# Some sharp inequalities of Mizohata–Takeuchi-type

Anthony Carbery, Marina Iliopoulou and Hong Wang  School of Mathematics and
Maxwell Institute for Mathematical Sciences
University of Edinburgh
Edinburgh EH9 3FD
Scotland<EMAIL_ADDRESS>Department of Mathematics
National and Kapodistrian University of Athens
Zografou
Athens 157 84
Greece<EMAIL_ADDRESS>Courant institute of Mathematical Sciences
New York University
New York, NY 10012
USA<EMAIL_ADDRESS>
###### Abstract.

Let $\Sigma$ be a strictly convex, compact patch of a $C^{2}$ hypersurface in
$\mathbb{R}^{n}$, with non-vanishing Gaussian curvature and surface measure
$d\sigma$ induced by the Lebesgue measure in $\mathbb{R}^{n}$. The
Mizohata–Takeuchi conjecture states that

$\int|\widehat{gd\sigma}|^{2}w\leq C\|Xw\|_{\infty}\int|g|^{2}$

for all $g\in L^{2}(\Sigma)$ and all weights
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$, where $X$ denotes the $X$-ray
transform. As partial progress towards the conjecture, we show, as a
straightforward consequence of recently-established decoupling inequalities,
that for every $\epsilon>0$, there exists a positive constant $C_{\epsilon}$,
which depends only on $\Sigma$ and $\epsilon$, such that for all $R\geq 1$ and
all weights $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ we have

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\leq
C_{\epsilon}R^{\epsilon}\sup_{T}\left(\int_{T}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int|g|^{2},$

where $T$ ranges over the family of all tubes in $\mathbb{R}^{n}$ of
dimensions $R^{1/2}\times\dots\times R^{1/2}\times R$. From this we deduce the
Mizohata–Takeuchi conjecture with an $R^{\frac{n-1}{n+1}}$-loss; i.e., that

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\leq
C_{\epsilon}R^{\frac{n-1}{n+1}+\epsilon}\|Xw\|_{\infty}\int|g|^{2}$

for any ball $B_{R}$ of radius $R$ and any $\epsilon>0$. The power
$(n-1)/(n+1)$ here cannot be replaced by anything smaller unless properties of
$\widehat{gd\sigma}$ beyond ‘decoupling axioms’ are exploited. We also provide
estimates which improve this inequality under various conditions on the
weight, and discuss some new cases where the conjecture holds.

##  1\. Introduction

Let $n\geq 2$, and henceforth fix $\Sigma$ to be a strictly convex, compact
patch of a $C^{2}$ hypersurface in $\mathbb{R}^{n}$ with non-vanishing
Gaussian curvature; a prototypical example is the sphere $\mathbb{S}^{n-1}$.
Let $d\sigma$ be the surface measure on $\Sigma$, induced by the Lebesgue
measure in $\mathbb{R}^{n}$. The Fourier extension operator associated to
$\Sigma$ is defined by

$g\mapsto\widehat{gd\sigma}$

where

$\widehat{gd\sigma}(x):=\int e^{2\pi i\langle
x,\xi\rangle}g(\xi)d\sigma(\xi)\text{ for }x\in\mathbb{R}^{n}.$

The Fourier restriction or extension conjecture [St78], which lies at the
heart of harmonic analysis, aims to understand the extension operator by
determining its $L^{p}\rightarrow L^{q}$ mapping properties. However, while
Fourier extension estimates provide information on the size of the level sets
of $|\widehat{gd\sigma}|$, they do not reveal much about their shape. The
Mizohata–Takeuchi conjecture aims to shed light in this direction,
specifically regarding the clustering of level sets along lines. The
conjecture arose in the study of dispersive PDE; see [Mi85] for some
background. In that setting, hypersurfaces such as the paraboloid and the cone
are particularly relevant. Although the conjecture stated below arose first in
the context of hypersurfaces with non-vanishing Gaussian curvature, it is
nevertheless expected that it should hold for arbitrary sufficiently smooth
hypersurfaces.

######  Conjecture 1.1.

_(Mizohata–Takeuchi)_ For any $C^{2}$ compact convex hypersurface $\Sigma$ in
$\mathbb{R}^{n}$, the inequality

$\int|\widehat{gd\sigma}|^{2}w\leq C\|Xw\|_{\infty}\int|g|^{2}$

holds for all $g\in L^{2}(\Sigma)$ and all weights
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$, for some $C>0$ that only depends on
$\Sigma$.

Here, $X$ denotes the $X$-ray transform, so that

$\|Xw\|_{\infty}=\sup_{\ell}\int_{\ell}w,$

where the supremum is taken over all lines $\ell$ in $\mathbb{R}^{n}$. By the
compactness of $\Sigma$ and uncertainty principle considerations, the
Mizohata–Takeuchi conjecture is equivalent to

$\int|\widehat{gd\sigma}|^{2}w\leq C\sup_{T}w(T)\int|g|^{2}$

where the supremum is taken over all $1$-neighbourhoods $T$ of doubly-infinite
lines in $\mathbb{R}^{n}$. In particular we may – and indeed we shall – assume
that $w$ is roughly constant at scale $1$.

The Mizohata–Takeuchi conjecture is open in all dimensions, including $n=2$
(where the Fourier extension conjecture has been resolved).111It is a nice
observation of Bennett and Nakamura [BN21, p.129] that when $n=2$, the
Mizohata–Takeuchi conjecture implies the Fourier extension conjecture.. It
would directly follow from the truth of the stronger conjecture

$\int|\widehat{gd\sigma}|^{2}w\leq\;\;C\int|g(\xi)|^{2}\sup_{\ell\parallel
N(\xi)}Xw(\ell)\;d\sigma(\xi),$ (1)

a formulation of which in the related context of the disc multipliers is due
to Stein [St78]; here, $N(\xi)$ denotes the normal to $\Sigma$ at $\xi$.

When $\Sigma=\mathbb{S}^{n-1}$ and the weight is radial, the Mizohata–Takeuchi
conjecture is known to hold (see [CRS92, BRV97, CS97a, CS97b, CSV07]), and the
Stein-like conjecture in the same setting is a trivial consequence of this.
When the weight is constant on parallel hyperplanes and the hypersurface is
arbitrary, both conjectures are true. This can be seen by using an affine
change of variables to reduce to the case of horizontal hyperplanes and a
hypersurface parametrised as $(t,\gamma(t))$ for $t\in\mathbb{R}^{n-1}$, and
in this case Plancherel’s theorem in $\mathbb{R}^{n-1}$ gives the result
directly. When $\Sigma=\mathbb{S}^{1}$ and the weight is a measure supported
on $\mathbb{S}^{1}$, both conjectures are also known [BCSV06]. Little is known
beyond these three cases.

One way to measure partial progress on the Mizohata–Takeuchi conjecture is to
consider inequalities of the form

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\leq
CR^{\alpha}\|Xw\|_{\infty}\int|g|^{2}$

where $B_{R}$ is the ball of radius $R$ centred at $0$, and to attempt to
establish such inequalities with the exponent $\alpha$ as small as possible.
By the Agmon–Hörmander trace inequality and the local constancy of $w$ at
scale $1$ we have

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\leq CR\|w\|_{\infty}\int|g|^{2}\leq
CR\|Xw\|_{\infty}\int|g|^{2}$ (2)

in all dimensions $n\geq 2$, and it is known that

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\leq
CR^{1/2}\|Xw\|_{\infty}\int|g|^{2}\;\;\text{ for }n=2.$ (3)

The latter inequality can be traced back to works of Bourgain [B94], Erdog̃an
[E04] and also Carbery and Seeger [CS00] – see [BBC08, Section 4] for further
details of inequalities which can be found in the literature and which have
(3) as a consequence. We give a more direct proof of this in Section 3 below.
In more recent developments, it is a consequence of the main result in Du and
Zhang [DZ19] that one may take any $\alpha>(n-1)/n$ (in fact, with the
significantly smaller functional $\sup_{x,\,1\leq r\leq R}w(B(x,r))/r^{n-1}$
in place of $\|Xw\|_{\infty}$) for arbitrary $n$. (See also Shayya [Sh21] and
Du et al [DGO+21], who gave alternative arguments when $n=3$ for $\alpha>6/7$
and $\alpha>2/3$ respectively.) In Theorem 1.2 below we show that one may take
any $\alpha>(n-1)/(n+1)$ in all dimensions.

See also [BN21, BNS22] for a tomographic approach to the Mizohata–Takeuchi
conjecture, [Sh22] for related weighted $L^{2}\rightarrow L^{4}$ estimates on
the extension operator, and [GWZ22] for variants of the conjecture when the
supports of $g$ and $w$ are respectively contained in and equal to
neighbourhoods of algebraic varieties.

Notation. The control we shall obtain on
$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w$ will be accompanied by multiplicative
losses of the form $C_{\epsilon}R^{\epsilon}$ for any $\epsilon>0$. In order
to facilitate expression of this, we adopt the following notation.

For any non-negative quantities $A$ and $B$ (which may depend on $R$),
$A\lesssim B$ means that $A\leq cB$ for some constant $c$ that depends only on
$\Sigma$ and the ambient dimension. Likewise, $A\gtrsim B$ means that
$B\lesssim A$, while $A\sim B$ means that $A\lesssim B$ and $A\gtrsim B$. With
$R\geq 1$ fixed, $A\lessapprox B$ means that, for every $\epsilon>0$, there
exists a constant $C_{\epsilon}$, depending only on $\epsilon$, $\Sigma$ and
the ambient dimension, such that $A\leq C_{\epsilon}R^{\epsilon}B$. Similarly,
$A\gtrapprox B$ means that $B\lessapprox A$, while $A\approx B$ means that
$A\lessapprox B$ and $A\gtrapprox B$.

For a weight $w$ on $\mathbb{R}^{n}$ and $A\subset\mathbb{R}^{n}$, we denote
by $w(A)$ the integral $\int_{A}w$ with respect to Lebesgue measure on
$\mathbb{R}^{n}$.

For $n\geq 2$, an $n$-dimensional ball of radius $r$ will be referred to as an
$r$-ball. A tube of length $r$ and cross section an $(n-1)$-dimensional ball
of radius $r^{1/2}$ will be referred to as an $r^{1/2}$-tube. With $R\geq 1$
fixed and $1\leq r\leq R$, we let $\mathbb{T}_{r}$ be the set of
$r^{1/2}$-tubes intersecting $B_{R}$.

For a line $\ell$ in $\mathbb{R}^{n}$ and $g\in L^{2}(\Sigma)$, we write
$\ell\perp{\rm supp}\;g$ if the direction of $\ell$ is parallel to one of the
normals to ${\rm supp}\;g\subset\Sigma$.

For a tube $T$ in $\mathbb{R}^{n}$, we write $T\perp{\rm supp}\;g$ if the
central line of $T$ is parallel to one of the normals to ${\rm
supp}\;g\subset\Sigma$. $\blacksquare$

Statement of results. In this paper, we present several $L^{2}$-weighted
inequalities for the Fourier extension operator which are related to the
Mizohata–Takeuchi conjecture. To place our results in context, we first
observe that the Stein–Tomas inequality

$\|\widehat{gd\sigma}\|_{L^{\frac{2(n+1)}{n-1}}(\mathbb{R}^{n})}\lesssim\|g\|_{2}$

together with Hölder’s inequality implies that

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\lesssim\left(\int_{B_{R}}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int|g|^{2}$

for all $g$ and all non-negative $w$. The first Mizohata–Takeuchi-type
estimates that we present give a significant improvement over this inequality,
and follow from the refined Stein–Tomas-type estimate in [GIOW20]. They are
given in Theorem 1.2 below. The main inequality of this result, (4), is
closely related to, but logically independent from, the Mizohata–Takeuchi
conjecture, and it is sharp in the sense we discuss below the statement. Its
consequence (5) is also sharp given the techniques that we employ; see [Gu22],
the remarks at the end of this section and Section 7. Estimates which improve
on Theorem 1.2 appear in Lemma 1.4 (for $g$ with small support), as well as in
Theorems 1.6 and 1.8 (for weights that are constant on slabs), and arise as
consequences of Theorem 1.2.

######  Theorem 1.2.

Let $n\geq 2$. For every $\epsilon>0$, there exists a positive constant
$C_{\epsilon}$, which depends only on $\Sigma$ and $\epsilon$, such that

$\displaystyle\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\leq
C_{\epsilon}R^{\epsilon}\sup_{T\in\mathbb{T}_{R}:\;T\perp{\rm
supp}\;g}\left(\int_{T}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int|g|^{2},$
(4)

and in particular

$\displaystyle\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\lessapprox
R^{\frac{n-1}{n+1}}\left(\sup_{\ell\perp{\rm
supp}\;g}Xw(\ell)\right)\int|g|^{2}$ (5)

for all $R\geq 1$, $g\in L^{2}(\Sigma)$ and weights
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$.

The second statement follows from the first upon noting that

$\displaystyle\begin{aligned} \sup_{T\in\mathbb{T}_{R}:\;T\perp{\rm
supp}\;g}\left(\int_{T}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}&\leq\|w\|_{\infty}^{\frac{n-1}{n+1}}\left(\sup_{T\in\mathbb{T}_{R}:\;T\perp{\rm
supp}\;g}w(T)\right)^{\frac{2}{n+1}}\\\ &\lesssim
R^{\frac{n-1}{n+1}}\;\|w\|_{\infty}^{\frac{n-1}{n+1}}\;\left(\sup_{\ell\perp{\rm
supp}\;g}Xw(\ell)\right)^{\frac{2}{n+1}}\\\ \end{aligned}$

and using the approximate constancy of $w$ at scale $1$.

Notice that Theorem 1.2, unlike the Mizohata–Takeuchi conjecture itself,
requires non-vanishing curvature of $\Sigma$.

######  Remark 1.3.

Inequality (4) of Theorem 1.2 is sharp in the following senses. Firstly, if
the exponent $r$ is such that

$\int_{B_{R}}|\widehat{gd\sigma}|^{q}w\lesssim\left(\int_{B_{R}}w^{r}\right)^{1/r}\left(\int|g|^{p}\right)^{q/p}$

(which, by duality, is equivalent to an $L^{p}-L^{qr^{\prime}}$ Fourier
extension estimate) holds, then necessarily
$1/qr^{\prime}\leq(n-1)/(n+1)p^{\prime}$; so the exponent $(n+1)/2$ appearing
in (4) (in which $p=q=2$) cannot be increased, irrespective of the size of the
tubes $T\subset B_{R}$. Secondly, fixing $r=(n+1)/2$ in (4), we cannot reduce
the width of the tubes appearing to be significantly smaller than $R^{1/2}$.
These two assertions can both be seen by testing as usual on $g$ the indicator
function of an $R^{-1/2}$-cap and $w$ the indicator of the dual
$R^{1/2}$-tube. On the other hand, we do not know whether one may take
$\epsilon=0$ in (4) and (5). It is likely that when $n=2$, we may be able to
replace the $R^{\epsilon}$ term by a power of $\log R$; see Remark 4.3 below.

Theorem 1.2 will follow from the more precise Theorem 4.1, in which
$\mathbb{T}_{R}$ is replaced by the set of tubes featuring in the wave packet
decomposition of $g$ at scale $R$.

