# The space-time estimates for the Schrödinger equation

Junfeng Li  School of Mathematical Sciences, Dalian University of Technology,
Dalian, LN, 116024, China<EMAIL_ADDRESS>,  Changxing Miao  Institute
of Applied Physics and Computational Mathematics, Beijing, 100088, China
<EMAIL_ADDRESS>and  Ankang Yu  School of Mathematical Sciences,
Dalian University of Technology, Dalian, LN, 116024, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we studied the space-time estimates for the solution to the
Schrödinger equation. By polynomial partitioning, induction arguments,
bilinear to linear arguments and broad norm estimates, we set up several
maximal estimates for the Schrödinger equation with high frequency input data.
By these maximal estimates we obtain the sharp global space-time estimate when
$n=2$ and improve the known results in the critical cases when $n\geq 3$. The
maximal estimate for $n=2$ is also used to extend the results of the local
space-time estimates for the solution to the Schrödinger equation.

###### Key words and phrases:

Schrödinger group; space-time estimate; polynomial partitioning; refined
Strichartz estimate

###### 2010 Mathematics Subject Classification:

42B20, 42B25

This work was supported by the National Key R&D program of China; No.
2022YFA1005700. Junfeng Li was partially supported by NSF of China grant: No.
12071052 and the fundamental research funds for the central universities.

##  1\. Introduction

Given the free Schr$\ddot{\text{o}}$dinger equation:

$i\partial_{t}u+\Delta u=0,\ \ \text{in}\ \ \mathbb{R}^{n+1}$

with an initial data $f\in\mathcal{S}(\mathbb{R}^{n})$, its solution can be
written as

$u(t,x):=e^{it\Delta}f(x)=\int_{\mathbb{R}^{n}}\widehat{f}(\xi)e^{2\pi
i(x\cdot\xi-2\pi t|\xi|^{2})}d\xi,$

where $\widehat{f}(\xi):=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\xi}dx$. A
nature problem is to understand the solution $u(t,x)$ with an initial data
$f\in H^{s}(\mathbb{R}^{n})$, the Sobolev space. The well-known Srichartz
estimate is a typical estimate. It is a time-space type estimate which weight
the solution in $L^{r}_{t}L^{p}_{x}(\mathbb{R}^{n+1})$. Such a estimate is
crucial in the study of the linear and nonlinear Schrödinger equations. By the
end of the last century, it was fully established.

######  Theorem 1.1.

[21, 29]. Let $r\geq 2$, $2\leq p<\infty$ and
$\frac{n}{p}+\frac{2}{r}\leq\frac{n}{2}$. Then

(1.1) $\|e^{it\Delta}f\|_{L_{t}^{r}(\mathbb{R},L_{x}^{p}(\mathbb{R}^{n}))}\leq
C\|f\|_{\dot{H}^{s}({\mathbb{R}^{n}})},\ \ \ \
s=\frac{n}{2}-\frac{n}{p}-\frac{2}{r}.$

In this paper, we first consider the space-time counter part of the Strichartz
estimate. That is weight the function $u(t,x)$ in the space
$L^{p}_{x}L^{r}_{t}(\mathbb{R}^{n+1})$. In [27] Planchon formulated the
following space-time conjecture.

######  Conjecture 1.2.

Let $2\leq r,p<\infty$ and $\frac{n+1}{p}+\frac{1}{r}\leq\frac{n}{2}$. Then

(1.2) $\|e^{it\Delta}f\|_{L_{x}^{p}L^{r}_{t}(\mathbb{R}^{n+1})}\leq
C\|f\|_{\dot{H}^{s}({\mathbb{R}^{n}})},\ \ \ \
s=\frac{n}{2}-\frac{n}{p}-\frac{2}{r}.$

Both inequalities (1.1) and (1.2) are the anisotropic generalization of the
adjoint estimate of the Fourier restriction estimate. They satisfy some common
symmetries such as translation invariance and scaling invariance. The former
allows us to do translations over $x$ and $t$, and the latter predicts the
relation $s=\frac{n}{2}-\frac{n}{p}-\frac{2}{r}$. The necessary conditions for
the index pairs $(p,r)$ can be verified by Knapp’s example.

