) and (1.17)
that the solution $\bm{g}$ to VMB/VML system (1.6) and (1.7) can be written as
the solution to

$\partial_{t}\bm{g}=L_{1}\bm{g}+N\bm{g},\quad\nabla_{x}\cdot
E=\int_{\mathbb{R}^{3}}\big{(}f_{+}(v)-f_{-}(v)\big{)}\,dv,\quad\nabla_{x}\cdot
B=0,\quad\bm{g}(0)=\bm{g}_{0},$

where $N(\bm{g})=((E+v\times
B)\cdot\nabla_{v}f+Q(f_{+},f_{+})+Q(f_{-},f_{+}),\,-(E+v\times
B)\cdot\nabla_{v}f+Q(f_{-},f_{-})+Q(f_{+},f_{-}),0,0)$. Note that
$N(\bm{g})_{E,B}=(0,0)$. Thus, if $s\geq\frac{1}{2}$, we can easily compute
that for any $\eta>0$,

$\displaystyle\int_{0}^{\infty}(S_{L_{1}}(\tau)N\bm{g},S_{L_{1}}(\tau)\bm{g})_{H^{K}_{x}L^{2}_{v}}d\tau$
$\displaystyle=\int_{0}^{\infty}((S_{L_{1}}(\tau)Ng)_{f},(S_{L_{1}}(\tau)g)_{f})_{H^{K}_{x}L^{2}_{v}}d\tau$
$\displaystyle(\text{because of}\,\,\eqref{equationl10})$
$\displaystyle\lesssim\|Ng\|_{H^{K}_{x}H^{-s}_{10-\gamma/2}}\|f\|_{H^{K}_{x}H^{-s}_{10-\gamma/2}}$
$\displaystyle\lesssim\|(E+v\times
B)\cdot\nabla_{v}f+Q(f,f)\|_{H^{K}_{x}H^{-s}_{10-\gamma/2}}\|f\|_{H^{K}_{x}L^{2}_{12}}$
$\displaystyle(\text{because of}\,\,\eqref{equationbols1})$
$\displaystyle\lesssim\|f\|_{H^{K}_{x}L^{2}_{12}}\|f\|_{H^{6}_{x}L^{2}_{12}}(\|f\|_{H^{K}_{x}H^{s}_{12}}+\|E\|_{H^{K}_{x}})$
$\displaystyle\lesssim\frac{1}{2\eta}\lambda D_{K,k}+C\eta
E_{6,k}(\|[E,B]\|_{H^{K}_{x}}^{2}+\|f\|_{H^{K}_{x}L^{2}_{v}}^{2}).$ (5.14)

The above proof relies on the crucial observation that the coordinates $E$ and
$B$ of $N(g)$ are both set to 0. Since the dissipation is exclusively in $E$
and $B$ and not in $f$, there is no loss in the $x$ derivatives for $f$. This
feature greatly simplifies the proof. So we deduce from (5), (5.12), (5) that

$\frac{1}{2}\frac{d}{dt}{{|\kern-1.07639pt|\kern-1.07639pt|}}\bm{g}{{|\kern-1.07639pt|\kern-1.07639pt|}}_{K,k}^{2}+\frac{\lambda}{2}D_{K,k}(f)+\left(\frac{\eta}{2}-C-\eta^{2}CE_{6,k}\right)(\|[E,B]\|_{H^{K}_{x}}^{2}+\|f\|_{H^{K}_{x}L^{2}_{v}}^{2})\leq
0.$

Therefore, if we take $\eta\gg C$ large enough and $E_{6,k}\ll\frac{1}{C^{3}}$
small enough, there exists $\lambda_{0}>0$, such that
$\frac{\eta}{2}-C-\eta^{2}CE_{6,k}\geq\lambda_{0}$. Then we have that for
$2\leq K\leq 6$,

$\frac{d}{dt}{{|\kern-1.07639pt|\kern-1.07639pt|}}\bm{g}{{|\kern-1.07639pt|\kern-1.07639pt|}}_{K,k}^{2}+\frac{\lambda_{0}}{2}G_{K,k}\leq
0.$ (5.15)

Choose $K=6$ in (5.15), we have
$\frac{d}{dt}{{|\kern-1.07639pt|\kern-1.07639pt|}}\bm{g}{{|\kern-1.07639pt|\kern-1.07639pt|}}_{6,k}^{2}\leq
0$. This implies that for any $t\in[0,T^{*}]$, we have

$E_{6,k}(t)\leq{{|\kern-1.07639pt|\kern-1.07639pt|}}\bm{g}{{|\kern-1.07639pt|\kern-1.07639pt|}}^{2}\leq
E_{6,k}(0)+\eta\int_{0}^{\infty}(S_{L_{1}}(\tau)\bm{g}_{0},S_{L_{1}}(\tau)\bm{g}_{0})_{H^{6}_{x}L^{2}_{v}}d\tau\lesssim
E_{7,k}(0).$

Thus if we assume the initial data $E_{7,k}(0)\leq\epsilon_{0}$ is small, we
are able to close the first argument $E_{6,k}(t)\leq\epsilon/2$ in the _a
priori_ assumption (5.3).

Finally, we come to prove the convergence. For any
$k\geq\frac{9}{2}\geq-\frac{3\gamma}{2}$, Take $K=2$ in (5.15), we obtain

$\frac{d}{dt}{{|\kern-1.07639pt|\kern-1.07639pt|}}f{{|\kern-1.07639pt|\kern-1.07639pt|}}_{2,k-\frac{9}{2}}^{2}+\frac{\lambda_{0}}{2}G_{2,k-\frac{9}{2}}(f)\leq
0.$ (5.16)

Next, from the Gagliardo-Nirenberg interpolation inequality, we have

$\|[E,B]\|_{H^{3}_{x}}\lesssim\|[E,B]\|^{\frac{1}{4}}_{H^{6}_{x}}\|[E,B]\|^{\frac{3}{4}}_{H^{2}_{x}}\leq
E_{6}^{\frac{1}{4}}D_{2}^{\frac{3}{4}}.$ (5.17)

Since $k\geq\frac{9}{2}\geq\frac{3\gamma}{2}$, we obtain from interpolation
that $\|f\|_{L^{2}_{x}L^{2}_{v}}\lesssim\|\langle
v\rangle^{k}f\|_{L^{2}_{x}L^{2}_{v}}^{\frac{1}{4}}\|\langle
v\rangle^{\gamma/2}f\|_{L^{2}_{x}L^{2}_{v}}^{\frac{3}{4}}$. Next, using the _a
priori_ assumption $E_{6,k}(f)\leq\varepsilon$, we obtain from (5.17) that

${{|\kern-1.07639pt|\kern-1.07639pt|}}f{{|\kern-1.07639pt|\kern-1.07639pt|}}_{2,k-\frac{9}{2}}^{2}\lesssim
E_{3,k-\frac{9}{2}}(f)\lesssim
E_{6,k}^{\frac{1}{4}}(f)G_{2,k-\frac{9}{2}}^{\frac{3}{4}}(f)\lesssim
G_{2,k-\frac{9}{2}}^{\frac{3}{4}}(f).$

Together with (5.16), and recalling that $E_{6,k}(f)\leq\varepsilon$, there
exists a constant $\lambda_{1}>0$ such that

$\frac{d}{dt}{{|\kern-1.07639pt|\kern-1.07639pt|}}f{{|\kern-1.07639pt|\kern-1.07639pt|}}_{2,k-\frac{9}{2}}^{2}+\lambda_{1}{{|\kern-1.07639pt|\kern-1.07639pt|}}f{{|\kern-1.07639pt|\kern-1.07639pt|}}_{2,k-3}^{\frac{8}{3}}\leq
0.$

Since $E_{5,k}(f_{0})\leq\epsilon_{0}$, solving this ODE, we finally obtain
that

$E_{2,k-\frac{9}{2}}(f)\leq{{|\kern-1.07639pt|\kern-1.07639pt|}}f{{|\kern-1.07639pt|\kern-1.07639pt|}}_{2,k-\frac{9}{2}}^{2}\lesssim\epsilon_{0}(1+t)^{-3}.$

This finally closes the _a priori_ assumption and implies that the lifespan
time $T^{*}=\infty$. This completes the proof of Theorem 1.2. ∎

Acknowledgments. C. Cao is supported by grants from Department of applied
mathematics, The Hong Kong Polytechnic University. D.-Q. Deng was partially
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (MSIT) No. RS-2023-00210484 and No. RS-2023-00212304.
X.-Y. Li has been partially supported by the Basque Government through the
BERC 2022- 2025 program and by the Spanish State Research Agency through BCAM
Severo Ochoa excellence accreditation SEV-2017-0718 and through project
PID2020-114189RB-I00 funded by Agencia Estatal de Investigaci´on (PID2020-
114189RB-I00 / AEI / 10.13039/501100011033). He has also been supported by IMT
(Institut de Mathematiqués), Université de Toulouse III Paul Sabatier.