We now turn to our other results. Theorems 1.6 and 1.8 below are improvements
of Theorem 1.2 for weights that exhibit a level of local constancy along
slabs. In the extreme case where there is no such local constancy beyond on
unit scale, both theorems reduce to Theorem 1.2. Theorem 1.6 involves slabs
that are ‘roughly parallel’ to caps of $\Sigma$, while Theorem 1.8 addresses
the general case.

Both theorems (and, in fact, the more precise Theorems 6.1 and 6.2) will
follow from a strengthened version of Theorem 1.2 for functions $g$ with small
support (Lemma 1.4 below) which we will prove for all weights.

In order to state Theorems 1.6 and 1.8, we first establish some further
notation, and introduce a quantity which is intermediate between the quantity

$\sup_{T\in\mathbb{T}_{R}:\;T\perp{\rm
supp}\;g}\left(\int_{T}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}$

occuring in Theorem 1.2 and a quantity more directly geared towards that
occuring in the Mizohata–Takeuchi conjecture itself. This will involve
considering an amalgam of ‘running averages’ of $w$ at certain scales related
to the level of constancy that we are assuming, which is measured by a
parameter $1\leq\rho\leq R$ which we now fix. Let $E\subset\Sigma$. For each
$T_{R}\in\mathbb{T}_{R}$ such that $T_{R}\perp E$, we cover $T_{R}$ by
essentially disjoint tubes $S_{\rho}\in\mathbb{T}_{\rho}$ which are parallel
to and contained in $T_{R}$. For $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ and
$E\subset\Sigma$ we define

$A_{\rho,R,E}(w):=\frac{1}{\rho^{\frac{n-1}{2}}}\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp
E}\left(\sum_{S_{\rho}\subset
T_{R}}w(S_{\rho})^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}},$

a quantity which can be expressed more geometrically as

$\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp E}\left(\sum_{S_{\rho}\subset
T_{R}}\left(\frac{w(S_{\rho})}{|S_{\rho}|}\right)^{\frac{n+1}{2}}|S_{\rho}|\right)^{\frac{2}{n+1}}$

and thus is seen to increase as $\rho$ gets smaller.222By Hölder’s inequality
we have, for $\lambda\geq 1$ and a tessellation of an $S_{\lambda\rho}$ by
$S_{\rho}$’s,
$\left(\frac{w(S_{\lambda\rho})}{|S_{\lambda\rho}|}\right)^{\frac{n+1}{2}}|S_{\lambda\rho}|\lesssim\sum_{S_{\rho}\subset
S_{\lambda\rho}}\left(\frac{w(S_{\rho})}{|S_{\rho}|}\right)^{\frac{n+1}{2}}|S_{\rho}|.$
For $\rho=1$,

$A_{1,R,E}(w)\sim\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp
E}\left(\int_{T_{R}}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}$

is the quantity appearing on the right-hand side of Theorem 1.2, controlling
the $L^{2}(E)\rightarrow L^{2}(w)$-norm of the extension operator. Theorem 1.2
fails in general for $g$ supported on $E$ if the above quantity is replaced by
the smaller

$A_{R,R,E}(w)=\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp
E}\;\frac{w(T_{R})}{R^{\frac{n-1}{2}}}$

(and in fact by $A_{\rho,R,E}(w)$ for any $\rho\gg 1$, as can be seen by
taking $g$ to be the indicator function of a $1$-cap and $w$ the indicator
function of the unit ball). In the results which follow, however, we shall
show that under certain auxiliary conditions ($g$ being supported on a small
cap, or the weight being the indicator function of a union of small slabs),
Theorem 1.2 nevertheless does hold for $g\in L^{2}(E)$ if we replace the
quantity $A_{1,R,E}(w)$ with $A_{\rho,R,E}(w)$ for an appropriate choice of
$\rho$. To further compare these two quantities, observe that

$A_{\rho,R,E}(w)\leq\sup_{S_{\rho}\in\mathbb{T}_{\rho}:\;S_{\rho}\perp
E}\left(\frac{w(S_{\rho})}{|S_{\rho}|}\right)^{\frac{n-1}{n+1}}\sup_{T_{R}\in\mathbb{T}_{R},T_{R}\perp
E}w(T_{R})^{\frac{2}{n+1}},$ (6)

which becomes

$A_{\rho,R,E}(w)\leq\sup_{S_{\rho}\in\mathbb{T}_{\rho}:\;S_{\rho}\perp
E}\left(\frac{w(S_{\rho})}{|S_{\rho}|}\right)^{\frac{n-1}{n+1}}\;A_{1,R,E}(w)$

when $w$ is an indicator function (which we may well assume for our purposes).

In situations in which we are able to bound the $L^{2}(E)\rightarrow
L^{2}(w)$-norm of the extension operator by $A_{\rho,R,E}(w)$, inequality (6)
leads to improved bounds in terms of $\|Xw\|_{\infty}$; in particular, to a
gain on Theorem 1.2 by a factor $\rho^{-\frac{n-1}{n+1}}$. Indeed, by (6),

$A_{\rho,R,E}(w)\leq\left(\frac{\|Xw\|_{\infty}}{\rho}\right)^{\frac{n-1}{n+1}}(R^{\frac{n-1}{2}}\|Xw\|_{\infty})^{\frac{2}{n+1}}\lesssim\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp
E}Xw(\ell).$

A situation such as this arises when $g$ is supported in a $\rho^{-1/2}$-cap
of $\Sigma$ (that is, the intersection of $\Sigma$ with a $\rho^{-1/2}$-ball),
and is summarised in Lemma 1.4 below. The lemma will in turn be used in
conjunction with a decoupling argument to derive Theorems 1.6 and 1.8 for all
functions $g$ and restricted classes of weights. Note that, in Lemma 1.4
below, the subscript $\tau$ on $g_{\tau}$ is not strictly needed, but we
retain it to emphasise its support.

######  Lemma 1.4.

_(Small caps)_ For every $\epsilon>0$, there exists $C_{\epsilon}>0$ such that
for all weights $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$, whenever
$1\leq\rho\leq R$, $\tau$ is a $\rho^{-1/2}$-cap of $\Sigma$ and $g_{\tau}\in
L^{2}(B^{n-1})$ is supported in $\tau$, we have

$\int_{B_{R}}|\widehat{g_{\tau}d\sigma}|^{2}w\leq
C_{\epsilon}R^{\epsilon}\;A_{\rho,R,\;{\rm
supp}\;g_{\tau}}(w)\int|g_{\tau}|^{2},$

and therefore also

$\int_{B_{R}}|\widehat{g_{\tau}d\sigma}|^{2}w\lessapprox\;\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp{\rm
supp}\;g_{\tau}}Xw(\ell)\int|g_{\tau}|^{2}.$ (7)

In order to state Theorems 1.6 and 1.8, we need to make precise what we mean
by a slab, and by a slab being ‘roughly parallel’ to caps of $\Sigma$.

######  Definition 1.5.

Fix $R\geq 1$, $1\leq\rho\leq R$ and $0\leq\nu\leq\pi/2$. We define a
$\rho^{1/2}$-slab to be any affine copy of the 1-neighbourhood of an
$(n-1)$-dimensional $\rho^{1/2}$-ball in $\mathbb{R}^{n}$. We say that a slab
is $\nu$-parallel to $\Sigma$ if all normals to $\Sigma$ create angle at least
$\nu$ with the slab (that is, they create angle at most $\frac{\pi}{2}-\nu$
with the normal to the slab).

In this definition, $\nu$ is a measure of how large the angles are between the
slab and the normals to $\Sigma$. The larger $\nu$ is, the larger these angles
are, and the more ‘parallel’ $\Sigma$ and the slab look.

With these preliminaries in hand, we are now ready to state our remaining
results. In the first two results which follow, the implicit constant blows up
as $\nu\downarrow 0$. Thus, the interesting cases of these two results are
those in which $\nu$ is large, i.e. when the slabs create large angles with
the normals to $\Sigma$. If for instance $\Sigma$ is roughly horizontal (i.e.
all normals to $\Sigma$ are within angle $\leq 1/100$ from the vertical
direction), then Theorem 1.6 gives meaningful results for slabs that are also
nearly horizontal (e.g. creating angle $\geq 2/100$ with the vertical
direction).

######  Theorem 1.6.

_(Slabs $\nu$-parallel to $\Sigma$)_ For every $0<\nu\leq\pi/2$ and
$\epsilon>0$, there exists $C_{\epsilon,\nu}>0$ such that the following hold.
Let $g\in L^{2}(\Sigma)$. For $R\geq 1$ and
$R^{\epsilon}\lesssim_{\epsilon}\rho\leq R$, let
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ be a weight of the form
$\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $\rho^{1/2}$-slabs $\nu$-parallel to $\Sigma$. Then the inequality

$\displaystyle\int_{B_{R}}|\widehat{gd\sigma}|^{2}w$ $\displaystyle\leq
C_{\epsilon,\nu}R^{\epsilon}A_{\rho,R,\;{\rm supp}\;g}(w)\int|g|^{2}$
$\displaystyle\lessapprox_{\nu}\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp{\rm
supp}\;g}Xw(\ell)\int|g|^{2}$

holds. In fact, if

$g=\sum_{\tau\in\mathfrak{T}}g_{\tau},\qquad{{\rm supp}\;g_{\tau}\subset\tau}$

for some boundedly overlapping family $\mathfrak{T}$ of $\rho^{-1/2}$-caps
$\tau$ of $\Sigma$, then

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\lessapprox_{\nu}\sum_{\tau\in\mathfrak{T}}A_{\rho,R,\;{\rm
supp}\;g_{\tau}}(w)\int|g_{\tau}|^{2}\lessapprox_{\nu}\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sum_{\tau\in\mathfrak{T}}\;\sup_{\ell\perp{\rm
supp}\;g_{\tau}}Xw(\ell)\int|g_{\tau}|^{2}.$

It follows that Stein’s stronger conjecture (1) (and thus the
Mizohata–Takeuchi conjecture) holds under the conditions of Theorem 1.6 when
the slabs involved are $R^{1/2}$-slabs. We single this out explicitly as a
corollary.

######  Corollary 1.7.

Let $R\geq 1$ and suppose that $w$ is a weight of the form
$\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $R^{1/2}$-slabs which are $\nu$-parallel to $\Sigma$ for some
$0<\nu\leq\pi/2$. Then

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\lessapprox_{\nu}\int|g(\xi)|^{2}\sup_{\ell\parallel
N(\xi)}Xw(\ell)d\sigma(\xi).$

for all $g\in L^{2}(\Sigma)$.

Stein’s conjecture continues to hold even when the slabs are curved. The
precise formulation of this appears in Corollary 3.4, and it is proved using a
direct method, which does not rely on Theorem 1.2, and which also featured in
[Gu22].

A substitute result for Theorem 1.6 in the case where there is no restriction
on $\nu$ (i.e. when the slabs can create arbitrarily small angles with normals
to $\Sigma$) is as follows.

######  Theorem 1.8.

_(All slabs)_ For every $\epsilon>0$, there exists $C_{\epsilon}>0$ such that
the following hold. Let $g\in L^{2}(\Sigma)$. For $R\geq 1$ and
$R^{\epsilon}\lesssim_{\epsilon}\rho\leq R$, let
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ be a weight of the form
$\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $\rho^{1/2}$-slabs with no conditions on their directions. Then the
inequality

$\displaystyle\int_{B_{R}}|\widehat{gd\sigma}|^{2}w$ $\displaystyle\leq
C_{\epsilon}R^{\epsilon}A_{\rho^{1/2},R,\;{\rm supp}\;g}(w)\int|g|^{2}$
$\displaystyle\lessapprox\left(\frac{R}{\rho^{1/2}}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp{\rm
supp}\;g}Xw(\ell)\int|g|^{2}$

holds. In fact, if

$g=\sum_{\tau\in\mathfrak{T}}g_{\tau},\qquad{{\rm supp}\;g_{\tau}\subset\tau}$

for some boundedly overlapping family $\mathfrak{T}$ of $\rho^{-1/4}$-caps
$\tau$ of $\Sigma$, then

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\lessapprox\sum_{\tau\in\mathfrak{T}}A_{\rho^{1/2},R,\;{\rm
supp}\;g_{\tau}}(w)\int|g_{\tau}|^{2}\lessapprox\left(\frac{R}{\rho^{1/2}}\right)^{\frac{n-1}{n+1}}\sum_{\tau\in\mathfrak{T}}\;\sup_{\ell\perp{\rm
supp}\;g_{\tau}}Xw(\ell)\int|g_{\tau}|^{2}.$

######  Corollary 1.9.

_( $R^{1/2}$-slabs)_ Let $R\geq 1$ and suppose that $w$ is a weight of the
form $\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $R^{1/2}$-slabs. Then

$\int_{B_{R}}|\widehat{gd\sigma}|^{2}w\lessapprox
R^{\frac{n-1}{2(n+1)}}\sup_{\ell\perp{\rm supp}\;g}Xw(\ell)\int|g|^{2}.$

for all $g\in L^{2}(\Sigma)$.

Sharpness of inequality (5) given the choice of technique. During the recent
talk [Gu22], which in fact partially inspired the work in this paper, Guth
explained that, using only basic local constancy and local
$L^{2}$-orthogonality properties of the functions $\widehat{gd\sigma}$ – which
are indeed the only properties that we exploit in proving Theorem 1.2 – one
cannot prove the Mizohata–Takeuchi conjecture for $B_{R}$ with a loss better
than $\sim(\log R)^{-3}R^{\frac{n-1}{n+1}}$.

This means that inequality (5) of Theorem 1.2, which establishes the
conjecture with a loss of $\lessapprox R^{\frac{n-1}{n+1}}$, is essentially
sharp given the techniques used.

Guth’s argument is discussed in Section 7 for purposes of self-containment.

Acknowledgements. We would like to thank Larry Guth, whose inspiring talk
[Gu22] partially motivated the work in this paper, for giving us permission to
present here a version of his main argument from that talk. We also thank
Jonathan Bennett for many illuminating conversations on this topic. The first
author would like to acknowledge support from a Leverhulme Fellowship while
part of this research was undertaken, and to thank David Beltrán and Bassam
Shayya for some helpful conversations. The third author would like to
acknowledge support from NSF Grant DMS-2238818 and DMS-2055544.