The estimates (1.1) and (1.2) behave differently under Galilean transforms.
The Strichartz estimate (1.1) is Galilean invariant. By which we can make
translation in the frequency space. While the space-time estimate (1.2) is not
Galilean invariant which makes space-time estimate more difficult. And the
estimate would depend on both the size and the position of the support of
$\widehat{f}$.

###  1.1. Maximal estimates

The space-time estimate and its variant has close relation with several
important problems in harmonic analysis. One of the endpoint index pair
$(\frac{2(n+1)}{n},\infty)$ of its critical case appears naturally in the
study of the well-known Carleson’s problem. Here and after, the index pair
$(p,r)$ is critical if $\frac{n+1}{p}+\frac{1}{r}=\frac{n}{2}$. And the pair
in the set {$(p,r):\frac{n+1}{p}+\frac{1}{r}<\frac{n}{2}$} will be called
subcritical. In [4] Carleson formulated the problem to find the the infimum
regularity $s_{c}$ such that

$\lim_{t\to 0}e^{it\Delta}f(x)=f(x)\quad a.e.\,x\in\mathbb{R}^{n},\,\forall
f\in H^{s}(\mathbb{R}^{n}).$

In the last few decades, there were abundant literature such as [22, 5, 9, 10,
13, 26] on this problem. So far, Carleson’s problem is well understood, and
the critical index is $s_{c}=\frac{n}{2(n+1)}$ for $n\geq 1$. The one
dimensional case follows from the results of Carleson [4] and Dahlberg and
Kenig [11]. For the high dimensional case, the necessary part follows from the
counterexample formulated by Bourgain in [2]. To achieve the positive result,
Du, Guth, and Li [9] set up the following $\epsilon$ lost estimate

(1.3)
$\|e^{it\Delta}f\|_{L_{x}^{3}L_{t}^{\infty}(B^{3}(0,R))}\lesssim_{\epsilon}R^{\epsilon}\|f\|_{L^{2}},$

for $\text{supp}\widehat{f}\subset\mathbb{B}^{2}(0,1)$ when $n=2$. Motivated
by the two dimensional case result, one may expect to set up the $\epsilon$
lost estimate

(1.4)
$\|e^{it\Delta}f(x)\|_{L_{x}^{\frac{2(n+1)}{n}}L^{\infty}_{t}(\mathbb{B}^{n+1}(0,R))}\lesssim_{\epsilon}R^{\epsilon}\|f\|_{L^{2}(\mathbb{R}^{n})}$

for $\text{supp}\widehat{f}\subset\mathbb{B}^{n}(0,1)$. Which corresponding to
endpoint index of Conjecture 1.2. With (1.4), combining the $\epsilon$ removal
argument in [6], one can obtain (1.2) for the whole subcritical case.

While Du, Kim, Wang, and Zhang [12] showed that the estimate (1.4) fails for
$n\geq 3$. As a replacement, Du and Zhang [13] proved

(1.5) $\|\sup\limits_{0<t\leq
R}|e^{it\Delta}f|\|_{L^{2}(\mathbb{B}^{n}(0,R))}\leq
R^{\frac{n}{2(n+1)}+\epsilon}\|f\|_{L^{2}(\mathbb{R}^{n})},\ \ \ \
\forall\epsilon>0,$

for $\text{supp}\widehat{f}\subset\mathbb{B}^{n}(0,1)$. It is a meaningful
problem to find the sharp range of $p$ such that

(1.6) $\|\sup_{0<t\leq
R}|e^{it\Delta}f(x)|\|_{L^{p}(\mathbb{B}^{n}(0,R))}\lesssim_{\epsilon}R^{\epsilon}\|f\|_{L^{2}(\mathbb{R}^{n})}$

for $\text{supp}\widehat{f}\subset\mathbb{B}^{n}(0,1)$. Recently, Wu [34],
Cao, Miao, and Wang [7] have maken great improvement on this problem.