## References

  * [1] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, “Regularizing effect and local existence for the non-cutoff Boltzmann equation,” _Arch. Ration. Mech. Anal._ , vol. 198, no. 1, pp. 39–123, 2010.
  * [2] ——, “The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential,” _Journal of Functional Analysis_ , vol. 262, no. 3, pp. 915–1010, 2012.
  * [3] ——, “The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential,” _Anal. Appl. (Singap.)_ , vol. 9, no. 2, pp. 113–134, 2011.
  * [4] ——, “Global existence and full regularity of the Boltzmann equation without angular cutoff,” _Comm. Math. Phys._ , vol. 304, no. 2, pp. 513–581, 2011.
  * [5] R. Alexandre and C. Villani, “On the Boltzmann equation for long-range interactions,” _Comm. Pure Appl. Math._ , vol. 55, no. 1, pp. 30–70, 2002\.
  * [6] R. Alonso, Y. Morimoto, W. Sun, and T. Yang, “Non-cutoff Boltzmann equation with polynomial decay perturbation,” _Revista Matematica Iberoamericana_ , vol. 37, no. 1, pp. 189–292, 2021.
  * [7] C. Cao, D. Deng, and X. Li, “The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian.”
  * [8] C. Cao, H. L. He, and J. Ji, “Non cutoff Boltzmann equation with soft potentials.”
  * [9] Y. Cao, C. Kim, and D. Lee, “Global Strong Solutions of the Vlasov–Poisson–Boltzmann System in Bounded Domains,” _Arch. Ration. Mech. Anal._ , vol. 233, pp. 1027–1130, 2019.
  * [10] K. Carrapatoso, I. Tristani, and K.-C. Wu, “Cauchy problem and exponential stability for the inhomogeneous Landau equation,” _Arch. Ration. Mech. Anal._ , vol. 221, no. 1, pp. 363–418, 2016.
  * [11] K. Carrapatoso and S. Mischler, “Landau equation for very soft and Coulomb potentials near Maxwellians,” _Annals of PDE_ , no. 3, pp. 1–65, 2017.
  * [12] S. Chaturvedi, “Stability of vacuum for the Landau equation with hard potentials,” _Probability and Mathematical Physics_ , vol. 3, pp. 793–838, 2023.
  * [13] ——, “Stability of Vacuum for the Boltzmann Equation with Moderately Soft Potentials,” _Ann. PDE_ , vol. 7, no. 15, 2021.
  * [14] Y. Chen and L. He, “Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case,” _Archive for rational mechanics and analysis_ , vol. 201, no. 2, pp. 501–548, 2011.
  * [15] ——, “Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case,” _Archive for Rational Mechanics and Analysis_ , vol. 203, no. 2, pp. 343–377, 2012.
  * [16] D.-Q. Deng, “Regularity of the Vlasov–Poisson–Boltzmann System Without Angular Cutoff,” _Comm. Math. Phys._ , vol. 387, pp. 1603–1654, 2021.
  * [17] ——, “The non-cutoff Vlasov-Poisson-Boltzmann and Vlasov-Poisson-Landau Systems in Union of Cubes.”
  * [18] L. Desvillettes and C. Villani, “On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation,” _Invent. Math._ , vol. 159, no. 2, pp. 245–316, 2005.
  * [19] R. J. DiPerna and P.-L. Lions, “On the Fokker-Planck-Boltzmann equation,” _Comm. Math. Phys._ , vol. 120, no. 1, pp. 1–23, 1988.
  * [20] ——, “On the Cauchy problem for Boltzmann equations: global existence and weak stability,” _Ann. of Math. (2)_ , vol. 130, no. 2, pp. 321–366, 1989\.
  * [21] R. Duan, Y. Lei, T. Yang, and Z. Zhao, “The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials,” _Comm. Math. Phys_ , vol. 351, no. 1, pp. 95–153, 2017.
  * [22] R. Duan, “Global smooth dynamics of a fully ionized plasma with long-range collisions,” _Ann Inst H Poincar’e Anal Non Lin’eaire_ , vol. 31, pp. 751–778, 2014.
  * [23] Y. Guo, C. Kim, D. Tonon, and A. Trescases, “Regularity of the Boltzmann equation in convex domains,” _Invent. Math._ , vol. 207, no. 1, pp. 115–290, Jul. 2016.
  * [24] Y. Guo, “Decay and Continuity of the Boltzmann Equation in Bounded Domains,” _Arch. Ration. Mech. Anal._ , vol. 197, no. 3, pp. 713–809, Dec. 2009.
  * [25] Y. Guo, H. J. Hwang, J. W. Jang, and Z. Ouyang, “The Landau Equation with the Specular Reflection Boundary Condition,” _Arch. Ration. Mech. Anal._ , vol. 236, no. 3, pp. 1389–1454, Feb. 2020.
  * [26] D.-Q. Deng, “The Landau and Non-cutoff Boltzmann Equation in Union of Cubes,” _Preprint_ , 2021.
  * [27] R. Duan and S. Liu, “The Vlasov-Poisson-Boltzmann system without angular cutoff,” _Commun. Math. Phys._ , vol. 324, no. 1, pp. 1–45, 2013.
  * [28] R. Duan, S. Liu, S. Sakamoto, and R. Strain, “Global mild solutions of the Landau and non-cutoff Boltzmann equations,” _Communications on Pure and Applied Mathematics_ , vol. 74, no. 5, pp. 932–1020, 2021.
  * [29] R. Duan, S. Liu, T. Yang, and H. Zhao, “Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials,” _Kinet. Relat. Models_ , vol. 6, no. 1, pp. 159–204, 2013.
  * [30] H. Dong, Y. Guo, and Z. Ouyang, “The Vlasov-Poisson-Landau System with the Specular-Reflection Boundary Condition,” _Arch. Ration. Mech. Anal._ , vol. 246, pp. 333–396, 2022.
  * [31] R. Duan and R. M. Strain, “Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” _Comm Pure Appl Math_ , vol. 24, pp. 1497–1546, 2011.
  * [32] R. Duan, T. Yang, and H. Zhao, “The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case,” _J. Differential Equations_ , vol. 252, pp. 6356–6386, 2012.
  * [33] ——, “The Vlasov-Poisson-Boltzmann system for soft potentials,” _Math Models Methods Appl Sci_ , vol. 23, pp. 979–1028, 2013.
  * [34] Y. Fan and Y. Lei, “The Vlasov–Maxwell–Landau System with Coulomb Potential and Strong Background Magnetic Field,” _Journal of Statistical Physics_ , vol. 190, no. 1, 2023.
  * [35] F. Golse, C. Imbert, C. Mouhot, and A. F. Vasseur, “Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation,” _Annali della Scuola Normale Superiore di Pisa 5_ , vol. 19, no. 1, pp. 253–295, 2019.
  * [36] P. T. Gressman and R. M. Strain, “Global classical solutions of the Boltzmann equation without angular cut-off,” _J. Amer. Math. Soc._ , vol. 24, no. 3, pp. 771–847, 2011.
  * [37] ——, “Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production,” _Adv. Math_ , vol. 227, no. 6, pp. 2349–2384, 2011\.
  * [38] M. P. Gualdani, S. Mischler, and C. Mouhot, “Factorization of non-symmetric operators and exponential H-theorem,” _M’em. Soc. Math. Fr. (N.S.)_ , no. 153, p. 137, 2017.
  * [39] Y. Guo, “The Landau equation in a periodic box,” _Comm. Math. Phys._ , vol. 231, no. 3, pp. 391–434, 2002.
  * [40] ——, “Classical solutions to the Boltzmann equation for molecules with an angular cutoff,” _Arch. Ration. Mech. Anal._ , vol. 169, no. 4, pp. 305–353, 2003.
  * [41] ——, “The Boltzmann equation in the whole space,” _Indiana Univ. Math. J._ , vol. 53, no. 4, pp. 1081–1094, 2004.
  * [42] ——, “Boltzmann diffusive limit beyond the Navier-Stokes approximation,” _Comm. Pure Appl. Math._ , vol. 59, no. 5, pp. 626–687, 2006.
  * [43] ——, “The Vlasov-Poisson-Boltzmann system near vacuum,” _Comm. Math. Phys._ , vol. 218, no. 2, pp. 293–313, 2001.
  * [44] ——, “The Vlasov-Poisson-Boltzmann system near Maxwellians,” _Comm. Pure Appl. Math._ , vol. 55, no. 9, pp. 1104–1135, 2002.
  * [45] ——, “The Vlasov-Maxwell-Boltzmann system near Maxwellians,” _Invent. Math._ , vol. 153, no. 3, pp. 593–630, 2003.
  * [46] ——, “The Vlasov-Poisson-Landau system in a periodic box,” _J. Amer. Math. Soc._ , vol. 25, no. 3, pp. 759–812, 2012.
  * [47] L. He, “Sharp bounds for Boltzmann and Landau collision operators,” _Annales Scientifiques de l’École Normale Supérieure_ , vol. 51, no. 5, 2018.
  * [48] F. Hérau, D. Tonon, and I. Tristani, “Regularity estimates and Cauchy Theory for inhomogeneous Boltzmann equation for hard potentials without cut-off,” _Commun. Math. Phys._ , vol. 377, pp. 697–771, 2020.
  * [49] C. Henderson and S. Snelson, “$C^{\infty}$ smoothing for weak solutions of the inhomogeneous Landau equation,” _Arch. Ration. Mech. Anal._ , vol. 236, no. 1, pp. 113–143, 2020.
  * [50] C. Imbert, C. Mouhot, and L. Silvestre, “Decay estimates for large velocities in the Boltzmann equation without cutoff,” _Journal de l’École Polytechnique Mathématiques_ , vol. 7, pp. 143–184, 2020.
  * [51] ——, “Gaussian lower bounds for the Boltzmann equation without cut-off,” _SIAM J. Math. Anal._ , vol. 52, no. 3, pp. 2930–2944.
  * [52] C. Imbert and L. Silvestre, “The weak Harnack inequality for the Boltzmann equation without cut-off,” _J. Eur. Math. Soc._ , vol. 22, no. 2, pp. 507–592, 2020.
  * [53] ——, “Global regularity estimates for the Boltzmann equation without cut-off,” _Journal of the American Mathematical Society_ , Year, accepted.
  * [54] ——, “The Schauder estimate for kinetic integral equations,” _Analysis and PDE_ , vol. 14, no. 1, pp. 171–204.
  * [55] P.-L. Lions, “On Boltzmann and Landau equations,” _Philos. Trans. Roy. Soc. London Ser. A_ , vol. 346, no. 1679, pp. 191–204, 1994.
  * [56] J. Luk, “Stability of vacuum for the Landau equation with moderately soft potentials,” _Ann. PDE_ , vol. 5, no. 1, p. 11, 2019.
  * [57] C. Mouhot and R. M. Strain, “Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff,” _J. Math. Pures Appl. (9)_ , vol. 87, no. 5, pp. 515–535, 2007.
  * [58] L. Silvestre, “A new regularization mechanism for the Boltzmann equation without cut-off,” _Comm. Math. Phys._ , vol. 348, no. 1, pp. 69–100, 2016\.
  * [59] R. M. Strain, “The Vlasov-Maxwell-Boltzmann system in the whole space,” _Comm. Math. Phys_ , vol. 268, no. 2, pp. 543–567, 2006.
  * [60] R. M. Strain and Y. Guo, “Almost exponential decay near Maxwellian,” _Comm. Partial Differential Equations_ , vol. 31, no. 1-3, pp. 417–429, 2006\.
  * [61] ——, “Exponential decay for soft potentials near Maxwellian,” _Arch. Ration. Mech. Anal._ , vol. 187, no. 2, pp. 287–339, 2008.
  * [62] R. M. Strain and K. Zhu, “The Vlasov-Poisson-Landau system in $\mathbb{R}^{3}_{x}$,” _Arch. Ration. Mech. Anal._ , vol. 210, no. 2, pp. 615–671, 2013.
  * [63] C. Villani, “On the Cauchy problem for Landau equation: sequential stability, global existence,” _Adv. Differential Equations_ , vol. 1, no. 5, pp. 793–816, 1996.
  * [64] ——, “On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,” _Arch. Ration. Mech. Anal._ , vol. 143, no. 3, pp. 273–307, 1998.
  * [65] Y. Wang, “Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^{3}_{x}$,” _SIAM J. Math Anal_ , vol. 44, no. 5, pp. 3281–3323, 2012.
  * [66] ——, “The two-species Vlasov–Maxwell–Landau system in $\mathbb{R}^{3}_{x}$,” _Annales de l’Institut Henri Poincaré C, Analyse non linéaire_ , vol. 32, no. 5, pp. 1099–1123, Year.
  * [67] Q. Xiao, L. Xiong, and H. Zhao, “The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials,” _Sci China Math_ , vol. 57, pp. 515–540, 2014\.
  * [68] H. Yu, “Global classical solution of the Vlasov–Maxwell–Landau system near Maxwellians,” _J. Math. Phys._ , vol. 45, p. 4360–4376, 2004.