##  2\. Preliminaries

For our purposes, we may assume that all normals to $\Sigma$ have angle at
most $1/100$ from the vertical direction, and that the projection of $\Sigma$
on the hyperplane $\mathbb{R}^{n-1}\times\\{0\\}$ is contained in the unit
ball $B^{n-1}$ centred at 0. This convention allows us to assume that $\Sigma$
has a parametrisation

$\Sigma=\\{\Sigma(\omega):=(\omega,h(\omega))\text{, for }\omega\in
B^{n-1}\\}$

for some $h:B^{n-1}\rightarrow\mathbb{R}$, and to work with the operator $E$
instead of $\widehat{\cdot\;d\sigma}$, where

$Eg(x):=\int_{B^{n-1}}e^{2\pi i\langle
x,\Sigma(\omega)\rangle}g(\omega)d\omega\;\text{ for }x\in\mathbb{R}^{n}.$

From now on, for fixed $\Sigma$ and$\epsilon>0$, we say that a quantity
$C(R,\epsilon)$ satisfies

$C(R,\epsilon)={\rm RapDec}_{\epsilon}(R)$

if for every $N\in\mathbb{N}$ there exists a non-negative constant
$C_{N,\epsilon}$ such that uniformly in $R\geq 1$ we have

$|C(R,\epsilon)|\leq C_{N,\epsilon}R^{-N}.$

Wave packet decomposition adapted to $\bm{B_{R}}$. Let $\epsilon>0$ and
$0<\delta\ll\epsilon$. Fix $R\gg 1$, and cover $B^{n-1}$ by boundedly
overlapping balls $\theta$ of radius $R^{-1/2}$. The set of these balls will
be denoted by $\Theta_{R}$, and the balls will be referred to as
$R^{-1/2}$-caps. Let $\\{\psi_{\theta}\\}_{\theta\in\Theta_{R}}$ be a smooth
partition of unity adapted to this cover. Thus,

$g=\sum_{\theta\in\Theta_{R}}\psi_{\theta}g$

for any $g:\mathbb{R}^{n-1}\rightarrow\mathbb{C}$ supported in $B^{n-1}$ (and
belonging to some suitable class). Now, cover $\mathbb{R}^{n-1}$ by boundedly
overlapping balls of radius $CR^{(1+\delta)/2}$ and centres on the lattice
$V_{R}:=R^{(1+\delta)/2}\mathbb{Z}^{n-1}$. There exists a bump function
$\eta$, adapted to the ball $B(0,R^{(1+\delta)/2})$, so that the bump
functions $\eta_{v}:=\eta(\cdot-v)$, over $v\in V_{R}$, form a partition of
unity for this cover. It follows that, with $\widehat{\cdot}$ and
$\widecheck{\cdot}$ denoting the $(n-1)$-dimensional Fourier transform and its
inverse respectively,

$\widecheck{g}=\sum_{(\theta,v)}\eta_{v}(\psi_{\theta}g)^{\widecheck{}}$

and thus

$g=\sum_{(\theta,v)}\widehat{\eta_{v}}\ast(\psi_{\theta}g)$

for all $g$ as above. Finally, restrict each of the above summands to the
corresponding cap $\theta$. In particular, let

$g_{\theta,v}:=\widetilde{\psi}_{\theta}\cdot(\widehat{\eta_{v}}\ast(\psi_{\theta}g)),$

where
$\widetilde{\psi}_{\theta}:=\widetilde{\psi}(R^{1/2}(\cdot-\omega_{\theta}))$
for some fixed smooth bump function $\widetilde{\psi}$ (where
$\omega_{\theta}$ is the centre of the cap $\theta$), chosen so that
$\widetilde{\psi}_{\theta}$ is supported in $\theta$ and equals 1 on the
$cR^{1/2}$-neighbourhood of ${\rm supp}\;\psi_{\theta}$, for some small $c>0$.

The $g_{\theta,v}$ are the wave packets of $g$ at scale $R$, while
$\\{g_{\theta,v}\\}_{(\theta,v)\in\Theta_{R}\times V_{R}}$ constitutes the
wave packet decomposition of $g$ at this scale. Note that the decomposition is
$\epsilon$-dependent.

The function $g$ is roughly the sum of its wave packets, all of which are
roughly orthogonal. More precisely, note that the function
$\widehat{\eta_{v}}$ is rapidly decaying when $|\omega|\gg R^{-(1+\delta)/2}$,
so

$\|g_{\theta,v}-\widehat{\eta_{v}}\ast(\psi_{\theta}g)\|_{\infty}\leq{\rm
RapDec}_{\epsilon}(R)\|g\|_{2}\text{ for each }(\theta,v),$

hence

$\|g-\sum_{(\theta,v)\in\Theta_{R}\times V_{R}}g_{\theta,v}\|_{\infty}\leq{\rm
RapDec}_{\epsilon}(R)\|g\|_{2}.$ (wp1)

The functions $g_{\theta,v}$ are almost orthogonal, in the sense that

$\|\sum_{(\theta,v)\in\mathbb{W}}g_{\theta,v}\|_{2}^{2}\sim\sum_{(\theta,v)\in\mathbb{W}}\|g_{\theta,v}\|_{2}^{2}$
(wp2)

for every subset $\mathbb{W}$ of $\Theta_{R}\times V_{R}$.

It turns out that, for every $(\theta,v)$, $Eg_{\theta,v}$ is essentially
supported in

$T_{\theta,v}:=\big{\\{}x\in
B_{R}:|x^{\prime}+x_{n}\partial_{\omega}h(\omega_{\theta})-v|\leq
R^{1/2+\delta}\big{\\}},$

the $R^{1/2+\delta}$-tube in $B_{R}$ whose central line passes through $(v,0)$
and has direction the normal
$N(\theta):=(\partial_{\omega}h(\omega_{\theta}),-1)$ to the cap
$\Sigma(\theta)$. Indeed, it follows by a non-stationary phase argument that

$|Eg_{\theta,v}(x)|\leq(1+R^{-1/2}|x^{\prime}+x_{n}\partial_{\omega}h(\omega_{\theta})-v|)^{-(n+1)}{\rm
RapDec}_{\epsilon}(R)\|g\|_{2},\;\;\;\forall x\in B_{R}\setminus
T_{\theta,v};$ (wp3)

a detailed analysis can be found in [Gu18].

Due to the curvature of $\Sigma$, different surface caps $\Sigma(\theta)$ have
different normals, so there is a one-to-one correspondence between the pairs
$(\theta,v)$ and the tubes $T_{\theta,v}$. We may thus denote each wave packet
$g_{\theta,v}$ by $g_{T}$, for the tube $T=T_{\theta,v}$.

Henceforth, denote

$\mathbb{T}_{\epsilon}(B_{R}):=\\{T_{\theta,v}:\;(\theta,v)\in\Theta_{R}\times
V_{R}\text{ and }T_{\theta,v}\cap B_{R}\neq\varnothing\\}$

and

$\mathbb{T}^{\theta}_{\epsilon}(B_{R}):=\\{T_{\overline{\theta},v}:|\overline{\theta}-\theta|\lesssim
R^{-1/2},\;v\in V_{R}\text{ and }T_{\theta,v}\cap B_{R}\neq\varnothing\\}$

for each $\theta\in\Theta_{R}$, where the implicit multiplicative constant is
sufficiently large. The above analysis ensures that

$Eg(x)=\sum_{T\in\mathbb{T}_{\epsilon}(B_{R})}Eg_{T}(x)+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\;\;\text{ for all }x\in B_{R},$ (wp4)

while also that any function $g_{\theta}$ supported on $\theta\in\Theta_{R}$
satisfies

$Eg_{\theta}(x)=\sum_{T\in\mathbb{T}_{\epsilon}^{\theta}(B_{R})}Eg_{T}(x)+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\;\;\text{ for all }x\in B_{R}.$ (wp5)

We will be referring to $\\{g_{T}\\}_{T\in\mathbb{T}_{\epsilon}(B_{R})}$ as
the wave packet decomposition of $g$ adapted to $B_{R}$.

Wave packet decompositions adapted to other balls. Let
$R^{\epsilon}\lesssim_{\epsilon}\rho\leq R$, and fix a ball $B=B(y,\rho)$. For
$x\in\mathbb{R}^{n}$, set $\widetilde{x}:=x-y$. It holds that

$\displaystyle\begin{aligned} Eg(x)&=\int e^{2\pi i\langle
x,\Sigma(\omega)\rangle}g(\omega)d\omega\\\ &=\int e^{2\pi
i\langle\widetilde{x},\Sigma(\omega)\rangle}e^{2\pi i\langle
y,\Sigma(\omega)\rangle}g(\omega)d\omega\\\
&=E\widetilde{g}(\widetilde{x}),\end{aligned}$

where $\widetilde{g}(\omega)=e^{2\pi i\langle
y,\Sigma(\omega)\rangle}g(\omega)$. For every $x\in B$, $\widetilde{x}$ lives
in $B_{\rho}$; therefore, by the earlier discussion,

$\displaystyle Eg(x)$
$\displaystyle=\sum_{T\in\mathbb{T}_{\epsilon}(B_{\rho})}E\widetilde{g}_{T}(\widetilde{x})+{\rm
RapDec}_{\epsilon}(\rho)\int|\widetilde{g}|^{2}$
$\displaystyle=\sum_{T\in\mathbb{T}_{\epsilon}(B_{\rho})}E\widetilde{g}_{T}(x-y)+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\qquad{\text{for all }}x\in B.$ (wp6)

From now on, we will be referring to
$\\{\widetilde{g}_{T}\\}_{T\in\mathbb{T}_{\epsilon}(B_{\rho})}$ as the wave
packet decomposition of $g$ adapted to $B$. Note that this decomposition is
$y$-dependent.

By the above analysis, for every $\rho^{-1/2}$-cap $\tau$ we have

$Eg_{\tau}(x)=\sum_{T\in\mathbb{T}^{\tau}_{\epsilon}(B_{\rho})}E\widetilde{g}_{T}(x-y)+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\qquad{\text{for all }}x\in B.$ (wp7)

Each of the wave packets in the above summand is essentially constant in
magnitude; this is made rigorous in the subsection below.

Fourier localisation and local constancy. Let $\epsilon>0$ and
$R^{\epsilon}\lesssim_{\epsilon}\rho\leq R$. Fix $g\in L^{2}(B^{n-1})$ and a
$\rho^{-1/2}$-cap $\tau$.

Roughly speaking, since $g_{\tau}$ is supported in $\tau$, the Fourier
transform of $Eg_{\tau}$ is supported in the $\rho^{-1}$-neighbourghood of
$\Sigma(\tau)$. The uncertainty principle then dictates that $|Eg_{\tau}|$ is
essentially constant on each dual object, i.e. on each $\rho^{1/2}$-tube
pointing in the direction the normal to $\Sigma(\tau)$.

The above heuristic is made rigorous as follows. Let $\omega(\tau)$ be the
centre of $\tau$. The patch of the tangent space to $\Sigma$ at
$\Sigma(\omega_{\tau})$ that lives over $\tau$ is the set

$T_{\tau}\Sigma:=\Big{\\{}\Sigma(\omega_{\tau})+M_{\tau}\cdot(\omega-\omega_{\tau}):\;\omega\in\tau\Big{\\}},\;\;\text{where
}M_{\tau}:=\begin{bmatrix}I_{n-1}&0\\\
\partial_{\omega}h(\omega_{\tau})&1\end{bmatrix}.$

The convex set

$S(\tau):=\left\\{\Sigma(\omega_{\tau})+M_{\tau}\cdot(\omega-\omega_{\tau})+t\cdot\frac{N(\omega_{\tau})}{\|N(\omega_{\tau})\|}:\;\;\omega\in\tau,\;t\in\big{[}-\rho^{-1},\rho^{-1}\big{]}\right\\}$

is a ‘thickening’ of the above tangent patch by $\rho^{-1}$ in the direction
normal to $\Sigma(\tau)$. The Fourier transform of ${Eg_{\tau}}_{|_{B_{R}}}$
is essentially supported in a dilation of $S(\tau)$. We are interested in a
precise version of this for appropriate cut-offs of $Eg_{\tau}$.

In particular, let $\zeta:\mathbb{R}^{n}\rightarrow\mathbb{R}$ with $\zeta=1$
on $B_{1}$ and $\zeta=0$ outside $B_{2}$. For every ball
$B=B(\overline{x},\rho)$ in $\mathbb{R}^{n}$ define

$\zeta_{B}(x):=\zeta\Big{(}\frac{x-\overline{x}}{\rho}\Big{)}.$

There exists a constant $C$, depending only on the dimension $n$, such that
the following holds.

######  Proposition 2.1.

_(Fourier localisation)_ Let $R^{\epsilon}\lesssim_{\epsilon}\rho\leq R$, and
let $g_{\tau}$ be supported in a $\rho^{-1/2}$-cap $\tau$. Then, for every
$\rho$-ball $B$ in $\mathbb{R}^{n}$,

$Eg_{\tau}\cdot\zeta_{B}=G_{\tau}+{\rm
RapDec}_{\epsilon}(\rho)\|g_{\tau}\|_{2},$

for some $G_{\tau}:\mathbb{R}^{n}\rightarrow\mathbb{C}$ with the property that
$\widehat{G_{\tau}}$ is supported in $S(C\cdot\tau)$.

The set $C\cdot\tau$ is the $C\rho^{-1/2}$-cap with the same centre as $\tau$.
The proof of Proposition 2.1 is exposed in full detail in [HI22].

When a function $f$ is Fourier localised on a convex set (such as the slab
$S(\tau)$), then to some extent it can be treated as a constant function on
objects dual to that convex set. The precise statement appears in Lemmas 6.1
and 6.2 in [GWZ20]. For our purposes, we only need the following corollary.

######  Proposition 2.2.

_(Local constancy)_ Let $R^{\epsilon}\lesssim_{\epsilon}\rho\leq R$. Let
$\tau$ be a $\rho^{-1/2}$-cap, and consider a function
$f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ with $\widehat{f}\subset S(\tau)$.
Then, every tube $T$ in $\mathbb{R}^{n}$ with direction $N(\tau)$, radius
$\rho^{1/2}$ and length $\rho$ satisfies

$\sup_{x\in T}|f(x)|^{2}\lesssim\frac{1}{|T|}\int|f|^{2}w_{T},$

for some non-negative function
$\omega_{T}:\mathbb{R}^{n}\rightarrow\mathbb{R}$, with $\omega_{T}=1$ on $T$
and $\omega(x)\sim C_{N}(1+n(x,T))^{-N}$ for all $x\in\mathbb{R}^{n}$ and
$N\in\mathbb{N}$, where $n(x,T)$ is the smallest $n\in\mathbb{N}$ such that
$x\in nT$. In particular, if $g\in L^{2}(B^{n-1})$ and $B$ is a $\rho$-ball
intersecting $T$, then

$\sup_{x\in
T}|Eg_{\tau}(x)|^{2}\lesssim\rho^{\delta}\frac{1}{|2\widetilde{T}|}\int_{2\widetilde{T}}|Eg_{\tau}|^{2}+{\rm
RapDec}_{\epsilon}(R)\int|g_{\tau}|^{2}.$

for all $\widetilde{T}$ in $\mathbb{T}_{\epsilon}^{\tau}(B)$ intersecting $T$.