Since the local estimate (1.4) does not hold therefore the corresponding
global estimate cannot hold either. Thus, Conjecture 1.2 becomes a meaningful
replacement of the maximal estimate. To achieve this aim, we set up several
maximal estimates with high frequency input functions. Which also reflect the
fact that the frequency of the input function affects the space-time
estimates. Given $z_{0}=(x_{0},t_{0})$, $N\geq 1$ and a usually very large
number $R\geq 1$, $Q_{R}(x_{0})\subset\mathbb{R}^{n}$ denotes a cube centered
at $x_{0}$ with side length $R$. A parallelepiped with direction $(4\pi
Ne_{1},1)$ and center $z_{0}$ is

$P_{R}(z_{0}):=\\{(x,t)\in\mathbb{R}^{n}\times[-R+t_{0},R+t_{0}]:x-4\pi
tNe_{1}\in Q_{R}(x_{0})\\},$

here $e_{1}=(1,0,\dots,0)$. The parameter $R$ will be called as its size of
$P_{R}(z_{0})$. Let $P_{R}=P_{R}(z_{0})$ with $z_{0}=(0,0)$. We now state our
main results in this part.

######  Theorem 1.3 (High frequency input endpoint estimate for $n=2$).

Let $\widehat{f}$ be supported in $\mathbb{B}^{2}(Ne_{1},1)$ with $N\geq 1$.
For any $\epsilon>0$, we have

(1.7)
$\|e^{it\Delta}f\|_{L_{x}^{3}L_{t}^{\infty}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{3}}R^{\epsilon}||f||_{L^{2}}.$

######  Theorem 1.4 (High frequency input estimate for $n\geq 3$).

Let $\widehat{f}$ be supported in $\mathbb{B}^{n}(Ne_{1},1)$ with $N\geq 1$.
For any $\epsilon>0$, we have

(1.8)
$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{\infty}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{p}}R^{\epsilon}||f||_{L^{2}},$

whenever

$p\geq\tilde{p}_{n}=2+\frac{4}{n+1+s_{2}^{n}}\ \text{with}\ \
s_{2}^{n}=\sum_{i=2}^{n}\frac{1}{i}.$

######  Remark 1.5.

The propagation speed of the Schrödinger wave $e^{it\Delta}f$ depends on the
location of $\widehat{f}$. In this case, the Schrödinger wave $e^{it\Delta}f$
roughly propagates at speed $N$ in the direction $(4\pi Ne_{1},1)$. Then for
$|t|\leq R$, the propagation region of the wave is essentially concentrated in
$P_{R}$. Since there is no Galilean invariant, the estimate with different
frequency input is inevitable.

###  1.2. Global space-time estimates

We turn back to Conjecture 1.2. For $n=1$, it was proved by Kenig, Ponce, and
Vega [28]. For $n\geq 2$, Vega [33] set up (1.2) for
$(p,r)=(\frac{2(n+1)}{n-1},2)$ by Plancherel’s theorem (in time) and the $n-1$
dimensional Stein-Tomas restriction theorem. Combining his result with the
$n$-dimensional Stein-Tomas restriction theorem, one can verify the estimate
(1.2) for $p\geq\frac{2(n+2)}{n}$ in the critical case. It remains to verify
Conjecture 1.2 when $\frac{2(n+1)}{n}\leq p\leq\frac{2(n+2)}{n}$. Combining
Tao’s sharp bilinear restriction estimates for the paraboloid [32] with a
bilinear-to-linear argument of Planchon [27], one can obtain (1.2) in the
subcritical region and $p>\frac{2(n+3)}{n+1}$. This result was latter extended
to the critical case by Lee, Rogers, and Vargas [23]. When $n=2$, there are
further results that breakthrough the index
$\frac{2(n+3)}{(n+1)}=\frac{10}{3}$. Rogers [30] obtained (1.2) for
$p>\frac{16}{5}$ in the subcritical case. Lee, Rogers, and Vargas [23] extend
it to the critical case. Very recently, Cho, Koh, and Lee [6] set up (1.2) in
the whole subcritical region. With theorems 1.3 and 1.4 at hand, we can obtain
the following space-time estimates in the critical case.

######  Theorem 1.6.

Let $2\leq r<\infty$ and $\frac{n+1}{p}+\frac{1}{r}=\frac{n}{2}$. Then

  * (i)

when $n=2$, (1.2) holds for $p>3$;

  * (ii)

when $n\geq 3$, (1.2) holds for $p>p_{n}$, where

$p_{n}=\frac{2(n+1)[(n+2)\tilde{p}_{n}-2(n+3)]}{(n+1)n\tilde{p}_{n}-2(n^{2}+2n-1)},$

with the same $\tilde{p}_{n}$ as in Theorem 1.4.