Let $\mu, \nu >0$ and let $\theta,\theta' \in \mathbb{C}[s,t]$ be linear forms that are $\mathbb{C}$-linearly independent. Then for every $\delta>0$

\[ |\{ s,t : |s|,|t|\lesssim 1, |\theta(s,t)^{\mu}\theta'(s,t)^{\nu}|\lesssim \delta\}| \lesssim_{\theta,\theta'} \begin{cases} \delta^{1/\max\{\mu,\nu\}} & \quad \text{ if } \mu \neq \nu \\
\delta^{1/\mu} \log^{+} 1/\delta & \quad \text{ if } \mu = \nu.
\end{cases} \]

From $\mathbb{C}$-linear independence we see in fact that we can pick real linear forms $\hat{\theta} \in \{ \operatorname{Re}\theta, \operatorname{Im}\theta\}$ and $\hat{\theta}' \in \{ \operatorname{Re}\theta', \operatorname{Im}\theta'\}$ so that $\hat{\theta}, \hat{\theta}'$ are $\mathbb{R}$-linearly independent. Since $|\hat{\theta}|\leq |\theta|$ and $|\hat{\theta}'|\leq |\theta'|$ we have then

\[ |\{ s,t : |s|,|t|\lesssim 1, |\theta(s,t)^{\mu}\theta'(s,t)^{\nu}|\lesssim \delta\}| \leq |\{ s,t : |s|,|t|\lesssim 1, |\hat{\theta}(s,t)|^{\mu}|\hat{\theta}'(s,t)|^{\nu} \lesssim \delta\}|, \]

and by a linear change of variables the latter is

\[ \lesssim_{\theta,\theta'} |\{ s,t : |s|,|t|\lesssim_{\theta,\theta'} 1, |s|^{\mu} |t|^{\nu}\lesssim \delta\}|. \]

By a simple integration we see that if $\mu \neq \nu$ then the last expression is dominated by $\lesssim \delta^{1/\max\{\mu,\nu\}}$, and if $\mu = \nu$ then it is dominated by $\lesssim \delta^{1/\mu} \log^{+} 1/\delta$.

The implicit constant in the estimate of Lemma <ref> can be made explicit: it is simply $O(|\det \begin{pmatrix} \hat{\theta} & \hat{\theta}' \end{pmatrix}|^{-1})$, where $\det \begin{pmatrix} \hat{\theta} & \hat{\theta}' \end{pmatrix}$ denotes the Jacobian determinant of the map $(s,t) \mapsto (\hat{\theta}(s,t), \hat{\theta}'(s,t))$, and $\hat{\theta}, \hat{\theta}'$ are as in the proof just given.

We will show that for a general polynomial that satisfies the multiplicity assumption of Proposition <ref>, we can always reduce at least to the second case of the lemma.As a step in the direction indicated, we claim that we can always rewrite the polynomial $P$ as a product of pairs of the form $(\theta_j\theta_k)^\mu$: more precisely, we will show that there exist quantities $\mu_{jk} \geq 0$ such that

\begin{equation}\label{eq:alternative_factorisation}
\prod_{j=1}^{\ell} \theta_j(s,t)^{m_j} = \prod_{j=1}^{\ell} \prod_{j < k \leq \ell} (\theta_j(s,t) \theta_k(s,t))^{\mu_{jk}}.
\end{equation}

Indeed, looking at the exponents, the equality translates immediately into the existence of a non-negative solution $(\mu_{jk})_{1\leq j < k \leq \ell}$ to the linear equations[Notice that the resulting system of equations has $\ell(\ell-1)/2$ variables and $\ell$ equations, and is therefore severely underdetermined.]

\begin{equation}\label{eq:system_positive_mu_jk}
L_j : \quad \sum_{i:\; i < j} \mu_{ij} + \sum_{k:\; k>j} \mu_{jk} = m_j, \qquad j \in \{1,\ldots, \ell\}.
\end{equation}

In order to treat such a system of linear equations, we recall the following fundamental linear programming lemma. For convenience, given a vector $\bm{v}$ we write $\bm{v} \geq 0$ to denote the fact that all components of $\bm{v}$ are non-negative.

If $M$ is an $m \times n$ real matrix and $\bm{b} \in \mathbb{R}^m$, then exactly one of the following mutually exclusive cases holds:

* there exists $\bm{x} \in \mathbb{R}^n$ such that $M\bm{x} = \bm{b}$ with $\bm{x}\geq 0$, or

*  there exists $\bm{y} \in \mathbb{R}^m$ such that $M^\top \bm{y} \geq 0$ and $\bm{b} \cdot \bm{y} < 0$.

The statement might appear somewhat cryptic at first, but the geometric content is actually elementary: if we let $\Gamma_{+} := \{ \bm{x} \in \mathbb{R}^n : \bm{x} \geq 0\}$, we observe that $\Gamma_{+}$ is a closed convex cone and therefore so is $M \Gamma_{+}$; then Farkas' lemma simply states that either $\bm{b}$ belongs to $M \Gamma_{+}$ or not, in which case the two can be separated by a hyperplane ($\bm{y}$ is an element orthogonal to this hyperplane and on the opposite side to $\bm{b}$).