###### Proof.

The first conclusion is a direct application of Lemmas 6.1 and 6.2 in [GWZ20].
We now in turn apply this conclusion to the function
$Eg_{\tau}\cdot\zeta_{B}$, which is essentially Fourier supported in
$S(C\cdot\tau)$ by Proposition 2.1. Respecting the notation of Proposition
2.1, denote by $T_{C}$ the tube with the same central line as $T$, radius
$(C^{-2}\rho)^{1/2}$ and length $C^{-2}\rho$. We obtain

$\sup_{x\in T}|Eg_{\tau}(x)|^{2}=\sup_{x\in
T}|Eg_{\tau}(x)\cdot\zeta_{B}(x)|^{2}\lesssim\frac{1}{|T_{C}|}\int|Eg_{\tau}\cdot\zeta_{B}|^{2}w_{T_{C}}+{\rm
RapDec}_{\epsilon}(\rho)\int|g_{\tau}|^{2}.$

Since $w_{T}(x)\sim w_{T_{C}}(x)$ for all $x\in\mathbb{R}^{n}$, it holds that

$\displaystyle\begin{aligned}
\frac{1}{|T_{C}|}\int|Eg_{\tau}\cdot\zeta_{B}|^{2}w_{T_{C}}&\lesssim\frac{1}{|T|}\int|Eg_{\tau}\cdot\zeta_{B}|^{2}w_{T}\\\
&\lesssim\frac{1}{|T|}\int_{2B}|Eg_{\tau}|^{2}w_{T}\\\
&\sim\frac{1}{|T|}\int_{2B\cap
2\widetilde{T}}|Eg_{\tau}|^{2}w_{T}+\frac{1}{|T|}\int_{2B\setminus
2\widetilde{T}}|Eg_{\tau}|^{2}w_{T}\\\
&\lesssim\frac{\rho^{\delta}}{|2\widetilde{T}|}\int_{2\widetilde{T}}|Eg_{\tau}|^{2}w_{T}+{\rm
RapDec}_{\epsilon}(\rho)\frac{1}{|T|}\int_{2B\setminus
2\widetilde{T}}|Eg_{\tau}|^{2}w_{T}.\end{aligned}$

The result follows as, due to the decay properties of $w_{T}$,

${\rm RapDec}_{\epsilon}(\rho)\frac{1}{|T|}\int_{2B\setminus
2\widetilde{T}}|Eg_{\tau}|^{2}w_{T}={\rm RapDec}_{\epsilon}(\rho)={\rm
RapDec}_{\epsilon}(R)\int|g_{\tau}|^{2}.$

∎

##  3\. Some new cases where Mizohata–Takeuchi holds.

In this section, $\Sigma:=\\{(\omega,h(\omega)):\omega\in B^{n-1}\\}$ is a
fixed hypersurface in $\mathbb{R}^{n}$, all of whose normals point within
angle $1/100$ from the vertical direction. There is no requirement that
$\Sigma$ have non-vanishing Gaussian curvature.

The truth of the Mizohata–Takeuchi conjecture for some simple weights (such as
indicator functions of neighbourhoods of roughly horizontal hyperplanes or
hypersurfaces) implies that the conjecture holds for more complicated weights
(superpositions of appropriately large patches of such surfaces). For
instance, the Mizohata–Takeuchi conjecture holds for nearly horizontal
$R^{1/2}$-slabs (case $\rho=R$ of Theorem 1.6) because it holds for horizontal
hyperplanes (Plancherel).

######  Definition 3.1.

A $\rho$-flake (or simply a flake) in $\mathbb{R}^{n}$ is the 1-neighbourhood
of any hypersurface of the form $\\{(\omega,\Gamma(\omega)):\omega\in
B^{n-1}_{\rho}\\}$, where $B^{n-1}_{\rho}$ is a $\rho$-ball in
$\mathbb{R}^{n-1}$ and $\Gamma:B^{n-1}_{\rho}\rightarrow\mathbb{R}$. A flake
is nearly horizontal if all its tangent spaces create angle larger than
$2/100$ with the vertical direction.

Note that $\rho$-slabs are $\rho$-flakes. We will usually be taking $\rho\geq
1$. We emphasise that $\Gamma$ and $h$ are unrelated.

Every line normal to $\Sigma$ which intersects a nearly horizontal flake will
do so along a line segment of length about $1$. Therefore, the following lemma
states that the Mizohata–Takeuchi conjecture holds when the weight is the
indicator of a single nearly horizontal flake.

######  Lemma 3.2.

Let $\gamma$ be a nearly horizontal flake in $\mathbb{R}^{n}$. Then, for all
$R\geq 1$ and $g\in L^{2}(B^{n-1})$,

$\int_{B_{R}\cap\gamma}|Eg|^{2}\lessapprox\int|g|^{2}.$

###### Proof.

The proof easily follows by induction on scales, and only a sketch is provided
here. In particular, the estimate trivially holds when $R\lesssim 1$. For
arbitrary larger $R$, we cover the flake $\gamma$ by finitely overlapping
$R^{1/2}$-balls $B$. For every one of these balls $B$, we may assume that

$\int_{B\cap\gamma}|Eg|^{2}\lessapprox\int|g_{B}|^{2},$

where $g_{B}$ is the sum of the wave packets $g_{T}$ of $g$ at scale $R$ that
intersect $B$. The functions $g_{B}$ are essentially orthogonal, as each of
the tubes $T$ in question has width $R^{1/2+\delta}$ (where as in Section 2,
$0<\delta\ll\epsilon$) and creates angle $\gtrsim 1$ with the flake, hence it
intersects $R^{O(\delta)}$ of the balls $B$. Adding up the above estimate over
all $B$ completes the proof. ∎

######  Remark 3.3.

We emphasise that when $\gamma$ is specifically a horizontal hyperplane, then
the stronger estimate

$\int_{\gamma}|Eg|^{2}=\int|g|^{2}$

directly follows by Plancherel’s theorem. Indeed, for every
$(x,t)\in\mathbb{R}^{n-1}\times\mathbb{R}$,

$Eg(x,t)=\int e^{2\pi i\langle x,\omega\rangle}e^{2\pi
ith(\omega)}g(\omega)d\omega=\widehat{g_{t}}(x),$

where $g_{t}:=e^{2\pi ith(\cdot)}g$ and $\widehat{\cdot}$ denotes the standard
Fourier transform on $\mathbb{R}^{n-1}$. Therefore,

$\int|Eg(\cdot,t)|^{2}=\int|\widehat{g_{t}}|^{2}=\int|g_{t}|^{2}=\int|g|^{2}$

for all $t\in\mathbb{R}$. (Note that this directly yields (2).) After an
appropriate change of variables, a similar argument resolves the
Mizohata–Takeuchi conjecture when the weight is the indicator function of the
$1$-neighbourhood of any hyperplane (independently of orientation), and
subsequently when the weight is a sum of indicator functions of such
$1$-neighbourhoods. See [BNS22, Corollary 3] for a stronger estimate (a
certain identity) in this specific scenario.

$\blacksquare$

Lemma 3.2 easily implies the Mizohata–Takeuchi conjecture for superpositions
of appropriately large flakes, and in fact an estimate stronger than Stein’s
conjecture (1).

######  Corollary 3.4.

_(MT holds for $R^{1/2}$-flakes)_ The inequality

$\int_{B_{R}}|Eg|^{2}w\lessapprox\|Xw\|_{\infty}\int|g|^{2}$

holds for every $g\in L^{2}(B^{n-1})$ and any weight
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ of the form
$\sum_{\gamma\in\mathcal{F}}c_{\gamma}\chi_{\gamma}$, where $\mathcal{F}$ is a
family of $R^{1/2}$-flakes. In fact, the stronger estimate

$\int_{B_{R}}|Eg|^{2}w\lessapprox\sum_{T\in\mathbb{T}}\;\sup_{\ell\subset
T}Xw(\ell)\int|g_{T}|^{2}$

holds, where $\\{g_{T}\\}_{T\in\mathbb{T}}$ is the wave packet decomposition
of $g$ at scale $R$.

###### Proof.

Fix $g:L^{2}(B^{n-1})$ and $\gamma\in\mathcal{F}$, and denote by
$\mathbb{T}_{\gamma}$ the set of tubes in $\mathbb{T}$ that intersect
$\gamma$. For all $x\in\gamma$,

$Eg(x)=Eg_{\gamma}(x)+{\rm RapDec}_{\epsilon}(R)\int|g|^{2},$

where $g_{\gamma}:=\sum_{T\in\mathbb{T}_{\gamma}}g_{T}$. Hence, by Lemma 3.2,

$\int_{\gamma}|Eg|^{2}\lessapprox\int|g_{\gamma}|^{2}\sim\sum_{T\in\mathbb{T}_{\gamma}}\int|g_{T}|^{2}$

up to an error of ${\rm RapDec}_{\epsilon}(R)\int|g|^{2}$. Adding up over all
$\gamma\in\mathcal{F}$, we obtain

$\displaystyle\begin{aligned}
\int|Eg|^{2}w&\lessapprox\sum_{\gamma\in\mathcal{F}}c_{\gamma}\sum_{T\in\mathbb{T}_{\gamma}}\int|g_{T}|^{2}\\\
&=\sum_{T\in\mathbb{T}}\left(\sum_{\gamma\in\mathcal{F}:\;\gamma\cap
T\neq\varnothing}c_{\gamma}\right)\int|g_{T}|^{2}\\\
&\lessapprox\sum_{T\in\mathbb{T}}\;\sup_{\ell\subset
T}Xw(\ell)\int|g_{T}|^{2}\end{aligned}$

up to an error of ${\rm RapDec}_{\epsilon}(R)\int|g|^{2}$ (where the final
$\approx 1$-loss is due to the fact that the tubes in $\mathbb{T}$ have width
$R^{1/2+\delta}$, rather than $R^{1/2}$). The last quantity is at most
$\|Xw\|_{\infty}\int|g|^{2}$.

∎

######  Remark 3.5.

The idea behind the proof of Corollary 3.4 also appeared in [Gu22], where the
same result was presented in the special case where the flakes are horizontal
slabs. Moreover, it was there pointed out that the statement of the corollary
also implies (3), i.e. that the Mizohata–Takeuchi conjecture holds with loss
$\lessapprox R^{1/2}$ in $\mathbb{R}^{2}$, by replacing each point in ${\rm
supp}\;w$ by a horizontal $R^{1/2}$-slab (a process which enlarges the maximal
line occupancy of $w$ by $\lesssim R^{1/2}$). Perhaps an easier way to derive
(3) is to observe that, by Proposition 2.2, the Mizohata–Takeuchi conjecture
holds with $\approx 1$-loss for each function $g_{\theta}$ supported in an
$R^{1/2}$-cap $\theta$; so (3) follows by the Cauchy–Schwarz inequality, as
$B^{1}$ consists of $\sim R^{1/2}$ such caps.

##  4\. Mizohata–Takeuchi with $R^{\frac{n-1}{n+1}}$-loss: Theorem 1.2

Theorem 1.2 immediately follows from the stronger Theorem 4.1 below, which
takes into account the directions in which the waves propagate. In particular,
fix $n\geq 2$. For $g\in L^{2}(B^{n-1})$ and
$\mathbb{T}\subset\mathbb{T}_{\epsilon}(B_{R})$, define

$g_{\mathbb{T}}:=\sum_{T\in\mathbb{T}}g_{T},$

where $\\{g_{T}\\}_{T\in\mathbb{T}_{\epsilon}(R)}$ is the wave-packet
decomposition of $g$ adapted to $B_{R}$ (at scale $R$).

######  Theorem 4.1.

For every $\epsilon>0$, there exists a positive constant $C_{\epsilon}$, which
depends only on $\Sigma$ and $\epsilon$, such that

$\displaystyle\begin{aligned}
\int_{B_{R}}|Eg_{\mathbb{T}}|^{2}w\leq&C_{\epsilon}R^{\epsilon}\left(\sum_{T\in\mathbb{T}}\Big{[}\sum_{B\in\mathcal{B}:\;B\cap
T\neq\emptyset}w^{\frac{n+1}{2}}(B)\Big{]}\|g_{T}\|_{2}^{2}\right)^{\frac{2}{n+1}}\|g_{\mathbb{T}}\|_{2}^{\frac{2(n-1)}{n+1}}\\\
+&{\rm
RapDec}_{\epsilon}(R)\|w\|_{\infty}\int|g_{\mathbb{T}}|^{2}\end{aligned}$ (8)

for all $R\geq 1$, $g\in L^{2}(\Sigma)$,
$\mathbb{T}\subset\mathbb{T}_{\epsilon}(R)$ and weights
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ on $\mathbb{R}^{n}$, and for every
family $\mathcal{B}$ of boundedly overlapping $R^{1/2}$-balls.

As an immediate consequence of this we have:

######  Corollary 4.2.

For every $\epsilon>0$, there exists a positive constant $C_{\epsilon}$, which
depends only on $\Sigma$ and $\epsilon$, such that

$\displaystyle\begin{aligned}
\int_{B_{R}}|Eg_{\mathbb{T}}|^{2}w\leq&C_{\epsilon}R^{\epsilon}\left(\int|g_{\mathbb{T}}(s)|^{2}\sup_{T\parallel
N(s),\,T\in\mathbb{T}}w^{\frac{n+1}{2}}(2T)ds\right)^{\frac{2}{n+1}}\|g_{\mathbb{T}}\|_{2}^{\frac{2(n-1)}{n+1}}\\\
\leq&C_{\epsilon}R^{\epsilon}\;\sup_{T\in{\mathbb{T}}}\left(\int_{2T}w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\|g_{\mathbb{T}}\|_{2}^{2}\\\
\end{aligned}$ (9)

up to a ${\rm RapDec}_{\epsilon}(R)\|w\|_{\infty}\int|g_{\mathbb{T}}|^{2}$
error term, for all $R\geq 1$, $g\in L^{2}(\Sigma)$,
$\mathbb{T}\subset\mathbb{T}_{\epsilon}(R)$ and weights
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ on $\mathbb{R}^{n}$.

######  Remark 4.3.

We need the error term ${\rm
RapDec}_{\epsilon}(R)\|w\|_{\infty}\int|g_{\mathbb{T}}|^{2}$ in these results
because $w$ may be large at some points of ${\rm supp}\;Eg_{\mathbb{T}}$ which
are outside $\bigcup_{T\in\mathbb{T}}T$. Theorem 4.1 manifestly implies
Theorem 1.2 directly, since the error term is easily absorbed into the right-
hand side of the first inequality of Theorem 1.2. However, unlike in the case
of Theorem 1.2, it is definitively not possible to take $\epsilon=0$ in
Theorem 4.1. This is because of the example (see [V81, p.104], [R86], [B93] or
[V97, pp.125–126]) demonstrating the necessity of a logarithmic term in the
discrete $\ell^{2}-L^{6}$ restriction theorem for the paraboloid. For the
argument linking the two phenomena see [BD15, pp.355–358]. As we observe
below, Theorem 4.1 is essentially a reformulation of the refined decoupling
theorem [GIOW20]. It is furthermore closely related to the improved decoupling
theorem of [GMW20]. More precisely, if one takes the natural weight
$w=|Eg_{\mathbb{T}}|^{4/(n-1)}$ in Theorem 4.1, one obtains an inequality
slightly stronger than the one considered in [GMW20, Theorem 1.2], but with
$R^{\epsilon}$ loss rather than the logarithmic loss obtained there when
$n=2$. Notice the Stein-like nature of the middle term appearing in (9).

Theorem 4.1 is actually a reformulation of the following refined Stein–Tomas
or decoupling estimate. Theorem 4.4 was also discovered independently by
Xiumin Du and Ruixiang Zhang (personal communication).