We will show that case (<ref>) of Lemma <ref> is impossible in our situation (in which $\bm{b} = (m_1, \ldots, m_\ell)$ and $M$ can be read off of the system of equations (<ref>)), and thus the desired $(\mu_{jk})_{j<k}$ exist. Assume by contradiction that there is such a vector $\bm{y} = (y_1,\ldots, y_\ell)$ as in case (<ref>). Inspecting the system (<ref>) we see that the condition $M^\top \bm{y} \geq 0$ translates into the system of inequalities

\begin{equation} \label{eq:Farkas_inequalities_yj_yk}
y_j + y_k \geq 0
\end{equation}

for all $1 \leq j < k \leq \ell$ (indeed, observe that each variable $\mu_{jk}$ appears only in equations $L_j$ and $L_k$, always with coefficient $+1$); the condition $\bm{b} \cdot \bm{y} < 0$ is simply the statement that

\[ y_1 m_1 + \ldots + y_\ell m_\ell < 0. \]

On the one hand, since the $m_j$ are all positive, from the last inequality we see that at least one of the $y_j$ must be negative. On the other hand, from inequalities (<ref>) we see that there can be at most a single index $j_{\ast}$ such that $y_{j_{\ast}} < 0$ and that all other $y_j$ must be strictly positive instead; in particular, $y_j \geq |y_{j_{\ast}}| >0$. But then we have

\[ \sum_{j \neq j_{\ast}} m_j \leq \sum_{j \neq j_{\ast}} \frac{y_j}{|y_{j_{\ast}}|} m_j < m_{j_{\ast}}, \]

and this implies that $m_{j_{\ast}} > d/2$, which is a contradiction.The above has shown that the desired structural factorisation of $P$ can be achieved – and notice in particular that we have necessarily $\sum_{j<k} \mu_{jk} = d/2$. Now consider only those indices $j,k$ such that $\mu_{jk} > 0$. By the pigeonhole principle and factorisation (<ref>) we have that if $|P(s,t)|\leq \delta$ then for at least one pair of indices $j < k$ we have

\[ |\theta_j(s,t)\theta_k(s,t)|^{\mu_{jk}} \lesssim_P \,\delta^{2\mu_{jk}/d}; \]

it follows that $|\{ s,t : |s|,|t|\lesssim 1, |P(s,t)|\leq \delta\}|$ is dominated by the sum in indices $j<k$ of

\[ |\{ s,t : |s|,|t|\lesssim 1, |\theta_j(s,t)\theta_k(s,t)|^{\mu_{jk}} \lesssim_P \, \delta^{2\mu_{jk}/d} \}|. \]

However, since the $\theta_j$'s are normalised and distinct, they are linearly independent in pairs; by Lemma <ref> this measure is dominated by $\lesssim_P \, (\delta^{2\mu_{jk}/d})^{1/\mu_{jk}}\log^{+} (1/\delta^{2\mu_{jk}/d}) \allowbreak \sim \delta^{2/d} \log^{+} 1/\delta$, and we are done.

The proof of (<ref>) follows similar lines but is much simpler.

As in the proof of (<ref>), we can factorise $P$ as

\[ P(s,t) = C \theta_{\ast}(s,t)^{m_{\ast}} \prod_{j=1}^{\ell} \theta_{j}(s,t)^{m_j}, \]

where $m_{\ast} > d/2$ is the largest multiplicity and $\theta_{\ast}, \theta_1, \ldots, \theta_{\ell}$ are linearly independent linear forms. Since $m_{\ast} > \sum_{j=1}^{\ell} m_j$, we can find $\mu_{j}$ such that $\mu_j > m_j$ and $\sum_{j=1}^{\ell} \mu_j = m_{\ast}$; as a consequence, we can rearrange the factorisation of $P$ as

\[ P(s,t) = C \prod_{j=1}^{\ell} ( \theta_{\ast}(s,t)^{\mu_j} \theta_j(s,t)^{m_j}). \]

By the pigeonhole principle, if $|P(s,t)| < \delta$ then for at least one index $j$ we have

\[ |\theta_{\ast}(s,t)^{\mu_j} \theta_j(s,t)^{m_j}| \lesssim_P \delta^{\mu_j / m_{\ast}}; \]

therefore the sublevel set $\{s,t : |s|,|t|\lesssim 1, |P(s,t)|< \delta\}$ is contained in the union over $j$ of sublevel sets

\[ \{ s,t : |s|,|t| \lesssim 1, |\theta_{\ast}(s,t)^{\mu_j} \theta_j(s,t)^{m_j}| \lesssim_P \delta^{\mu_j / m_{\ast}} \}. \]

By Lemma <ref>, each of these has measure $\lesssim_P (\delta^{\mu_j / m_{\ast}})^{1/ \max\{\mu_j, m_j\}} = \delta^{1/{m_{\ast}}}$, and thus the proof is concluded.

While it is not possible in general to remove the logarithmic loss in (<ref>) even in the case of polynomials $P(s,t) = \det(sA + tB)$, the class of polynomials for which we incur such a loss can be narrowed down significantly. Indeed, with a more precise argument (such as e.g. the aforementioned integration argument in polar coordinates using Corollary 3.4 of [32]) one incurs logarithmic losses only when the polynomial $P$ has a root of multiplicity exactly equal to $d/2$. It follows that for well-curved surfaces $\Sigma(Q_1,Q_2)$ we can always obtain the restricted weak-type endpoint $L^{(d+4)/4} \to L^{(d+4)/2}$, provided all the roots have multiplicity strictly smaller than $d/2$. In particular, one recovers in these cases the critical line that is missing from the statement of Theorem <ref>.

§ FLAT SURFACES

In this final section we will give counterexamples that show the necessity of the curvature assumptions of Theorems <ref> and <ref>. More specifically, for flat $\Sigma(Q_1,Q_2)$ surfaces:

* We will show that if $\det(s \nabla^2 Q_1 + t \nabla^2 Q_2)$ does not vanish identically but has a root of multiplicity $m_{\ast} >d/2$, then for any $(p,q)$ sufficiently close to the endpoint $\big(\tfrac{d+4}{4}, \tfrac{d+4}{2}\big)$ the $L^p \to L^q$ estimate for operator $\mathcal{T}$ given by (<ref>) is false; in particular, we will show that any estimate with $2/q = 1/p$ and $q < m_{\ast} + 2$ is false.

* We will show that if $\det(s \nabla^2 Q_1 + t \nabla^2 Q_2)$ vanishes identically then any estimate with $2/q = 1/p$ is false (except for $p = q = \infty$); more in general, we will rule out every estimate for which $(2 - \epsilon)/q < 1/p$ for some $\epsilon > 0$ (this range intersects non-trivially the conjectural non-mixed range given by (<ref>)).

We will deal with each case in a separate subsection. Once again we resort to writing $A,B$ for $\nabla^2 Q_1, \nabla^2 Q_2$.

§.§ Case I: $\det(sA+tB)$ is not identically vanishing

In order to allow for a cleaner argument, we begin by making some reductions that are entirely analogous to those operated in Section <ref>; some care is needed because of the local nature of $\mathcal{T}$. For added precision, we introduce operators

\[ \mathcal{T}_{\Omega}^{A,B}f(x,\xi) := \iint_{\Omega} f(x - (sA + tB)\xi, s,t) \,ds\,dt, \]

in which the subscript $\Omega$ specifies the integration domain; thus for the operator given by (<ref>) we have $\mathcal{T} = \mathcal{T}_{[-1,1]^2}^{A,B}$.First of all, we claim that we can assume that $B$ is invertible. Indeed, otherwise there exists some $\tau_0$ such that $B_0 := -A - \tau_0 B$ is invertible, and we can write

\[ sA + tB = s' A_0 + t' B_0 \]

for $A_0 := B$ and $s',t'$ given by

\[ \begin{pmatrix} s' \\ t' \end{pmatrix} = N \begin{pmatrix}
s \\ t
\end{pmatrix}, \qquad N = \begin{pmatrix}
-\tau_0 & 1 \\ -1 & 0
\end{pmatrix} \in SL(\mathbb{R}^2). \]

If for any function $f$ we let $f_{\tau_0}(y,s,t):= f(y, t - s \tau_0, -s)$, we see by a change of variables that

\[ \mathcal{T}_{[-1,1]^2}^{A,B} f_{\tau_0} = \mathcal{T}_{N([-1,1]^2)}^{A_0,B_0} f; \]

therefore it will suffice to show that $\mathcal{T}_{N([-1,1]^2)}^{A_0,B_0}$ is unbounded, where now $B_0$ is invertible. Notice that since the operators are positive it will suffice to show that $\mathcal{T}_{[-\epsilon,\epsilon]^2}^{A_0,B_0}$ is unbounded for some $\epsilon > 0$ such that $[-\epsilon,\epsilon]^2 \subset N([-1,1]^2)$; by a rescaling, it then suffices to show that $\mathcal{T}_{[-1,1]^2}^{\epsilon A_0, \epsilon B_0}$ is unbounded.Assuming then that $B$ is invertible, we further claim that we can assume that $(A,B)$ is in the form $(\widetilde{\bm{J}}, \widetilde{\bm{I}})$ given by (<ref>) of Section <ref>. Indeed, using the notation of that section, we see that

\begin{align*}
sA + tB &= (s AB^{-1} + t I) B = (s Q J Q^{-1} + t I)B \\
&= Q(s \widetilde{\bm{J}} \widetilde{\bm{I}} + t \widetilde{\bm{I}}^2) Q^{-1} B = Q(s \widetilde{\bm{J}} + t \widetilde{\bm{I}}) \widetilde{\bm{I}} Q^{-1} B
\end{align*}

(recall that $Q$ is an invertible matrix such that $Q^{-1}AB^{-1}Q$ is in Jordan normal form). If for any function $f$ we let $f_{Q^{-1}}(y,s,t) := f(Q^{-1} y, s,t)$, we see by a straightforward calculation that

\[ \mathcal{T}_{[-1,1]^2}^{A,B} f_{Q^{-1}} (Qx, B^{-1} Q\widetilde{\bm{I}} \xi) = \mathcal{T}_{[-1,1]^2}^{\widetilde{\bm{J}}, \widetilde{\bm{I}}} f(x,\xi). \]

As a consequence, it will suffice to show that $\mathcal{T}_{[-1,1]^2}^{\widetilde{\bm{J}}, \widetilde{\bm{I}}}$ is unbounded from $L^p(B(0,C)\times [-1,1]^2)$ to $L^q(\mathbb{R}^d \times [-\epsilon',\epsilon']^d)$, where $\epsilon'>0$ is chosen sufficiently small to ensure $B^{-1} Q \widetilde{\bm{I}}([-\epsilon',\epsilon']^d) \allowbreak \subset [-1,1]^d$.Finally, assuming that the matrices are of the form $(\widetilde{\bm{J}}, \widetilde{\bm{I}})$, we can further assume that the eigenvalue of $\widetilde{\bm{J}}$ of highest multiplicity is $\lambda_{\ast} = 0$: this can be achieved by a repetition of the argument given to show that we could assume $B$ to be invertible, and thus we omit the details. Associated to eigenvalue $0$ we have the generalised eigenspaces of $\widetilde{\bm{J}}$: let $V_0$ be the span of all the generalised eigenspaces of dimension $1$, and let $V_1, \ldots, V_\ell$ be the generalised eigenspaces of dimension larger than $1$. For $j \in \{0,\ldots, \ell\}$ we let $\bm{e}^{(j)}_{1}, \ldots, \bm{e}^{(j)}_{n_j}$ be the generalised eigenvectors that span $V_j$, where $n_j := \dim V_j$. Moreover, we let $W$ denote the span of the generalised eigenspaces of non-zero eigenvalue – thus $\mathbb{R}^d = V_0 \oplus \ldots \oplus V_{\ell} \oplus W$. In the resulting basis of generalised eigenvectors the matrix $\widetilde{\bm{J}}$ has the form

\[ \widetilde{\bm{J}} = \begin{pmatrix}
0 & & & & & & & & \\
& \ddots & & & & & & & \\
& & 0 & & & & & & \\
& & & \widetilde{J}_{n_1}(0) & & & & & \\
& & & & \ddots & & & & \\
& & & & & \widetilde{J}_{n_\ell}(0) & & & \\
& & & & & & \bm{\ast} & \cdots & \bm{\ast} \\
& & & & & & \vdots & \ddots & \vdots \\
& & & & & & \bm{\ast} & \cdots & \bm{\ast}
\end{pmatrix}, \]

where $\widetilde{J}_{r}(0)$ is given by (<ref>); matrix $\widetilde{\bm{I}}$ has analogous form but $\widetilde{J}_r(0)$ is replaced by $\widetilde{I}_r$ (also given by (<ref>)). In particular, we have

\begin{equation}\label{eq:J_I_acting_on_V_0} \widetilde{\bm{J}} \bm{e}^{(0)}_k = 0, \qquad \widetilde{\bm{I}}\bm{e}^{(0)}_k = \bm{e}^{(0)}_k
\end{equation}

for every $k \leq n_0$, and for $1 \leq j \leq \ell$

\begin{equation}\label{eq:J_I_acting_on_V_j}
\begin{aligned}
\widetilde{\bm{J}}\bm{e}^{(j)}_k &= \bm{e}^{(j)}_{n_j - k}, \\
\widetilde{\bm{J}}\bm{e}^{(j)}_{n_j} &=0,
\end{aligned}
\quad
\begin{aligned}
\widetilde{\bm{I}}\bm{e}^{(j)}_k &= \bm{e}^{(j)}_{n_j +1 - k}, \quad \text{ for } 1 \leq k \leq n_j - 1, \\
\widetilde{\bm{I}}\bm{e}^{(j)}_{n_j} &= \bm{e}^{(j)}_{1}.
\end{aligned}
\end{equation}
We introduce two types of parabolic boxes adapted to the generalised eigenspaces: for any $j \in \{0,\ldots, \ell\}$ and $\delta > 0$ (an arbitrarily small parameter) we let

\[ R(\delta,V_j) := \Big\{ \sum_{k=1}^{n_j} \alpha^{(j)}_k \delta^{n_j - k} \bm{e}^{(j)}_k : |\alpha^{(j)}_k| < \epsilon' \text{ for all } k \Big\}, \]

and for $\epsilon >0$ we let also

\[ \widetilde{R}(\delta, \epsilon, V_j) := \Big\{ \sum_{k=1}^{n_j} \beta^{(j)}_k \delta^{k} \bm{e}^{(j)}_k : |\beta^{(j)}_k| < \epsilon \text{ for all } k \Big\} \]

(notice how $R(\delta, V_j)$ is a parabolic box of dimensions $\sim \delta^{n_j - 1} \times \ldots \times \delta \times 1$, whereas $\widetilde{R}(\delta, \epsilon, V_j)$ is a parabolic box of dimensions $\sim \delta \times \delta^2 \times  \ldots \times \delta^{n_j}$). Consider now parameters $(s,t)$ restricted to the strip

\[ S_{\delta} := \{ (s,t) \in [-1,1]^2 : |t|<\delta\} \]

and let us study how $s \widetilde{\bm{J}} + t \widetilde{\bm{I}}$ acts on the parabolic boxes. If $\bm{v} \in R(1,V_0)$ we have $\bm{v} = \sum_{k=1}^{n_0} \alpha^{(0)}_k  \bm{e}^{(0)}_k$ and thus by (<ref>)

\[ (s \widetilde{\bm{J}} + t \widetilde{\bm{I}})\bm{v} = \sum_{k=1}^{n_0} t \alpha^{(0)}_{k}  \bm{e}^{(0)}_k; \]

as a consequence, we have $(s \widetilde{\bm{J}} + t \widetilde{\bm{I}}) R(1, V_0) \subset \widetilde{R}(1,\delta \epsilon', V_0)$. If $\bm{v} \in R(\delta, V_j)$ for $1 \leq j \leq \ell$ we have $\bm{v} = \sum_{k=1}^{n_j} \alpha^{(j)}_k \delta^{n_j - k} \bm{e}^{(j)}_k$ and thus by (<ref>)

\[ (s \widetilde{\bm{J}} + t \widetilde{\bm{I}})\bm{v} = t \alpha^{(j)}_1 \delta^{n_j - 1} \bm{e}^{(j)}_{n_j} + \sum_{k=1}^{n_j - 1} (s \alpha^{(j)}_{n_j - k} \delta + t \alpha^{(j)}_{n_j + 1 - k}) \delta^{k-1} \bm{e}^{(j)}_k; \]

therefore $(s \widetilde{\bm{J}} + t \widetilde{\bm{I}}) R(\delta, V_j) \subset \widetilde{R}(\delta, 2\epsilon', V_j)$. Such inclusions have the following consequences: define (with a little abuse of notation) subsets of $\mathbb{R}^d$

\begin{align*}
E_{\delta} &:= R(1,V_0) \times \Big(\prod_{j=1}^{\ell} R(\delta, V_j)\Big) \times \{ \bm{w} \in W : \|\bm{w}\|_{\ell^{\infty}} < \epsilon'\}, \\
F_{\delta} &:= \widetilde{R}(1,\delta\epsilon',V_0) \times \Big( \prod_{j=1}^{\ell} \widetilde{R}(\delta, 2\epsilon',V_j)\Big)\times \{\bm{w} \in W : \|\bm{w}\|_{\ell^{\infty}} \lesssim_{\widetilde{\bm{J}},\widetilde{\bm{I}}} \epsilon'\};
\end{align*}

then we have $(s \widetilde{\bm{J}} + t \widetilde{\bm{I}}) E_\delta \subset F_\delta$ and $F_\delta - F_\delta \subset 2F_{\delta}$, which in particular implies

\[ \mathcal{T} \mathbf{1}_{2F_{\delta} \times S_{\delta}} \geq |S_{\delta}| \mathbf{1}_{F_{\delta} \times E_{\delta}} \]

(where we wrote $\mathcal{T}$ for $\mathcal{T}_{[-1,1]^2}^{\widetilde{\bm{J}},\widetilde{\bm{I}}}$ to ease the notation a little). If $\mathcal{T}$ were $L^p \to L^q$ bounded, the last inequality would imply (with some rearranging)

\[ |S_{\delta}|^{1/{p'}} |E_{\delta}|^{1/q} \lesssim |F_{\delta}|^{1/p - 1/q}. \]

However, it is easy to see that in terms of $\delta$

\[ |S_{\delta}| \sim \delta, \quad |E_{\delta}| \sim \delta^{\sum_{j=1}^{\ell} n_j(n_j - 1)/2}, \quad |F_{\delta}| \sim \delta^{n_0 + \sum_{j=1}^{\ell} n_j(n_j + 1)/2}, \]

and letting $\delta \to 0$ we obtain the necessary condition (after further rearranging)

\begin{equation}\label{eq:flat_surfaces_necessary_condition_1}
1 + \Big(n_0 + \sum_{j=1}^{\ell} n_j^2\Big) \frac{1}{q} \geq \Big( 1 + n_0 + \sum_{j=1}^{\ell} \frac{n_j(n_j + 1)}{2} \Big) \frac{1}{p}.
\end{equation}

Observe that $m_{\ast} = n_0 + \sum_{j=1}^{\ell} n_j$, so that if we restrict ourselves to exponents such that $2/q = 1/p$ we see with some algebra that (<ref>) yields the same set of exponents as the condition

\[ 1 + \frac{m_{\ast}}{q} \geq \frac{m_{\ast} + 1}{p} \]

stated in Theorem <ref>. On the other hand, the general condition excludes a range of exponents beyond those strictly on the critical line $2/q = 1/p$, as illustrated in Figure <ref>. The figure also illustrates that the reduced range provided by (<ref>) does not quite coincide with the range of true estimates afforded by Theorem <ref>; notice however that the two ranges coincide when $m_{\ast} = n_0$, that is, when the generalised eigenspaces of eigenvalue $\lambda_{\ast}$ are all of dimension $1$ (Theorem <ref> is then sharp in such cases, save perhaps for the endpoint).