######  Theorem 4.4.

_(Refined decoupling[GIOW20])_ Let $\epsilon>0$, $g\in L^{2}(B^{n-1})$, and
let $\mathbb{T}$ be a subset of $\mathbb{T}_{\epsilon}(B_{R})$ with the
property that $\|g_{T}\|_{2}$ is roughly constant over all $T\in\mathbb{T}$.
For each $k\in\mathbb{N}$, denote by $U_{k}$ an essentially disjoint union of
$R^{1/2}$-balls in $B_{R}$ each intersecting $\sim k$ tubes in $\mathbb{T}$.
Then the function

$g_{\mathbb{T}}=\sum_{T\in\mathbb{T}}g_{T}$

satisfies

$\displaystyle\|Eg_{\mathbb{T}}\|_{L^{\frac{2(n+1)}{n-1}}(U_{k})}$
$\displaystyle\leq
C_{\epsilon}R^{\epsilon}\left(\frac{k}{\\#\mathbb{T}}\right)^{\frac{1}{n+1}}\left(\sum_{T\in\mathbb{T}}\|Eg_{T}\|_{L^{\frac{2(n+1)}{n-1}}}^{2}\right)^{1/2}$
(10) $\displaystyle\sim
C_{\epsilon}R^{\epsilon}\left(\frac{k}{\\#\mathbb{T}}\right)^{\frac{1}{n+1}}\left(\sum_{T\in\mathbb{T}}\|g_{T}\|_{2}^{2}\right)^{1/2}\sim
C_{\epsilon}R^{\epsilon}\left(\frac{k}{\\#\mathbb{T}}\right)^{\frac{1}{n+1}}\|g_{\mathbb{T}}\|_{2}.$

Since $k\leq\\#\mathbb{T}$, estimate (10) provides an improvement on the
classical Stein–Tomas inequality

$\|Eg_{\mathbb{T}}\|_{L^{\frac{2(n+1)}{{n-1}}}(\mathbb{R}^{n})}\lesssim\|g_{\mathbb{T}}\|_{2}$

on the ‘$k$-rich’ sets $U_{k}$ in $B_{R}$, according to their level $k$ of
richness.

If we assume Theorem 4.1, we can immediately deduce Theorem 4.4 by testing on
a weight $w\in L^{\frac{n+1}{2}}(U_{k})$. Indeed, under the hypotheses of 4.4,
we apply Theorem 4.1 and we have

$\displaystyle\begin{aligned}
\int_{B_{R}}|Eg_{\mathbb{T}}|^{2}w\leq&C_{\epsilon}R^{\epsilon}\left(\sum_{T\in\mathbb{T}}\Big{[}\sum_{B\in\mathcal{B}:\;B\cap
T\neq\emptyset}w^{\frac{n+1}{2}}(B)\Big{]}\|g_{T}\|_{2}^{2}\right)^{\frac{2}{n+1}}\|g_{\mathbb{T}}\|_{2}^{\frac{2(n-1)}{n+1}}\\\
+&{\rm
RapDec}_{\epsilon}(R)\|w\|_{\infty}\int|g_{\mathbb{T}}|^{2}\end{aligned}$

and, suppressing the error term (as we may) and letting
$\lambda=\|g_{\mathbb{T}}\|_{2}^{2}/\\#\mathbb{T}$ denote the common value of
$\|g_{T}\|_{2}^{2}$, the right hand side here equals

$C_{\epsilon}R^{\epsilon}\lambda^{\frac{2}{n+1}}\left(\sum_{T\in\mathbb{T}}\sum_{B\in\mathcal{B}:\;B\cap
T\neq\emptyset}w^{\frac{n+1}{2}}(B)\right)^{\frac{2}{n+1}}\|g_{\mathbb{T}}\|_{2}^{\frac{2(n-1)}{n+1}}$
$\sim C_{\epsilon}R^{\epsilon}(\lambda
k)^{\frac{2}{n+1}}\left(\sum_{B\in\mathcal{B}}w^{\frac{n+1}{2}}(B)\right)^{\frac{2}{n+1}}\|g_{\mathbb{T}}\|_{2}^{\frac{2(n-1)}{n+1}}$
$\sim
C_{\epsilon}R^{\epsilon}\left(\frac{k}{\\#\mathbb{T}}\right)^{\frac{2}{n+1}}\|w\|_{\frac{n+1}{2}}\|g_{\mathbb{T}}\|_{2}^{2},$

as needed to verify Theorem 4.4.

Likewise, Theorem 4.1 will in turn follow from (10), as the following simple
argument shows.

###### Proof of Theorem 4.1.

Let $\epsilon>0$, fix $g\in L^{2}(B^{n-1})$, $w:B_{R}\rightarrow[0,+\infty)$
and $\mathbb{T}\subset\mathbb{T}_{\epsilon}(B_{R})$.

In order to prove (8), we may assume that:

  1. (a)

$w$ is supported in $\bigcup_{T\in\mathbb{T}}T$.

  2. (b)

$\|g_{T}\|_{2}\sim 1$ for all $T\in\mathbb{T}$.

Indeed, assumption (a) is possible because, by (wp3), the part of the weight
supported outside $\bigcup_{T\in\mathbb{T}}T$ contributes at most ${\rm
RapDec}_{\epsilon}(R)\|w\|_{\infty}\int|g_{\mathbb{T}}|^{2}$ to
$\int_{B_{R}}|Eg_{\mathbb{T}}|^{2}w$. For (b), observe that, in terms of our
goal, it is trivial to control the contributions of the wave packets $g_{T}$
with $\|g_{T}\|_{2}<R^{-100n}\|g\|_{2}$. So, by dyadic pigeonholing, it
suffices to prove (8) under the additional assumption that the $g_{T}$ have
roughly the same $L^{2}$ norms over all $T\in\mathbb{T}$. By scaling we may
assume this common value is $1$.

We now fix a family $\mathcal{B}$ of boundedly overlapping $R^{1/2}$-balls
covering $B_{R}$. By the above it suffices to prove that

$\int|Eg_{\mathbb{T}}|^{2}w\lessapprox\left(\frac{1}{\\#\mathbb{T}}\sum_{T\in\mathbb{T}}\;\;\sum_{B\in\mathcal{B}:\;B\cap
T\neq\emptyset}w^{\frac{n+1}{2}}(B)\right)^{\frac{2}{n+1}}\int|g_{\mathbb{T}}|^{2}$
(11)

under assumptions (a) and (b).

Let $U_{k}$ be the union of the balls in this family which meet $\sim k$
members of $\mathbb{T}$.

Importantly, (a) ensures that there exists some dyadic $k\in\mathbb{N}$ for
which

$\int_{B_{R}}|Eg_{\mathbb{T}}|^{2}w\approx\int_{U_{k}}|Eg_{\mathbb{T}}|^{2}w,$

So by Hölder’s inequality and (10) we obtain

$\displaystyle\begin{aligned}
\int_{B_{R}}|Eg_{\mathbb{T}}|^{2}w&\lessapprox\left(\int_{U_{k}}|Eg_{\mathbb{T}}|^{\frac{2(n+1)}{n-1}}\right)^{\frac{n-1}{n+1}}\;(w^{\frac{n+1}{2}}(U_{k}))^{\frac{2}{n+1}}\\\
&\leq
C_{\epsilon}R^{\epsilon}\left(\frac{k}{\\#\mathbb{T}}\;w^{\frac{n+1}{2}}(U_{k})\right)^{\frac{2}{n+1}}\int|g_{\mathbb{T}}|^{2}\\\
&\sim
C_{\epsilon}R^{\epsilon}\left(k\;w^{\frac{n+1}{2}}(U_{k})\right)^{\frac{2}{n+1}}(\\#\mathbb{T})^{\frac{n-1}{n+1}}.\end{aligned}$

We conclude with a simple counting argument. Indeed, let $\mathcal{B}_{k}$ be
the set of $R^{1/2}$-balls comprising $U_{k}$. Then,

$\displaystyle\begin{aligned}
k\;w^{\frac{n+1}{2}}(U_{k})&\sim\sum_{B\in\mathcal{B}_{k}}w^{\frac{n+1}{2}}(B)\;k\\\
&\sim\sum_{B\in\mathcal{B}_{k}}\;\;\sum_{T\in\mathbb{T}:\;T\cap
B\neq\emptyset}w^{\frac{n+1}{2}}(B)\\\
&=\sum_{T\in\mathbb{T}}\;\;\sum_{B\in\mathcal{B}_{k}:\;B\cap
T\neq\emptyset}w^{\frac{n+1}{2}}(B),\end{aligned}$

establishing (11) and thus (8). ∎

##  5\. Improved Mizohata–Takeuchi estimates for small caps

In this section we prove Lemma 1.4, which will be key to the proofs of
Theorems 6.1 and 6.2. It is a Mizohata–Takeuchi-type estimate which holds for
functions supported in small caps, and it represents an improvement over what
we can obtain under no support hypothesis.

Towards proving the lemma, we may assume as in Section 2 that all normals to
$\Sigma$ have angle at most $1/100$ from the vertical direction, and that the
projection of $\Sigma$ on the hyperplane $\mathbb{R}^{n-1}\times\\{0\\}$ is
contained in the unit ball $B^{n-1}$ centred at 0. It thus suffices to
establish the analogous statement (Lemma 5.1 below) with $Eg_{\tau}$ in place
of $\widehat{g_{\tau}d\sigma}$, where $E$ is the extension operator associated
to $\Sigma$ and $g_{\tau}\in L^{2}(B^{n-1})$ is a function supported in a
$\rho^{-1/2}$-cap $\tau$ in $B^{n-1}$.

To simplify notation, for $E\subset B^{n-1}$ (rather than $E\subset\Sigma$),
and any line $\ell$ (or tube $T$ in $B_{R}$), we write $\ell\perp E$ if
$\ell\perp\Sigma(E)$ (similarly, we write $T\perp E$ if $T\perp\Sigma(E)$). We
also define

$A_{\rho,R,E}(w):=A_{\rho,R,\Sigma(E)}(w).$

######  Lemma 5.1.

For every $\epsilon>0$, there exists $C_{\epsilon}>0$ such that for all
weights $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$, whenever $1\leq\rho\leq R$,
$\tau$ is a $\rho^{-1/2}$-cap in $B^{n-1}$ and $g_{\tau}\in L^{2}(B^{n-1})$ is
supported in $\tau$, we have

$\int_{B_{R}}|Eg_{\tau}|^{2}w\leq C_{\epsilon}R^{\epsilon}\;A_{\rho,R,\;{\rm
supp}\;g_{\tau}}(w)\int|g_{\tau}|^{2},$

and therefore also

$\int_{B_{R}}|Eg_{\tau}|^{2}w\leq
C_{\epsilon}R^{\epsilon}\;\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp{\rm
supp}\;g_{\tau}}Xw(\ell)\int|g_{\tau}|^{2}.$

Notice that the tubes and lines featuring here have directions perpendicular
to the support of $g_{\tau}$.

###### Proof.

Let $\epsilon>0$ and $R\geq 1$. For $\rho\lesssim R^{\epsilon}$, the
conclusion of the lemma follows directly from Theorem 1.2. We therefore
consider $\rho\gtrsim R^{\epsilon}$.

In order to prove the lemma for arbitrary weights, it suffices by dyadic
pigeonholing to prove it for weights that are indicator functions. Indeed,
first observe that we may assume that $w(x)\geq R^{-2n}\|w\|_{\infty}$ for all
$x\in{\rm supp}\;w$. Therefore, after a dyadic pigeonholing causing losses of
$\sim\log R$, we may assume that $w(x)\sim q$ for some fixed $q>0$ over all
$x\in{\rm supp}\;w$; and hence that $w$ is an indicator function, due to the
scaling properties of our desired estimate.

So, let $w$ be an indicator function of a non-empty union of unit balls. Fix a
$\rho^{-1/2}$-cap $\tau$, and let $g$ be a function supported in $\tau$. Let
$\mathbb{T}$ be a family of boundedly overlapping parallel $\rho^{1/2}$-tubes
that cover ${\rm supp}\;w$, and point in some direction $N$ normal to ${\rm
supp}\;g$; observe that $\mathbb{T}\subset\mathbb{T}_{\rho}$. At a cost of a
$\log R$-loss, it may be further assumed that

$\frac{w(S_{\rho})}{|S_{\rho}|}\sim\lambda\text{ for all
}S_{\rho}\in\mathbb{T}$

for some $\lambda\leq 1$, hence

$\displaystyle\begin{aligned} A_{\rho,R,\;{\rm
supp}\;g}(w)&=\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp{\rm
supp}\;g}\left(\sum_{S_{\rho}\subset
T_{R}}\left(\frac{w(S_{\rho})}{|S_{\rho}|}\right)^{\frac{n+1}{2}}|S_{\rho}|\right)^{\frac{2}{n+1}}\\\
&\sim\lambda\rho\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp{\rm
supp}\;g}\\#\\{S_{\rho}\in\mathbb{T}\,:\,S_{\rho}\cap
T_{R}\neq\emptyset\\}^{\frac{2}{n+1}}.\end{aligned}$

It therefore suffices to prove that

$\int|Eg|^{2}w\leq
C_{\epsilon}R^{\epsilon}\lambda\rho\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp{\rm
supp}\;g}\\#\\{S_{\rho}\in\mathbb{T}\,:\,S_{\rho}\subset
T_{R}\\}^{\frac{2}{n+1}}\int|g|^{2}.$

Proposition 2.2 ensures that, roughly speaking, $|Eg|$ is constant on each
$S_{\rho}\in\mathbb{T}$. In particular, let $\mathbb{T}_{N}$ be a set of
boundedly overlapping tubes in direction $N$, of width $\rho^{1/2+\delta}$ and
length $\rho$, that cover $B_{R}$. For each $S_{\rho}\in\mathbb{T}$, fix
$\widetilde{S}_{\rho}\in\mathbb{T}_{N}$ that intersects $S_{\rho}$. By
Proposition 2.2,

$\displaystyle\int_{S_{\rho}}|Eg|^{2}w$
$\displaystyle\lesssim\frac{w(S_{\rho})}{|S_{\rho}|}\int_{2\widetilde{S}_{\rho}}|Eg|^{2}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}$
$\displaystyle\sim\lambda\int_{2\widetilde{S}_{\rho}}|Eg|^{2}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}.$

By adding over all $S_{\rho}\in\mathbb{T}$, we obtain

$\int|Eg|^{2}w\lesssim\lambda\int|Eg|^{2}\widetilde{w}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2},$

where

$\widetilde{w}:=\sum_{S_{\rho}\in\mathbb{T}}\chi_{2\widetilde{S}_{\rho}}.$

Now by Theorem 1.2 we have

$\int|Eg|^{2}\tilde{w}\lessapprox\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp{\rm
supp}\;g}\tilde{w}(T_{R})^{\frac{2}{n+1}}\int|g|^{2},$

and for $T_{R}\in\mathbb{T}_{R}$ with $T_{R}\perp{\rm supp}\;g$ we have

$\tilde{w}(T_{R})\lesssim\rho^{\frac{n+1}{2}}\\#\\{S_{\rho}\in\mathbb{T}\,:\,2S_{\rho}\cap
T_{R}\neq\varnothing\\}.$

Therefore,

$\int|Eg|^{2}w\lessapprox\lambda\left(\rho^{\frac{n+1}{2}}\sup_{T_{R}\in\mathbb{T}_{R}:\;T_{R}\perp{\rm
supp}\;g}\\#\\{S_{\rho}\in\mathbb{T}\,:\,S_{\rho}\subset
T_{R}\\}\right)^{\frac{2}{n+1}}\int|g|^{2},$

as required. ∎

##  6\. Weights constant on slabs: Theorems 1.6 and 1.8

In this section we will use the favourable estimates for functions $g_{\tau}$
supported in small caps which were established in Section 5 to obtain
Mizohata–Takeuchi estimates which improve on Theorem 1.2 for general functions
$g$ and weights possessing a certain measure of local constancy. In
particular, recall from (7) that if a function $g_{\tau}$ is supported in a
$\rho^{-1/2}$-cap $\tau$, then the Mizohata–Takeuchi conjecture holds for
$g_{\tau}$ with an improved $(R/\rho)^{(n-1)(n+1)}$-loss. Therefore, for any
fixed $g\in L^{2}(B^{n-1})$ and $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$, a
decoupling inequality of the form

$\int_{B_{R}}|Eg|^{2}w\lessapprox\sum_{\tau}\int_{B_{R}}|Eg_{\tau}|^{2}w$

for a boundedly overlapping collection of $\rho^{-1/2}$-caps $\tau$ (where
$g=\sum_{\tau}g_{\tau}$ and ${\rm supp}\;g_{\tau}\subset\tau$) would directly
imply that Mizohata–Takeuchi holds for $g$ with the inherited loss
$(R/\rho)^{(n-1)(n+1)}$. The smaller the caps we manage to decouple into, the
smaller the loss.