[line cap=round,line join=round,>=Stealth,x=1cm,y=1cm, scale=7]
(-0.15,-0.1) rectangle (1.55,1.1);
[->,line width=0.5pt] (0,-0.06) – (0,1.1);
[->,line width=0.5pt] (-0.06,0) – (1.1,0);
[line width=0.5pt] (-0.02,1)– (1,1);
[line width=0.5pt,domain=-0.1:1.2] plot(,(-0–1*)/2);
[line width=0.5pt,domain=-0.1:1.65] plot(,(-2–7*)/5);
[line width=0.5pt,domain=-0.1:0.95] plot(,(-1–8*)/10);
[line width=0.5pt] (1/3,1/6)– (1,1);
[line width=0.5pt] (1,-0.02)– (1,0.02);
(0.4,0.27) node[anchor=south east] $\Big(\frac{2}{m_{\ast} + 2}, \frac{1}{m_{\ast}+2}\Big)$;
(1.05,0.84) node[anchor=north west] $2+\frac{d}{q} = \frac{d+2}{p}$;
(0.48,0.21) node[anchor=north west] $1 + \Big(n_0 + \sum_{j} n_j^2\Big)\frac{1}{q} = \Big(1 + n_0 + \sum_{j} \frac{n_j(n_j+1)}{2}\Big)\frac{1}{p}$;
(0.532,-0.015) node[anchor=north west] $1/p$;
(-0.13,0.6) node[anchor=north west] $1/q$;
(0.95,0.36) node[anchor=north west] $\frac{2}{q} = \frac{1}{p}$;
(0.5,0.45) node[anchor=south east] $\Big(\frac{4}{d+4}, \frac{2}{d+4}\Big)$;
(-0.08,1.035) node[anchor=north west] $1$;
(0.97,-0.04) node[anchor=north west] $1$;

[fill=black] (4/9,2/9) circle (0.2pt);
[fill=black] (1/3,1/6) circle (0.2pt);
[fill=black,fill opacity=0.1] (0,0) – (1/3,1/6) – (1,1) – (0,1) – cycle;
[->,line width=0.5pt] (0.64,0.2) – (0.58,0.364);
[->,line width=0.5pt] (0.94,0.31) – (0.8,0.4);
[->,line width=0.5pt] (1.05,0.78) – (0.76,0.664);
[->,line width=0.5pt] (0.24,0.276) – (0.3295,0.168);
[->,line width=0.5pt] (0.385,0.45) – (0.4435,0.225);

The shaded area corresponds to the range of boundedness afforded by Theorem <ref>, that is, when the surface $\Sigma(Q_1,Q_2)$ is flat but $\det(s\nabla^2 Q_1 + t \nabla^2 Q_2)$ does not vanish identically. The critical lines given by (<ref>) and (<ref>) are indicated: as one can see, the range of Theorem <ref> is sharp when $2/q = 1/p$. The endpoint $\big(\tfrac{4}{d+4}, \tfrac{2}{d+4}\big)$ for the well-curved case is also indicated, and one can see that for these surfaces all $L^p \to L^q$ estimates for $(1/p,1/q)$ close to this endpoint are false.

§.§ Case II: $\det(sA+tB)$ vanishes identically

We consider first the case in which $\ker (s_1 A + t_1 B) \cap \ker (s_2 A + t_2 B) = \{0\}$ for any linearly independent $(s_1,t_1), (s_2, t_2)$ (equivalently, $\ker A \cap \ker B = \{0\}$). As in Section <ref>, we can locate a maximal non-vanishing minor $\det_{I_{\ast}, J_{\ast}}(sA + tB)$ (where $I_{\ast},J_{\ast} \subset \{1,\ldots, d\}$ and $|I_{\ast}|=|J_{\ast}|$) and use it to define the set of generic $(s,t)$:

\[ \mathscr{G} := \{ (s,t) \in \mathbb{R}^2 : \det\nolimits_{I_{\ast}, J_{\ast}}(sA + tB) \neq 0 \} \]

(notice that, unlike in Section <ref>, we are considering real parameters only). Observe that we can find a set $S \subset [-1,1]^2 \cap \mathscr{G}$ such that $|S|>1/2$, since $\mathscr{G}$ is simply $\mathbb{R}^2$ with some lines removed. We define then the subspace of $\mathbb{R}^d$

\[ V := \operatorname{Span}\Big\{ \bigcup_{(s,t) \in \mathscr{G}} \ker (sA + tB) \Big\}; \]

by the same arguments given in Section <ref> we have that for every $(s,t) \in \mathscr{G}$ the image $(sA + tB)V$ consists of a common subspace $H$, which is a strict subspace of $V$. As a consequence, if $\xi \in \mathcal{N}_{\delta}(V)$ (the $\delta$-neighbourhood of $V$) we see that for $(s,t) \in S$ we have $(sA + tB)\xi \in \mathcal{N}_{K \delta}(H)$, where $K := \|A\|+ \|B\|$. Define then sets

\begin{align*}
E_{\delta} &:= \mathcal{N}_{\delta}(V) \cap [-1,1]^d, \\
F_{\delta} &:= \mathcal{N}_{K \delta}(H) \cap [-K,K]^d;
\end{align*}

by the discussion above we have that

\[ \mathcal{T} \mathbf{1}_{2F_{\delta} \times S} \gtrsim \mathbf{1}_{F_{\delta} \times E_{\delta}}, \]

and therefore if $\mathcal{T}$ is $L^p \to L^q$ bounded we have from the last inequality (after some rearranging)

\[ |E_{\delta}|^{1/q} \lesssim |F_{\delta}|^{1/p - 1/q}. \]

It is easy to see that

\[ |E_{\delta}| \sim \delta^{d - \dim V}, \quad |F_{\delta}| \sim \delta^{d - \dim H}, \]

so that letting $\delta \to 0$ we obtain the necessary condition

\[ \frac{d - \dim V}{q} \geq (d - \dim H)\Big(\frac{1}{p} - \frac{1}{q}\Big), \]

which after some rearranging is rewritten as

\[ \Big( 2 - \frac{\dim V - \dim H}{d - \dim H}\Big) \frac{1}{q} \geq \frac{1}{p}, \]

as claimed in Theorem <ref>. Since $\dim V > \dim H$, the condition shows that every $L^p \to L^q$ estimate with $2/q = 1/p$ is false in this case (with the exclusion of $(p,q) = (\infty, \infty)$). It remains to treat the case in which $\ker A \cap \ker B \neq \{ 0\}$, in which case $\mathcal{T}$ does not satisfy any non-trivial estimate. Indeed, there exists a strict subspace $W \subsetneq \mathbb{R}^d$ such that $(sA + tB) \mathbb{R}^d \subset W$ for all $(s,t)$. If we let

\[ F_{\delta} := \mathcal{N}_{\delta}(W) \cap [-K,K]^d \]

we see easily that

\[ \mathcal{T} \mathbf{1}_{2F_{\delta} \times [-1,1]^2} \geq \mathbf{1}_{F_{\delta} \times [-1,1]^d}; \]

if $\mathcal{T}$ is $L^p \to L^q$ bounded we have then

\[ |F_{\delta}|^{1/q} \lesssim |F_{\delta}|^{1/p}, \]

and since $|F_{\delta}| \sim \delta^{d - \dim W}$ it is immediate to deduce the necessary condition $1/q \geq 1/p$. Thus every estimate beyond those obtained from interpolation of the trivial estimates of Remark <ref> is false.

§ GENERAL WELL-CURVED SURFACES

In this appendix we sketch the modifications of the arguments presented in this paper that allow to extend Theorem <ref> to Theorem <ref>, that is, to general well-curved surfaces $\Sigma(\varphi_1,\varphi_2)$ of the form

\[ (\xi, \varphi_1(\xi), \varphi_2(\xi)), \qquad \xi \in [-\epsilon, \epsilon]^d, \]

where $\varphi_1,\varphi_2$ are $C^2$ functions such that $\nabla \varphi_1(0) = \nabla \varphi_2(0) = 0$, and $\epsilon$ will be taken sufficiently small depending on $\varphi_1,\varphi_2$. We will borrow heavily from other sections and their notations to keep the appendix short.The first observation is that if $\Sigma(\varphi_1,\varphi_2)$ is well-curved at $\xi = 0$ then it is well-curved in a neighbourhood of $0$ as well. Indeed, this is a consequence of the fact that condition (<ref>) is stable under small perturbations: observe that the coefficients of the polynomial $\det(s \nabla^2\varphi_1(\xi) + t \nabla^2 \varphi_2(\xi))$ are continuous functions of $\xi$. It is well-known that the roots of a univariate polynomial are continuous functions of its coefficients, and it is not hard to see that this fact extends to homogeneous polynomials of two variables (for example, by passing to the projectivisation). Thus the roots  of $\det(s \nabla^2\varphi_1(\xi) + t \nabla^2 \varphi_2(\xi))$ are continuous functions of $\xi$ and we see that if (<ref>) is satisfied at $\xi = 0$ then it is satisfied for $\xi \in [-\epsilon,\epsilon]^d$ for some $\epsilon >0$ (this has to do with the fact that (<ref>) is an upperbound on the maximal algebraic multiplicity of the roots).The bulk of the argument of Section <ref> goes through without major changes: in particular, the Jacobian determinant of the map $\Psi$ is still given by (<ref>) – that is, by $\det((s-s_0) \nabla^2\varphi_1(\eta) + (t-t_0) \nabla^2 \varphi_2(\eta))$, which unlike  the quadratic case is now a function of $\eta$ too. For $\epsilon$ sufficiently small, the multiplicity $\mu_{\Psi}$ of the map $\Psi$ is still $1$. Indeed, we see that $\Psi(\eta,s,t) = \Psi(\eta',s',t')$ only if $s=s', t=t'$ and

\[ \hat{s}(\nabla \varphi_1(\eta) - \nabla \varphi_1(\eta')) + \hat{t} (\nabla \varphi_2(\eta) - \nabla \varphi_2(\eta')) = 0 \]

(where $\hat{s} = s - s_0$, $\hat{t} = t - t_0$ for shortness); this can be rewritten as

\[ \Big( \int_{0}^{1} [\hat{s} \nabla^2 \varphi_1 + \hat{t} \nabla^2 \varphi_2](\theta \eta + (1 - \theta)\eta') \,d\theta\Big) (\eta - \eta') = 0, \]

so that the matrix in brackets must have determinant zero if $\eta \neq \eta'$. However, expanding the determinant we see that it equals

\[ \int_{[0,1]^d} \sum_{\sigma \in S_d} \operatorname{sgn}\sigma \prod_{j=1}^{d} \partial_j \partial_{\sigma(j)} (\hat{s} \varphi_1 + \hat{t} \varphi_2)(\theta_j \eta + (1 - \theta_j) \eta') \,d\theta_1 \ldots \,d\theta_d; \]

the integrand is seen to be the determinant of a matrix that is a small perturbation of $\hat{s} \nabla^2 \varphi_1(\eta) + \hat{t} \nabla^2 \varphi_2(\eta)$. If we impose – as we do – that for $(\eta,s,t) \in S$ (where $S$ is given by (<ref>)) this is non-zero, then the integrand is never zero and in particular single-signed (provided $\epsilon$ is small), and therefore the determinant above is not zero and $\eta = \eta'$. To complete the proof given in Section <ref> all that remains to show is that we can make the sublevel set estimate (<ref>) uniform in $\eta$; this is the most delicate part. First of all, recall as observed in Remark <ref> that the implicit constant in Lemma <ref> can be made explicit: with $\theta,\theta'$ normalised linear forms (which for simplicity we assume real, without loss of generality), we have

\[ |\{ (s,t) : |s|,|t|\leq 1, |\theta(s,t)^{\mu} \theta'(s,t)^{\nu}|<\delta\}| \lesssim \frac{|\{ (s,t) : |s|,|t|\lesssim 1, |s^{\mu} t^{\nu}|<\delta\}|}{|\det \begin{pmatrix} \theta & \theta' \end{pmatrix}|}, \]

where $\det \begin{pmatrix} \theta & \theta' \end{pmatrix}$ is the Jacobian determinant of the map $(s,t) \mapsto (\theta(s,t), \theta'(s,t))$; thus the implicit constant is $O(|\det \begin{pmatrix} \theta & \theta' \end{pmatrix}|^{-1})$. Secondly, by continuity of the roots we have the following: if

\[ \theta_1^{m_1}, \ldots, \theta_{\ell}^{m_{\ell}} \]

are the distinct normalised roots of $\det(s \nabla^2 \varphi_1 (0) + t \nabla^2 \varphi_2 (0))$ with respective multiplicities, then for a fixed $\eta \in [-\epsilon, \epsilon]^d$ and $\epsilon$ sufficiently small the distinct normalised roots of $\det(s \nabla^2 \varphi_1 (\eta) + t \nabla^2 \varphi_2 (\eta))$ are

\[ \tilde{\theta}_{1 1}^{m_{1 1}}, \ldots, \tilde{\theta}_{1  n_1}^{m_{1 n_1}}, \ldots, \tilde{\theta}_{\ell 1}^{m_{\ell 1}}, \ldots, \tilde{\theta}_{\ell n_\ell}^{m_{\ell n_{\ell}}}, \]

where each $\tilde{\theta}_{ji}$ for $1\leq i \leq n_j$ is a small perturbation of $\theta_{j}$ and for each $j$ we have $\sum_{i=1}^{n_j} m_{ji} = m_j$. In particular, for any $j,k$ we have

\[ |\det \begin{pmatrix} \theta_j & \theta_k \end{pmatrix}| \sim |\det \begin{pmatrix} \tilde{\theta}_{ji} & \tilde{\theta}_{k i'} \end{pmatrix}| \]

for all $1\leq i \leq n_j$ and $1 \leq i' \leq n_k$. To obtain a sublevel set estimate that is uniform in $\eta \in [-\epsilon, \epsilon]^d$ it will then suffice to show that we can find coefficients $\mu_{j i k i'} \geq 0$ such that we have the structural factorisation

\[ \prod_{j=1}^{\ell}\prod_{i=1}^{n_j} \tilde{\theta}_{j i}^{m_{j i}} = \prod_{j=1}^{\ell} \prod_{j < k \leq \ell} \prod_{i=1}^{n_j} \prod_{i'=1}^{n_k} (\tilde{\theta}_{j i} \tilde{\theta}_{k i'})^{\mu_{j i k i'}} \]

(in this way in our constants we will avoid terms like $|\det(\begin{pmatrix} \tilde{\theta}_{j i} & \tilde{\theta}_{j i'} \end{pmatrix}|^{-1}$, which could be arbitrarily large). This can be achieved by a variation of the argument used in the proof of Proposition <ref>, as we now illustrate. As in there, the existence of such a factorisation translates into the existence of a non-negative solution to the equations

\[ L_{ji} : \quad \sum_{k < j} \sum_{i' = 1}^{n_k} \mu_{k i' j i} + \sum_{k'>j}\sum_{i''=1}^{n_{k'}} \mu_{j i k' i''} = m_{j i}, \]

for $1\leq j \leq \ell$ and $1\leq i \leq n_j$. Appealing once again to Lemma <ref>, it suffices to show that there is no simultaneous solution $(y_{j i})_{j\leq \ell, i\leq n_j}$ to the inequalities

\[ \left\{ \begin{aligned}
y_{j i} + y_{k i'} &\geq 0 \quad \text{ for all } 1\leq j < k \leq \ell,\, 1\leq i \leq n_j, \, 1\leq  i' \leq n_k, \\
\sum_{j=1}^{\ell} \sum_{i=1}^{n_j} m_{j i} y_{j i} & < 0.
\end{aligned} \right. \]

Since $m_{ji} >0$ the second inequality implies that for some $j_{\ast}$ one coefficient $y_{j_{\ast} i}$ is negative; let $y_{j_{\ast} i_{\ast}}$ be the most negative of such coefficients. From the first inequality we see that for every $j \neq j_{\ast}$ we must have $y_{ji} \geq |y_{j_{\ast} i_{\ast}}| > 0$ for all $1\leq i\leq n_j$, and therefore we have

\begin{align*}
&\sum_{j \neq j_{\ast}} m_j = \sum_{j \neq j_{\ast}} \sum_{i=1}^{n_j} m_{ji} \leq  \sum_{j \neq j_{\ast}} \sum_{i=1}^{n_j} m_{ji} \frac{y_{ji}}{|y_{j_{\ast} i_{\ast}}|} \\
&= \frac{1}{|y_{j_{\ast} i_{\ast}}|}\sum_{j=1}^{\ell} \sum_{i=1}^{n_j} m_{ji}y_{ji} - \frac{1}{|y_{j_{\ast} i_{\ast}}|}\sum_{i=1}^{n_{j_{\ast}}} m_{j_{\ast} i} (y_{j_{\ast} i} - y_{j_{\ast} i_{\ast}}) + \sum_{i=1}^{n_{j_{\ast}}} m_{j_{\ast} i} \\
& < - \frac{1}{|y_{j_{\ast} i_{\ast}}|}\sum_{i=1}^{n_{j_{\ast}}} m_{j_{\ast} i} (y_{j_{\ast} i} - y_{j_{\ast} i_{\ast}}) + \sum_{i=1}^{n_{j_{\ast}}} m_{j_{\ast} i} \leq \sum_{i=1}^{n_{j_{\ast}}} m_{j_{\ast} i} = m_{j_{\ast}};
\end{align*}

this would imply $m_{j_{\ast}} > d/2$, a contradiction because $\Sigma(\varphi_1,\varphi_2)$ is well-curved at $\xi = 0$. This concludes the proof.