In general, it is not possible to decouple into small caps. However, we can
indeed decouple into $\rho^{-1/2}$-caps when $w$ is a weight of the form
$\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $\rho^{1/2}$-slabs that are $\nu$-parallel to $\Sigma$; more
precisely, we show that (12) below holds. This yields Mizohata–Takeuchi for
such weights with an $(R/\rho)^{(n-1)(n+1)}$-loss. If the slabs in
$\mathcal{S}$ are allowed to point in any direction, then we can decouple into
larger $\rho^{-1/4}$-caps (14), inheriting Mizohata–Takeuchi with an
$(R/\rho^{1/2})^{(n-1)(n+1)}$-loss.

These results are given in Theorems 6.1 and 6.2 below, which are more precise
versions of Theorems 1.6 and 1.8, respectively. As per the above discussion,
the new ingredients here are the decoupling inequalities (12) and (14) which
follow. Note that, as in Section 5, we will be working with the extension
operator $E$ associated to $\Sigma$ (rather than with
$\widehat{\cdot\;d\sigma}$). When $E\subset B^{n-1}$, we will be using the
simpler the notation $A_{\rho,R,E}(w)$ in place of $A_{\rho,R,\Sigma(E)}(w)$,
and $\ell\perp E$ (or $T\perp E$) to mean $\ell\perp\Sigma(E)$ (similarly,
$T\perp\Sigma(E)$) for any line $\ell$ and tube $T$ in $\mathbb{R}^{n}$.

######  Theorem 6.1.

_(Roughly horizontal slabs)_ Fix $\nu>0$ and $\epsilon>0$. For $1\leq\rho\leq
R$, let $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ be a weight of the form
$\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $\rho^{1/2}$-slabs $\nu$-parallel to $\Sigma$, and let
$w^{\star}:=\sum_{s\in\mathcal{S}}c_{s}\chi_{3s}$. For $g\in L^{2}(B^{n-1})$,
write

$g=\sum_{\tau\in\mathfrak{T}}g_{\tau},\quad{{\rm supp}\;g_{\tau}\subset\tau},$

where $\mathfrak{T}$ is a family of boundedly overlapping $\rho^{-1/2}$-caps
$\tau$ in $B^{n-1}$. Then the decoupling inequality

$\int_{B_{R}}|Eg|^{2}w\lessapprox_{\nu}\sum_{\tau\in\mathfrak{T}}\int_{B_{R}}|Eg_{\tau}|^{2}w^{\star}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}$ (12)

holds. Consequently we have

$\displaystyle\begin{aligned} \int_{B_{R}}|Eg|^{2}w&\leq
C_{\epsilon,\nu}R^{\epsilon}\sum_{\tau\in\mathfrak{T}}A_{\rho,R,\;{\rm
supp}\;g_{\tau}}(w)\int|g_{\tau}|^{2}\\\
&\lessapprox_{\nu}\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sum_{\tau\in\mathfrak{T}}\;\sup_{\ell\perp{\rm
supp}\;g_{\tau}}Xw(\ell)\int|g_{\tau}|^{2}.\end{aligned}$ (13)

Note that an immediate consequence of (13) is

$\displaystyle\begin{aligned} \int_{B_{R}}|Eg|^{2}w&\leq
C_{\epsilon,\nu}R^{\epsilon}A_{\rho,R,\;{\rm supp}\;g}(w)\int|g|^{2}\\\
&\lessapprox_{\nu}\left(\frac{R}{\rho}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp{\rm
supp}\;g}Xw(\ell)\int|g|^{2}.\end{aligned}$

######  Theorem 6.2.

_(All slabs)_ Fix $\epsilon>0$. For $1\leq\rho\leq R$, let
$w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ be a weight of the form
$\sum_{s\in\mathcal{S}}c_{s}\chi_{s}$, where $\mathcal{S}$ is a set of
disjoint $\rho^{1/2}$-slabs. Let
$w^{\star}:=\sum_{s\in\mathcal{S}}c_{s}\chi_{3s}$. For $g\in L^{2}(B^{n-1})$,
write

$g=\sum_{\widetilde{\tau}\in\widetilde{\mathfrak{T}}}g_{\widetilde{\tau}},\quad{{\rm
supp}\;g_{\widetilde{\tau}}\subset\widetilde{\tau}}$

where $\widetilde{\mathfrak{T}}$ is a family of finitely overlapping
$\rho^{-1/4}$-caps $\widetilde{\tau}$ in $B^{n-1}$. Then the decoupling
inequality

$\int_{B_{R}}|Eg|^{2}w\lessapprox\sum_{\widetilde{\tau}\in\widetilde{\mathfrak{T}}}\int_{B_{R}}|Eg_{\widetilde{\tau}}|^{2}w^{\star}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}$ (14)

holds. Consequently we have,

$\displaystyle\begin{aligned} \int_{B_{R}}|Eg|^{2}w&\leq
C_{\epsilon}R^{\epsilon}\sum_{\widetilde{\tau}\in\widetilde{\mathfrak{T}}}A_{\rho^{1/2},R,\;\rm{supp}\,g_{\widetilde{\tau}}}(w)\int|g_{\widetilde{\tau}}|^{2}\\\
&\lessapprox\left(\frac{R}{\rho^{1/2}}\right)^{\frac{n-1}{n+1}}\sum_{\widetilde{\tau}\in\widetilde{\mathfrak{T}}}\;\sup_{\ell\perp{\rm
supp}\;g_{\widetilde{\tau}}}Xw(\ell)\int|g_{\widetilde{\tau}}|^{2}.\end{aligned}$
(15)

Note that an immediate consequence of (15) is

$\displaystyle\begin{aligned} \int_{B_{R}}|Eg|^{2}w&\leq
C_{\epsilon,\nu}R^{\epsilon}A_{\rho^{1/2},R,\;{\rm supp}\;g}(w)\int|g|^{2}\\\
&\lessapprox_{\nu}\left(\frac{R}{\rho^{1/2}}\right)^{\frac{n-1}{n+1}}\sup_{\ell\perp{\rm
supp}\;g}Xw(\ell)\int|g|^{2}.\end{aligned}$

###### Proofs of (12) and (14)..

Fix $\epsilon>0$ and $R\geq 1$. Let $s$ be a $\rho^{1/2}$ slab in $B_{R}$, and
fix $g\in L^{2}(B^{n-1})$. Let $\mathfrak{T}_{1}$, $\mathfrak{T}_{2}$ be
collections of finitely overlapping $\rho^{-1/4}$ and $\rho^{-1/2}$-caps,
respectively, that cover $B^{n-1}$. For $i=1,2$, write

$g=\sum_{\tau\in\mathfrak{T}_{i}}g_{\tau}\qquad{{\rm
supp}\,g_{\tau}\subset\tau}.$

We will show that

$\int_{s}|Eg|^{2}\leq
C_{\epsilon}R^{\epsilon}\sum_{\tau\in\mathfrak{T}_{1}}\int_{3s}|Eg_{\tau}|^{2}$

and that, if additionally $s$ is $\nu$-parallel to $\Sigma$ for some $\nu>0$,
then

$\int_{s}|Eg|^{2}\leq
C_{\nu,\epsilon}R^{\epsilon}\sum_{\tau\in\mathfrak{T}_{2}}\int_{3s}|Eg_{\tau}|^{2}.$

Note that henceforth we may assume that $\rho\gtrsim_{\epsilon}R^{\epsilon/n}$
(as otherwise (12) and (14) follow trivially by the Cauchy–Schwarz
inequality), and that $\nu\gtrsim_{\epsilon}R^{-\epsilon}$ (as otherwise
$C_{\epsilon,\nu}$ may be chosen to be an appropriately large power of $R$ for
(12) to follow).

For this proof, it will be useful to think of $g$ as truly supported on
$\Sigma$. And indeed, due to our assumption that the normals to $\Sigma$
create angles at most $1/100$ with the vertical direction, it suffices instead
to prove the above decoupling inequalities for $g\in L^{2}(\Sigma)$, for
$\widehat{gd\sigma}$ in place of $Eg$ and for $\mathfrak{T}_{i}$ collections
of finitely overlapping $\rho^{-1/4}$-caps and $\rho^{-1/2}$-caps,
respectively, of $\Sigma$.

Let $\eta:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a non-negative, smooth bump
function with $\eta(x)=1$ for all $x\in B_{1}$ and $\eta(x)=0$ for all $x\in
B_{2}$. Denote by $\eta_{s}$ a smooth bump function adapted to $s$. In
particular, if $s_{0}=[0,\rho^{1/2}]^{n-1}\times[0,1]$, define

$\eta_{s_{0}}(x):=\eta\left(\frac{x^{\prime}}{\rho^{1/2}},x_{n}\right),$

and let $\eta_{s}(x):=\eta_{s_{0}}(Mx)$, where $M$ is a rigid motion mapping
$s$ to $s_{0}$. Let $s^{\star}$ be a ‘dual’ object to $s$, specifically the
tube with centre 0, direction the normal to $s$, length 1 and cross section of
radius $\rho^{-1/2+\delta}$. It is easy to see by stationary phase that
$\widehat{\eta_{s}(x)}$ is essentially supported in $s^{\star}$; more
precisely,

$|\widehat{\eta_{s}}(y)|={\rm RapDec}_{\epsilon}(\rho)\|\eta_{s}\|_{1}={\rm
RapDec}_{\epsilon}(R)\qquad{\text{for all }y\in\mathbb{R}^{n}\setminus
s^{\star}}.$

Therefore, for $i=1,2$,

$\displaystyle\int_{s}|\widehat{gd\sigma}|^{2}\leq\int|\widehat{gd\sigma}|^{2}\eta_{s}$
$\displaystyle=\int\Big{|}\sum_{\tau\in\mathfrak{T}_{i}}\widehat{g_{\tau}d\sigma}\Big{|}^{2}\eta_{s}$
$\displaystyle=\int\Big{(}\sum_{\tau\in\mathfrak{T}_{i}}\widehat{g_{\tau}d\sigma}\Big{)}\Big{(}\sum_{\tau^{\prime}\in\mathfrak{T}_{i}}\overline{\widehat{g_{\tau^{\prime}}d\sigma}}\Big{)}\eta_{s}$
$\displaystyle=\sum_{\tau,\tau^{\prime}\in\mathfrak{T}_{i}}\int\left(\widehat{g_{\tau}d\sigma}\;\overline{\widehat{g_{\tau^{\prime}}d\sigma}}\right)\eta_{s}$
$\displaystyle=\sum_{\tau,\tau^{\prime}\in\mathfrak{T}_{i}}\int(g_{\tau}d\sigma)\ast(\widetilde{g_{\tau^{\prime}}d\sigma})\;\widehat{\eta_{s}},$

where, for every $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$, $\widetilde{f}$ is
defined by $\widetilde{f}(y):=\overline{f(-y)}$.

For every $\tau,\tau^{\prime}\in\mathfrak{T}_{i}$, the function
$(g_{\tau}d\sigma)\ast(\widetilde{g_{\tau^{\prime}}d\sigma})$ is supported in
$\tau-\tau^{\prime}$, and thus its contribution to the above sum is negligible
unless $\tau-\tau^{\prime}$ intersects $s^{\star}$. More precisely,

$\int(g_{\tau}d\sigma)\ast(\widetilde{g_{\tau^{\prime}}d\sigma})\;\widehat{\eta_{s}}=\int_{\mathbb{R}^{n}\setminus
s^{\star}}(g_{\tau}d\sigma)\ast(\widetilde{g_{\tau^{\prime}}d\sigma})\;\widehat{\eta_{s}}={\rm
RapDec}_{\epsilon}(R)\|g_{\tau}\|_{2}\|g_{\tau^{\prime}}\|_{2}$

whenever $\tau-\tau^{\prime}\cap s^{\star}=\varnothing$, whence

$\displaystyle\begin{aligned}
\int_{s}|\widehat{gd\sigma}|^{2}&=\sum_{\tau,\tau^{\prime}\in\mathfrak{T}_{i}:\;(\tau-\tau^{\prime})\cap
s^{\star}\neq\varnothing}\int(g_{\tau}d\sigma)\ast(\widetilde{g_{\tau^{\prime}}d\sigma})\;\widehat{\eta_{s}}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\\\
&=\sum_{\tau,\tau^{\prime}\in\mathfrak{T}_{i}:\;(\tau-\tau^{\prime})\cap
s^{\star}\neq\varnothing}\int\left(\widehat{g_{\tau}d\sigma}\;\overline{\widehat{g_{\tau^{\prime}}d\sigma}}\right)\eta_{s}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\\\
&\leq\sum_{\tau,\tau^{\prime}\in\mathfrak{T}_{i}:\;(\tau-\tau^{\prime})\cap
s^{\star}\neq\varnothing}\left(\int_{3s}|\widehat{g_{\tau}d\sigma}|^{2}+\int_{3s}|\widehat{g_{\tau^{\prime}}d\sigma}|^{2}\right)+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2}\\\ &\leq
N_{i}\cdot\sum_{\tau\in\mathfrak{T}_{i}}\int_{3s}|\widehat{g_{\tau}d\sigma}|^{2}+{\rm
RapDec}_{\epsilon}(R)\int|g|^{2},\end{aligned}$ (16)

where

$N_{i}:=\max_{\tau\in\mathfrak{T}_{i}}\;\\#\\{\tau^{\prime}\in\mathfrak{T}_{i}:(\tau-\tau^{\prime})\cap
s^{\star}\neq\varnothing\\}.$

Note that for the last inequality in (16) we used that $s^{\star}$ is
symmetric around 0.

It now suffices to show that

$N_{1}\leq C_{\epsilon}R^{\epsilon}$ (17)

and that, if additionally $s$ is $\nu$-parallel to $\Sigma$ for some
$\nu\gtrsim_{\epsilon}R^{\epsilon}$, then

$N_{2}\leq C_{\epsilon,\nu}R^{\epsilon}.$ (18)

We first focus on the case $i=1$. Fix $\tau\in\mathfrak{T}_{1}$, and let
$\omega(\tau)$ denote its centre. The family $\mathfrak{T}_{1}$ consists of
$\rho^{-1/4}$-caps, so the $\tau^{\prime}\in\mathfrak{T}_{1}$ with
$(\tau-\tau^{\prime})\cap s^{\star}\neq\varnothing$ cover the set

$A(\tau):=\\{\omega\in\Sigma:\;(\tau-\omega)\cap s^{\star}\neq\varnothing\\}.$

Let $e$ denote the direction of $s^{\star}$. For every $\omega\in A(\tau)$,
there exists $\omega_{0}\in\tau$ such that $\omega_{0}-\omega\in s^{\star}$,
which implies that

$|\omega_{0}-\omega|\lesssim\rho^{-1/4+\delta}\;\;\;\text{ or }\;\;\;{\rm
Angle}(\omega_{0}-\omega,e)\lesssim\rho^{-1/4+\delta},$

hence

$|\omega-\omega(\tau)|\lesssim\rho^{-1/4+\delta}\;\;\;\text{ or }\;\;\;{\rm
Angle}(\omega-\omega(\tau),e)\lesssim\rho^{-1/4+\delta}.$

It follows that $A(\tau)$ can be covered by two $\sim\rho^{-1/4+\delta}$-caps
of $\Sigma$, and thus by $O(\rho^{\delta})=O(R^{\epsilon})$ $\rho^{-1/4}$-caps
of $\Sigma$. This immediately implies (17), which in turn establishes the
desired estimate (14) when combined with (16).

For the case $i=2$, let $\nu\gtrsim_{\epsilon}R^{\epsilon}$. Fix
$\tau\in\mathfrak{T}_{2}$ and denote by $\omega(\tau)$ its centre. Similarly
to the previous case, the $\tau^{\prime}\in\mathfrak{T}_{2}$ with
$(\tau-\tau^{\prime})\cap s^{\star}\neq\varnothing$ cover the set

$A(\tau):=\\{\omega\in\Sigma:\;(\tau-\omega)\cap
s^{\star}\neq\varnothing\\}=\Sigma\cap(\tau-s^{\star}).$

Now however the family $\mathfrak{T}_{2}$ consists of $\rho^{-1/2}$-caps;
moreover, $s$ is $\nu$-parallel to $\Sigma$, which implies that all tangents
to $\tau$ create angles at least $\nu$ with the (roughly vertical) direction
$e$ of $s^{\star}$. Therefore,

$\tau-s^{\star}\subset R_{s^{\star}},$

for some vertical rectangle $R_{s^{\star}}$, with vertical side of length
$\sim_{\nu}1$ (roughly the length of $s^{\star}$) and all other sides of
length $\sim_{\nu}\rho^{-1/2+\delta}$ (approximately the sum of the width of
$s^{\star}$ and the radius of $\tau$).

Due to our assumption that all tangents to $\Sigma$ create angle at most
$1/100$ with the vertical direction, it follows that $\Sigma\cap
R_{s^{\star}}$ (and consequently $A(\tau)$) is contained in a single
$\sim_{\nu}\rho^{-1/2+\delta}$-cap of $\Sigma$, and can thus be covered by
$O(R^{\epsilon})$ $\rho^{-1/2}$-caps in $\mathfrak{T}_{2}$. This implies the
desired estimate (18) and hence completes the proof of (12). ∎

###### Proof of Theorem 6.1..

Let $\nu$, $\epsilon$, $R$, $\rho$, $w$ and $g$ be as in the statement of the
theorem. Now that (12) has been established, it suffices to prove the first
assertion in (13).

To that end, observe that $w^{\star}$ is the sum of $3^{n-1}$ weights: the
weight $w_{0}:=w$ (supported in $B_{R}$), and weights $w_{j}$ of the form
$w(\cdot-t_{j})$ (for appropriate $t_{j}\in\mathbb{R}^{n-1}\times\\{0\\}$,
with $|t_{j}|\leq R$, for $j=1,2\dots$). It thus suffices to show that

$\int|Eg|^{2}w_{j}\leq
C_{\epsilon,\nu}R^{\epsilon}\sum_{\tau\in\mathfrak{T}}A_{\rho,R,\,{\rm
supp}\,g_{\tau}}(w)\int|g_{\tau}|^{2}$

for all $j=1,2\dots$. For $j=0$ the inequality follows by Lemma 5.1. For
$j=1,2\dots$,

$Eg=Eg_{j}(\cdot-t_{j})\text{, where }g_{j}:=e^{2\pi i\langle
t_{j},\Sigma(\cdot)\rangle}g.$

Observe that, denoting $g_{j,\tau}:=e^{2\pi i\langle
t_{j},\Sigma(\cdot)\rangle}g_{\tau}$, we can write

$g_{j}=\sum_{\tau\in\mathfrak{T}}g_{j,\tau},\quad{{\rm supp}\;g_{j,\tau}={\rm
supp}\;g_{\tau}\subset\tau}.$

Therefore, by Lemma 5.1,

$\displaystyle\int|Eg|^{2}w_{j}$ $\displaystyle=\int|Eg_{j}(\cdot-
t_{j})|^{2}w(\cdot-t_{j})$ $\displaystyle=\int|Eg_{j}|^{2}w$
$\displaystyle\leq
C_{\epsilon,\nu}R^{\epsilon}\sum_{\tau\in\mathfrak{T}}A_{\rho,R,\,{\rm
supp}\,g_{j,\tau}}(w)\int|g_{j,\tau}|^{2}$
$\displaystyle=C_{\epsilon,\nu}R^{\epsilon}\sum_{\tau\in\mathfrak{T}}A_{\rho,R,\,{\rm
supp}\,g_{\tau}}(w)\int|g_{\tau}|^{2},$

completing the proof. ∎

###### Proof of Theorem 6.2..

The proof follows the same steps as that of Theorem 6.1, but with the family
$\mathfrak{T}$ replaced by $\widetilde{\mathfrak{T}}$. ∎

##  7\. Guth’s argument: the $R^{\frac{n-1}{n+1}}$ barrier

In his recent talk [Gu22]:

  1. (a)

Guth identified two ‘decoupling axioms’ (appropriate local constancy and local
$L^{2}$-orthogonality conditions) that are satisfied by all $Eg$, and are
sufficient to ensure that the Bourgain–Demeter decoupling inequality [BD15]
holds in $B_{R}$ for every function $F$ satisfying them.

  2. (b)

He then constructed a function $F:B_{R}\rightarrow\mathbb{C}$ which satisfies
the decoupling axioms, but for which the Mizohata–Takeuchi conjecture fails by
a factor of $\sim(\log R)^{-3}R^{\frac{n-1}{n+1}}$. Notably, $F_{|B_{R}}$ is
not of the form $Eg_{|B_{R}}$ for any $g\in L^{2}(B^{n-1})$.

Notably, Guth’s decoupling axioms for all $Eg$ are also sufficient to imply
the refined decoupling Theorem 4.4 (as a careful review of its proof reveals),
and thus its corollary Theorem 1.2, which established the conjecture with a
loss of $\lessapprox R^{\frac{n-1}{n+1}}$. Therefore, our main result is
essentially sharp given the techniques used.

In this section we outline Guth’s axiomatic approach and argument
demonstrating the existence of a counterexample [Gu22], and briefly review our
result within this context. We emphasise that these results are not ours, and
we present them only for self-containment.

Fix $R\geq 1$ and $\epsilon>0$. In this section, for every $g\in
L^{2}(B^{n-1})$ and every cap $\tau$ in $B^{n-1}$, we denote
$g_{\tau}:=g_{|\tau}$. In particular, $g_{B^{n-1}}=g$.

We call a cap $\tau$ in $B^{n-1}$ admissible if its diameter $d(\tau)$ is a
dyadic number in $[R^{-1/2},R^{-\epsilon}]\cup\\{2\\}$. In this analysis,
$B^{n-1}$ is the only admissible cap of diameter 2. Denote by
$\mathcal{D}_{R}$ the set of all admissible caps.

For every $\tau\in\mathcal{D}_{R}$, let
$F_{\tau}:\mathbb{R}^{n}\rightarrow\mathbb{C}$ be some function. Note that the
caps $\tau$ are simply used for enumeration here, and may be entirely
unrelated to properties of $F_{\tau}$. This is in contrast to, say, functions
of the form $Eg_{\tau}$, which are Fourier-localised close to $\Sigma(\tau)$.

Axiomatic decoupling. (Guth [Gu22]) If the decoupling axioms (DA1) and (DA2)
below hold for the full sequence $(F_{\tau})_{\tau\in D_{R}}$, then the
function $F:=F_{B^{n-1}}$ in $B_{R}$ can be decoupled into the functions
$F_{\theta}$ corresponding to the smallest possible scale, as follows:

$\|F\|_{L^{p}(B_{R})}\leq
C_{\epsilon}R^{O(\epsilon)}\left(\sum_{\theta\in\mathcal{D}_{R}:\;d(\theta)\sim
R^{-1/2}}\|F_{\theta}\|_{L^{p}(B_{R})}^{2}\right)^{1/2}\;\;\;\text{ for all
}2\leq p\leq\frac{2(n+1)}{n-1}.$

The decoupling axioms (DA1) and (DA2) for a sequence
$(F_{\tau})_{\tau\in\mathcal{D}_{R}}$ are the following statements.

(DA1) (Local constancy). For every $\tau\in\mathcal{D}_{R}$ with $d(\tau)\leq
R^{-\epsilon}$, the function $|F_{\tau}|$ is essentially constant on each
translate of

$\Sigma(\tau)^{\star}:=\\{x:|x\cdot(\xi-\xi_{\tau})|\leq 1\text{ for all
}\xi\in\Sigma(\tau)\\},$

where $\xi_{\tau}$ denotes the centre of $\Sigma(\tau)$.333Formally, a
function is essentially constant on translates of $\Sigma(\tau)^{\star}$ if it
satisfies estimate (24) in the statement of Lemma 6.1 in [GWZ22], with
$\theta$ replaced by the smallest rectangle containing $\Sigma(\tau)$.

(DA2) (Local $L^{2}$–orthogonality). Let $\gamma\in D_{R}$, and suppose that
$\gamma=\sqcup_{\tau\in\mathfrak{T}}\;\tau$, where $\mathfrak{T}$ is a family
of finitely overlapping caps in $\mathcal{D}_{R}$ with diameters smaller than
$d(\gamma)$. Then, the estimate

$\int_{K}|F_{\gamma}|^{2}\sim\sum_{\tau\subset\gamma}\int_{K}|F_{\tau}|^{2}+{\rm
RapDec}_{\epsilon}(R)\int|F_{\gamma}|^{2}$

holds for every convex $K\subset\mathbb{R}^{n}$ such that the sets
$\tau+K^{\star}$, over all $\tau\in\mathfrak{T}$, are finitely
overlapping.444Without (DA2), no relationship between the different $F_{\tau}$
would be imposed. Observe that, in contrast to the case where
$(F_{\tau})_{\tau\in\mathcal{D}_{R}}=(Eg_{\tau})_{\tau\in\mathcal{D}_{R}}$,
the equality $F_{\gamma}=\sum_{\tau\in\mathfrak{T}}F_{\tau}$ may not hold for
a sequence $(F_{\tau})_{\tau\in\mathcal{D}_{R}}$ satisfying the decoupling
axioms.

It is not hard to see that, for all $g\in L^{2}(B^{n-1})$, the sequence
$(Eg_{\tau})_{\tau\in\mathcal{D}_{R}}$ satisfies (DA1) and (DA2). Guth’s
axiomatic decoupling statement above, together with a careful review of the
proof [GWZ22] of the refined decoupling Theorem 4.4 (which directly led to our
Theorem 1.2, or equivalently to (19) below), reveal the following.

Fact A. (DA1 & DA2 $\Rightarrow$ MT with $\lessapprox
R^{\frac{n-1}{n+1}}$-loss for all $Eg$) The fact that

$(Eg_{\tau})_{\tau\in\mathcal{D}_{R}}\text{ {satisfies} (DA1) {and} (DA2) {for
all} }g\in L^{2}(B^{n-1})$

implies the inequality

$\int_{B_{R}}|Eg|^{2}w\leq
C_{\epsilon}R^{\frac{n-1}{n+1}+\epsilon}\;\|Xw\|_{\infty}\;\frac{1}{R}\int_{B_{R}}|Eg|^{2}$
(19)

for all $g\in L^{2}(B^{n-1})$ and $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$.

To improve on the Mizohata–Takeuchi conjecture, one needs to reduce the lossy
factor $R^{\frac{n-1}{n+1}}$ in (19) (and ideally to remove it altogether). Up
to $\approx 1$ factors, this is impossible if one insists on only using that
all $(Eg_{\tau})_{\tau\in\mathcal{D}_{R}}$ satisfy (DA1) and (DA2). Indeed,
Guth [Gu22] proved the following.

Fact B. (DA1 & DA2 $\not\Rightarrow$ MT with $\ll R^{\frac{n-1}{n+1}}$-loss
for general $F$) There exists $F:\mathbb{R}^{n}\rightarrow\mathbb{C}$ with

$F=F_{B^{n-1}}\textit{ for some }(F_{\tau})_{\tau\in\mathcal{D}_{R}}\textit{
satisfying}\text{ (DA1) }\textit{and}\text{ (DA2)}\textit{,}$ (20)

such that

$\int_{B_{R}}|F|^{2}w\gtrsim(\log
R)^{-3}R^{\frac{n-1}{n+1}}\;\|Xw\|_{\infty}\;\frac{1}{R}\int_{B_{R}}|F|^{2}$
(21)

for some $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$.

###### Proof.

Let $\Sigma$ be as earlier. The scale $R^{-\frac{1}{n+1}}$ plays a key role in
the upcoming argument; thus, denote by $\mathcal{D}$ the set of all
$\tau\in\mathcal{D}_{R}$ with $d(\tau)=R^{-\frac{1}{n+1}}$ (or, precisely,
with $d(\tau)$ equal to the smallest dyadic number that is at least
$R^{-\frac{1}{n+1}}$). For each $\tau\in\mathcal{D}$, let $\mathbb{T}_{\tau}$
be a family of finitely-overlapping parallel tubes in $\mathbb{R}^{n}$ that
intersect and cover $B_{R}$, of radius $R^{\frac{1}{n+1}}$, length
$R^{\frac{2}{n+1}}$ and direction the normal to $\Sigma(\tau)$ (these tubes
are essentially translates of $\Sigma(\tau)^{\star}$). Let

$\mathbb{T}:=\\{T\in\mathbb{T}_{\tau}:\;\tau\in\mathcal{D}\\}.$

There exists a weight $w:\mathbb{R}^{n}\rightarrow[0,+\infty)$ such that the
following hold.

  1. (1)

$w$ is the characteristic function of a union of $\sim R^{n-1}$ unit balls in
$B_{R}$.

  2. (2)

Each tube $L$ of radius $1$ satisfies $w(L)\lesssim\log R$.

  3. (3)

Each tube $T\in\mathbb{T}$ satisfies $w(T)\lesssim\log R$, and fully contains
every $1$-ball in ${\rm supp}\;w$ that it intersects.

This is the weight that will feature in (21), and its existence is guaranteed
by prior work of the first author [Ca09, Theorem 3] on aspects of the
Mizohata–Takeuchi conjecture. The details are omitted.

The function $F$ will be carefully defined as a sum of wave packets, so that
it is large on a big proportion of ${\rm supp}\;w$; more precisely, on a large
set $\mathcal{B}$ of unit balls in ${\rm supp}\;w$. The set $\mathcal{B}$ is
the one appearing in the claim below. The proof is postponed to the end of the
section. (Note that the claim would be trivial if each tube in $\mathbb{T}$
intersected and fully contained at most one $1$-ball in ${\rm supp}\;w$.)