§.§ Acknowledgements
The third author was supported in part by the Irish Research Council via the IRC Postdoctoral Fellowship GOIPD/2019/434. The authors are indebted to P. Gressman and M. Christ for enlightening conversations about their work, and to J. Bennett and M. Iliopoulou for equally enlightening conversations on the Mizohata-Takeuchi conjecture.

[1]
J. A. Barceló, J. M. Bennett, and A. Carbery.
A note on localised weighted inequalities for the extension operator.
J. Aust. Math. Soc., 84(3):289–299, 2008.

[2]
J. A. Barceló, A. Ruiz, and L. Vega.
Weighted estimates for the Helmholtz equation and some
J. Funct. Anal., 150(2):356–382, 1997.

[3]
A. Benedek and R. Panzone.
The space $L\sp{p}$, with mixed norm.
Duke Math. J., 28:301–324, 1961.

[4]
J. Bennett, A. Carbery, F. Soria, and A. Vargas.
A Stein conjecture for the circle.
Math. Ann., 336(3):671–695, 2006.

[5]
J. Bennett and S. Nakamura.
Tomography bounds for the Fourier extension operator and
Math. Ann., 380(1-2):119–159, 2021.

[6]
N. Bez and M. Sugimoto.
Remarks on the Mizohata-Takeuchi conjecture and related problems.
In Asymptotic analysis for nonlinear dispersive and wave
equations, volume 81 of Adv. Stud. Pure Math., pages 1–12. Math. Soc.
Japan, Tokyo, 2019.

[7]
D. Birkes.
Orbits of linear algebraic groups.
Ann. Math. (2), 93:459–475, 1971.

[8]
J. Bourgain and L. Guth.
Bounds on oscillatory integral operators based on multilinear
Geom. Funct. Anal., 21(6):1239–1295, 2011.

[9]
A. Carbery, M. Christ, and J. Wright.
Multidimensional van der Corput and sublevel set estimates.
J. Amer. Math. Soc., 12(4):981–1015, 1999.

[10]
A. Carbery, E. Romera, and F. Soria.
Radial weights and mixed norm inequalities for the disc multiplier.
J. Funct. Anal., 109(1):52–75, 1992.

[11]
A. Carbery and F. Soria.
Pointwise Fourier inversion and localisation in ${\bf R}^n$.
In Proceedings of the conference dedicated to Professor
Miguel de Guzmán (El Escorial, 1996), volume 3, pages 847–858,
Special Issue.

[12]
M. Christ.
Restriction of the Fourier Transform to submanifolds of low
ProQuest LLC, Ann Arbor, MI, 1982.
Thesis (Ph.D.)–The University of Chicago.

[13]
M. Christ.
Estimates for the $k$-plane transform.
Indiana Univ. Math. J., 33(6):891–910, 1984.

[14]
M. Christ.
Convolution, curvature, and combinatorics: a case study.
Internat. Math. Res. Notices, 19:1033–1048, 1998.

[15]
M. Christ.
Counting to ${L}^p$.
, 2002.
Edinburgh, Lecture Notes (accessed 12/9/22).

[16]
M. Christ.
Quasiextremals for a Radon-like transform, 2005.
preprint, arXiv:1106.0722.

[17]
M. Christ and M. B. Erdoǧan.
Mixed norm estimates for a restricted X-ray transform.
J. Anal. Math., 87:187–198, 2002.

[18]
L. De Carli and A. Iosevich.
A restriction theorem for flat manifolds of codimension two.
Illinois J. Math., 39(4):576–585, 1995.

[19]
S. Dendrinos and B. Stovall.
Uniform estimates for the X-ray transform restricted to polynomial
J. Funct. Anal., 262(12):4986–5020, 2012.

[20]
S. Dendrinos and B. Stovall.
Uniform bounds for convolution and restricted X-ray transforms
along degenerate curves.
J. Funct. Anal., 268(3):585–633, 2015.

[21]
S. W. Drury.
$L\sp{p}$ estimates for the X-ray transform.
Illinois J. Math., 27(1):125–129, 1983.

[22]
S. W. Drury.
Generalizations of Riesz potentials and $L\sp{p}$ estimates for
certain $k$-plane transforms.
Illinois J. Math., 28(3):495–512, 1984.

[23]
J. Duoandikoetxea and V. Naibo.
Mixed-norm estimates for the $k$-plane transform.
In Excursions in harmonic analysis. Volume 2, Appl. Numer.
Harmon. Anal., pages 211–228. Birkhäuser/Springer, New York, 2013.

[24]
M. B. Erdoğan and R. Oberlin.
Estimates for the $X$-ray transform restricted to 2-manifolds.
Rev. Mat. Iberoam., 26(1):91–114, 2010.

[25]
J. M. Fraser, T. L. J. Harris, and N. G. Kroon.
On the Fourier dimension of $(d,k)$-sets and Kakeya sets with
restricted directions.
Math. Z., 301(3):2497–2508, 2022.

[26]
M. Gilula.
A real analytic approach to estimating oscillatory integrals.
ProQuest LLC, Ann Arbor, MI, 2016.
Thesis (Ph.D.)–University of Pennsylvania.

[27]
J. H. Grace and A. Young.
The algebra of invariants.
Cambridge Library Collection. Cambridge University Press, Cambridge,
Reprint of the 1903 original.

[28]
P. T. Gressman.
On the Oberlin affine curvature condition.
Duke Math. J., 168(11):2075–2126, 2019.

[29]
P. T. Gressman.
Geometric averaging operators and nonconcentration inequalities.
Anal. PDE, 15(1):85–122, 2022.

[30]
S. Guo and C. Oh.
Fourier restriction estimates for surfaces of co-dimension two in
$\mathbb{R}^5$, 2020.

[31]
D. Hilbert.
Über die vollen Invariantensysteme.
Math. Ann., 42:313–373, 1893.

[32]
I. A. Ikromov and D. Müller.
On adapted coordinate systems.
Trans. Amer. Math. Soc., 363(6):2821–2848, 2011.

[33]
A. Iosevich and G. Lu.
Sharpness results and Knapp's homogeneity argument.
Canad. Math. Bull., 43(1):63–68, 2000.

[34]
N. H. Katz, I. Łaba, and T. Tao.
An improved bound on the Minkowski dimension of Besicovitch sets
in ${\bf R}^3$.
Ann. of Math. (2), 152(2):383–446, 2000.

[35]
I. Łaba and T. Tao.
An X-ray transform estimate in $\Bbb R^n$.
Rev. Mat. Iberoamericana, 17(2):375–407, 2001.

[36]
P. Mattila.
Fourier analysis and Hausdorff dimension, volume 150 of Camb. Stud. Adv. Math.
Cambridge: Cambridge University Press, 2015.

[37]
G. Mockenhaupt.
Bounds in Lebesgue spaces of oscillatory integral operators.
Siegen: Univ.-GHS Siegen, Fachbereich Mathematik (Habil.), 1996.

[38]
D. Mumford, J. Fogarty, and F. Kirwan.
Geometric Invariant Theory, volume 34.
Berlin: Springer-Verlag, 3rd enl. ed. edition, 1993.

[39]
D. M. Oberlin.
Convolution and restriction estimates for a 3-surface in $\Bbb
J. Fourier Anal. Appl., 10(4):377–382, 2004.

[40]
D. M. Oberlin.
Restricted Radon transforms and unions of hyperplanes.
Rev. Mat. Iberoam., 22(3):977–992, 2006.

[41]
D. M. Oberlin.
Exceptional sets of projections, unions of $k$-planes and
associated transforms.
Israel J. Math., 202(1):331–342, 2014.

[42]
R. Oberlin.
Two bounds for the X-ray transform.
Math. Z., 266(3):623–644, 2010.

[43]
R. T. Rockafellar.
Convex analysis.
Princeton Mathematical Series, No. 28. Princeton University Press,
Princeton, N.J., 1970.

[44]
K. M. Rogers.
On a planar variant of the Kakeya problem.
Math. Res. Lett., 13(2-3):199–213, 2006.

[45]
B. Shayya.
Mizohata-Takeuchi estimates in the plane, 2022.

[46]
A. N. Varčenko.
Newton polyhedra and estimates of oscillatory integrals.
Funkcional. Anal. i Priložen., 10(3):13–38, 1976.
English translation: Functional Anal. Appl. 18 (1976), no. 3,
175–196 (1977).

[47]
N. R. Wallach.
Geometric Invariant Theory - over the real and complex numbers.
Universitext. Springer, Cham, 2017.

[48]
T. Wolff.
A mixed norm estimate for the X-ray transform.
Rev. Mat. Iberoamericana, 14(3):561–600, 1998.

[49]
T. Wolff.
A sharp bilinear cone restriction estimate.
Ann. of Math. (2), 153(3):661–698, 2001.