######  Claim 7.1.

There exist
_(i)_ a set $\mathcal{B}=\\{B_{1},\ldots,B_{m}\\}$ of $\gtrsim(\log
R)^{-2}R^{n-1}$ disjoint unit balls in ${\rm supp}\;w$, and
_(ii)_ sets $\mathbb{T}_{j}\subset\mathbb{T}$ with
$\\#\mathbb{T}_{j}\gtrsim\\#\mathcal{D}$ for every $j=1,\ldots,m$,
such that the following hold.

_(P1)_ The tubes in $\mathbb{T}_{j}$ contain $B_{j}$, for all $j=1,\ldots,m$.
_(P2)_ For each $j=2,\ldots,m$, the tubes in $\mathbb{T}_{j}$ do not intersect
any of the balls $B_{1},\ldots,B_{j-1}$.

We now construct a sequence $(F_{\tau})_{\tau\in\mathcal{D}_{R}}$ of functions
$F_{\tau}:\mathbb{R}^{n}\rightarrow\mathbb{C}$ as follows.

  * •

For each $\tau\in\mathcal{D}_{R}$ with $R^{-1/2}\lesssim
d(\tau)<R^{-\frac{1}{n+1}}$, define
$F_{\tau}:=d(\tau)^{\frac{n-1}{2}}\chi_{B_{R}}$.

  * •

For $\tau\in\mathcal{D}_{R}$ with $d(\tau)=R^{-\frac{1}{n+1}}$ (or, precisely,
for each $\tau\in\mathcal{D}$), define

$F_{\tau}:=\sum_{T\in\mathbb{T}_{\tau}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau}\rangle}d(\tau)^{\frac{n-1}{2}}\phi_{T},$

where $\phi_{T}$ is a bump function on $T$ and $\xi_{\tau}$ is the centre of
$\Sigma(\tau)$. The coefficients $c_{T}\in\mathbb{C}$ are defined below.

  * •

For $\gamma\in\mathcal{D}_{R}$ with $R^{-\frac{1}{n+1}}<d(\tau)\leq 2$, define

$F_{\gamma}:=\sum_{\tau\in\mathcal{D},\tau\subset\gamma}F_{\tau}.$

Let $F:=F_{B^{n-1}}=\sum_{\tau\in\mathcal{D}}F_{\tau}$. The coefficients
$c_{T}$ will all have modulus 1, and will be chosen below so that

$|F|\gtrsim R^{\frac{n-1}{2(n+1)}}\text{ on }\bigcup_{B\in\mathcal{B}}B.$ (22)

Verifying (20) and (21). For each $\tau\in\mathcal{D}$, $F_{\tau}$ is Fourier
supported roughly in the smallest slab containing $\Sigma(\tau)$. It easily
follows that $(F_{\tau})_{\tau\in\mathcal{D}_{R}}$ satisfies the decoupling
axioms (DA1) and (DA2).

On the other hand, (22) and the small line occupancy of $w$ imply (21), so $F$
and $w$ do not respect the numerology of the Mizohata–Takeuchi conjecture.
Indeed,

$\int_{B_{R}}|F|^{2}w\gtrsim R^{\frac{n-1}{n+1}}\\#\mathcal{B}\gtrsim(\log
R)^{-2}R^{\frac{n-1}{n+1}}R^{n-1}$

by (22), while

$\displaystyle\begin{aligned}
\int|F|^{2}&\lesssim\sum_{\tau\in\mathcal{D}}|F_{\tau}|^{2}\lesssim\sum_{\tau\in\mathcal{D}}\sum_{T\in\mathbb{T}_{\tau}}\int_{T}|c_{T}d(\tau)^{\frac{n-1}{2}}|^{2}\sim\sum_{\tau\in\mathcal{D}}\sum_{T\in\mathbb{T}_{\tau}}|T|\cdot|\tau|=|B^{n-1}|\cdot|B_{R}|\sim
R^{n}\end{aligned}$

due to the essential disjointness of the Fourier supports of the $F_{\tau}$,
and therefore

$\|Xw\|_{\infty}\frac{1}{R}\int_{B_{R}}|F|^{2}\lesssim(\log
R)R^{n-1}\lesssim(\log R)^{3}R^{-\frac{n-1}{n+1}}\int_{B_{R}}|F|^{2}w.$

Defining the $c_{T}$. For $T\in\mathbb{T}$, let $\tau(T)$ be the cap
$\tau\in\mathcal{D}$ with $T\in\mathbb{T}_{\tau}$. For $B\in\mathcal{B}$, let

$\mathbb{T}_{B}:=\\{T\in\mathbb{T}:\;T\text{ intersects }B\\},$

and observe that, once the $c_{T}$ are defined for all $T\in\mathbb{T}$, it
will hold that

$F_{\;|B}=R^{\frac{-(n-1)}{2(n+1)}}\sum_{T\in\mathbb{T}_{B}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T_{\;|B}}\;\;\;\text{ for all
}B\in\mathcal{B}.$

The $c_{T}$ are thus defined via an iteration, the $j$-th step of which
ensures that the above sum has large magnitude for $B=B_{j}$. First, for all
$T\in\mathbb{T}_{B_{1}}$ define

$c_{T}:=e^{2\pi i\langle x_{1},\;\xi_{\tau(T)}\rangle},$

where $x_{1}$ is the centre of $B_{1}$. Due to the small radius of $B_{1}$,

${\rm Re}\left(c_{T}e^{-2\pi i\langle x,\;\xi_{\tau(T)}\rangle}\right)={\rm
Re}\left(e^{2\pi i\langle x_{1}-x,\;\xi_{\tau(T)}\rangle}\right)\gtrsim
1\text{ for all }x\in B_{1},$

hence

${\rm
Re}\left(R^{\frac{-(n-1)}{2(n+1)}}\sum_{T\in\mathbb{T}_{B_{1}}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T}\right)\gtrsim
R^{\frac{-(n-1)}{2(n+1)}}\\#\mathbb{T}_{1}\gtrsim R^{\frac{n-1}{2(n+1)}}$

on $B_{1}$. Therefore, once the remaining $c_{T}$ have been defined, we will
have that

$|F|\geq{\rm Re}\;F\gtrsim R^{\frac{n-1}{2(n+1)}}\text{ on }B_{1},$

as desired.

Now, fix $j=2,\ldots,m$. Suppose that, for each $i=1,\ldots,j-1$, we have
performed the $i$-th step of the iteration, by defining $c_{T}$ for all
$T\in\mathbb{T}_{B_{1}}$ (when $i=1$) and for all
$T\in\mathbb{T}_{B_{i}}\setminus(\mathbb{T}_{B_{1}}\cup\ldots\mathbb{T}_{B_{i-1}})$
(when $i\geq 2$) so that

$\left|{\rm
Re}\left(R^{\frac{-(n-1)}{2(n+1)}}\sum_{T\in\mathbb{T}_{B_{i}}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T}\right)\right|\gtrsim
R^{\frac{n-1}{2(n+1)}}$

on $B_{i}$ (which ensures that, once the remaining $c_{T}$ have been defined,
we will have that

$|F|\gtrsim R^{\frac{n-1}{2(n+1)}}\text{ on }B_{1},\ldots,B_{j-1}).$

During the $j$-th step of the iteration, we will define the $c_{T}$ for
$T\in\mathbb{T}_{B_{j}}\setminus(\mathbb{T}_{B_{1}}\cup\ldots\cup\mathbb{T}_{B_{j-1}})$
so that

$\left|{\rm
Re}\left(R^{\frac{-(n-1)}{2(n+1)}}\sum_{T\in\mathbb{T}_{B_{j}}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T}\right)\right|\gtrsim
R^{\frac{n-1}{2(n+1)}}$

on $B_{j}$ (ensuring that eventually $|F|\gtrsim R^{\frac{n-1}{2(n+1)}}$ on
$B_{j}$ as well). Write

$\mathbb{T}_{B_{j}}:=\mathbb{T}_{B_{j}}^{1}\sqcup\mathbb{T}_{B_{j}}^{2},$

where
$\mathbb{T}_{B_{j}}^{1}:=\mathbb{T}_{B_{j}}\setminus(\mathbb{T}_{B_{1}}\cup\ldots\mathbb{T}_{B_{j-1}})$
(the set of tubes through $B_{j}$ for which we still need to define the
$c_{T}$), while $\mathbb{T}_{B_{j}}^{2}$ consists of the tubes through $B_{j}$
for which the $c_{T}$ have already been defined. Importantly,
$\mathbb{T}_{B_{j}}^{1}\supset\mathbb{T}_{j}$.

Let $\sigma_{B_{j}}$ be the sign of $F^{2}_{j}:={\rm
Re}\left(\sum_{T\in\mathbb{T}_{B_{j}}^{2}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T}\right)$ on $B_{j}$
555Technically, this sign does not have to be uniform over all points of
$B_{j}$; we can however choose the dominant sign over $B_{j}$, and eventually
control the sum of the $F_{\tau}$ on a large subset of $B_{j}$. We omit this
additional technicality from our exposition., and define

$c_{T}:=\sigma_{B_{j}}e^{2\pi i\langle x_{j},\xi_{\tau(T)}\rangle}\text{ for
all }T\in\mathbb{T}_{B_{j}}^{1},$

where $x_{j}$ is the centre of $B_{j}$. As earlier,

$\left|{\rm Re}\left(c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\right)\right|\gtrsim 1\text{ on
}B_{j};$

and, crucially, ${\rm Re}\left(c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\right)$ also has sign
$\sigma_{B_{j}}$ on $B_{j}$, for all $T\in\mathbb{T}^{1}_{B_{j}}$. Therefore,
the functions $F_{j}^{2}$ and

$F_{j}^{1}:={\rm
Re}\left(R^{\frac{-(n-1)}{2(n+1)}}\sum_{T\in\mathbb{T}^{1}_{B_{j}}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T}\right)$

have the same sign on $B_{j}$, so

$\displaystyle\begin{aligned} \left|{\rm
Re}\left(R^{\frac{-(n-1)}{2(n+1)}}\sum_{T\in\mathbb{T}_{B_{j}}}c_{T}e^{-2\pi
i\langle\;\cdot\;,\;\xi_{\tau(T)}\rangle}\phi_{T}\right)\right|&=|F_{j}^{1}+F_{j}^{2}|\geq|F_{j}^{1}|\\\
&\gtrsim R^{\frac{-(n-1)}{2(n+1)}}\\#\mathbb{T}_{j}\gtrsim
R^{\frac{n-1}{2(n+1)}}\end{aligned}$

on $B_{j}$, as desired.

For all $T\in\mathbb{T}$ that do not contain any of the balls in
$\mathcal{B}$, we define $c_{T}=1$. By the end of the iteration, (22) holds. ∎

###### Proof of Claim 7.1..

Let $\mathcal{P}$ be a family of disjoint unit balls inside ${\rm supp}\;w$,
with

$\\#\mathcal{P}\sim|{\rm supp}\;w|\sim R^{n-1}.$

For each $B\in\mathcal{P}$, denote by $\mathbb{T}_{B}$ the set of tubes in
$\mathbb{T}$ through $B$, and observe that $\\#\mathbb{T}_{B}=\\#\mathcal{D}$.

Write $\mathcal{P}=\\{B_{1},B_{2},\ldots,B_{N}\\}$. To prove the claim, we
will show that there exist indices $k_{1}<k_{2}<\ldots<k_{m}$ such that:

  * •

$m\gtrsim(\log R)^{-10}R^{n-1}$,

  * •

$B_{k_{1}}=B_{1}$, and for each $j=2,3,\ldots,m$, at least $\\#\mathcal{D}/2$
tubes in $\mathbb{T}_{B_{k_{j}}}$ do not lie in
$\mathbb{T}_{B_{k_{1}}}\cup\mathbb{T}_{B_{k_{2}}}\cup\ldots\cup\mathbb{T}_{B_{j-1}}$.

Indeed,

  * •

let $k_{1}:=1$,

  * •

let $k_{2}$ be the smallest $j>k_{1}$ such that at most $\\#\mathcal{D}/2$
tubes through $B_{j}$ contain $B_{k_{1}}$,

  * •

let $k_{3}$ be the smallest $j>k_{2}$ such that at most $\\#\mathcal{D}/2$
tubes through $B_{j}$ contain $B_{k_{1}}$ or $B_{k_{2}}$,

and so on, until no further $k_{j}$ as above exists. Let $\mathcal{P}^{1}$ be
the set of balls $B_{k_{j}}$, over all the $k_{j}$ selected via the above
process. To complete the proof of the claim, it will now be shown that

$\\#\mathcal{P}^{1}\gtrsim(\log R)^{-2}R^{n-1},$

by studying the incidences between $\mathcal{P}$ and $\mathbb{T}$. For any
$\mathcal{S}\subset\mathcal{P}$ and $\mathbb{L}\subset\mathbb{T}$, denote

$I(\mathcal{S},\mathbb{L}):=\\#\\{(B,T)\in S\times\mathbb{L}:\;B\text{ is
contained in }T\\},$

the number of incidences between $\mathcal{S}$ and $\mathbb{L}$.

Assume for contradiction that

$\\#\mathcal{P}^{1}\lesssim(\log R)^{-2}\\#\mathcal{P}$ (23)

for an appropriately small implicit constant. Then, the set $\mathbb{T}^{1}$
of tubes in $\mathbb{T}$ that pass through balls in $\mathcal{P}^{1}$ is not
too large; in particular,

$\\#\mathbb{T}^{1}\leq
I(\mathcal{P}^{1},\mathbb{T}^{1})\leq\mathcal{P}^{1}\\#\mathcal{D}\lesssim(\log
R)^{-2}\\#\mathcal{P}\\#\mathcal{D}\sim(\log
R)^{-2}I(\mathcal{P},\mathbb{T})\lesssim(\log R)^{-1}\\#\mathbb{T},$

for a small implicit constant. Therefore, the tubes in $\mathbb{T}^{1}$ only
contribute a small fraction of the total incidences between $\mathbb{T}$ and
$\mathcal{P}$:

$I(\mathcal{P},\mathbb{T}^{1})\lesssim\\#\mathbb{T}^{1}\log
R\lesssim\mathbb{T}\sim(\log
R)^{-1}\\#\mathcal{D}R^{n-1}\sim\\#\mathcal{D}\\#\mathcal{P}\leq\frac{1}{10}\;I(\mathcal{P},\mathbb{T})$

(the implicit constant in (23) is chosen so that this is true).

This is a contradiction, as $\mathcal{P}^{1}$ was selected so that
$\mathbb{T}^{1}\;(=\cup_{j=1}^{m}\mathbb{T}_{B_{k_{j}}})$ contributes at least
half of the total incidences between $\mathbb{T}$ and $\mathcal{P}$. Indeed,
each $B_{i}\in\mathcal{P}\setminus\mathcal{P}^{1}$ is incident to at least
$\\#\mathcal{D}/2$ tubes in
$\cup_{k_{j}<i}\mathbb{T}_{B_{k_{j}}}\subset\mathbb{T}^{1}$; while each
$B_{i}\in\mathcal{P}$ has all the $\\#\mathcal{D}$ tubes in $\mathbb{T}$
through it in $\mathbb{T}^{1}$. Therefore,

$I(\mathcal{P},\mathbb{T}^{1})\geq\\#\mathcal{P}\\#\mathcal{D}/2=\frac{1}{2}\;I(\mathcal{P},\mathbb{T}),$

contradicting (23). ∎

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