######  Remark 1.7.

By this theorem, we confirm Conjecture 1.2 when $n=2$ (see Figure 1). For
$n\geq 3$, we improve the results obtained Lee, Rogers, and Vargas [23] (see
the green part of Figure 2). In Table 1, we list several values of $p_{n}$,
the conjectured sharp index $p_{s}$, and $p_{lrv}$ obtained by Lee, Rogers,
and Vargas [23].

######  Remark 1.8.

The local maximal estimate (1.3) is not enough to obtain critical estimate
(1.2) for $n=2$ (see Cho, Koh, and Lee [6]).

$\frac{1}{p}$$\frac{1}{r}$O$\frac{1}{6}$$\frac{1}{3}$$\frac{1}{2}$$\frac{1}{2}$
Figure 1. $n=2$

$\frac{1}{p}$$\frac{1}{r}$O$\frac{n-2}{2n}$$\frac{1}{p_{lrv}}$$\frac{1}{p_{n}}$$\frac{1}{p_{s}}$$\frac{1}{2}$$\frac{1}{2}$
Figure 2. $n\geq 3$

Table 1.  $n=$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---
$p_{\text{s}}=$ | 3 | $2+\frac{2}{3}$ | $2+\frac{1}{2}$ | $2+\frac{2}{5}$ | $2+\frac{1}{3}$ | $2+\frac{2}{7}$ | $2+\frac{1}{4}$ | $2+\frac{2}{9}$ | $2+\frac{1}{5}$ | $2+\frac{2}{11}$
$p_{n}\sim$ | 3 | $2+\frac{44}{50}$ | $2+\frac{7}{10}$ | $2+\frac{3}{5}$ | $2+\frac{1}{2}$ | $2+\frac{47}{100}$ | $2+\frac{2}{5}$ | $2+\frac{38}{100}$ | $2+\frac{7}{20}$ | $2+\frac{3}{10}$
$p_{lrv}=$ | $3+\frac{1}{5}$ | $3$ | $2+\frac{4}{5}$ | $2+\frac{2}{3}$ | $2+\frac{4}{7}$ | $2+\frac{1}{2}$ | $2+\frac{4}{9}$ | $2+\frac{2}{5}$ | $2+\frac{4}{11}$ | $2+\frac{1}{3}$

###  1.3. Local space-time estimates

Beside the global in time estimates as above, the local in time estimates are
also interesting. The following conjecture formulated by Lee, Rogers, and
Seeger [25] involving the Stein’s square function estimate for the Bochner-
Riesz mean and also the adjoint estimate of the Fourier restriction estimate.

######  Conjecture 1.9.

Let $2\leq p<\infty$, $2\leq r\leq\infty$ satisfy
$\frac{n}{p}+\frac{1}{r}<\frac{n}{2}$ and $\frac{2n+1}{p}+\frac{1}{r}<n$.
Then,

(1.9)
$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n}\times[0,1])}\lesssim\|f\|_{B_{\alpha,p}^{p}(\mathbb{R}^{n})}$

hold for $\alpha=n(1-\frac{2}{p})-\frac{2}{r}$. Here $B_{\alpha,p}^{p}$
denotes the non-homogeneous Besov space with the norm

$\|f\|_{B_{\alpha,p}^{p}}=(\sum_{k\geq 0}2^{k\alpha
p}\|P_{k}f\|_{p}^{p})^{1/p}.$

For $n=1$, the conjecture was proved by Lee, Rogers, and Seeger [25]. The
higher dimensional case is still open. Figure 3 is the index graph
corresponding to Conjecture 1.9. On the line
$(0,\frac{1}{2})-(\frac{n-1}{2n},\frac{1}{2})$, Conjecture 1.9 is the
corollary of the Stein’s square function conjecture (see [24]). Which was
proved by Lee, Rogers, and Seeger [24] to be true for $p>4$ when $n=2$. The
most recent progress in higher dimension can be found in Gan, Oh, and Wu [20].
Combing the knowledge of the Fourier restriction estimate, one can obtain the
estimates in the red region. For the yellow area, Lee, Rogers, and Seeger [25]
showed that Conjecture 1.9 can be reduced to the Fourier restriction estimate
conjecture that

(1.10)
$\Big{\|}\int_{\mathbb{B}^{n}(0,1)}e^{i(\xi_{1}y_{1}+\dots+\xi_{n}y_{n}+s|\xi|^{2})}f(\xi)d\xi\Big{\|}_{L^{p}(\mathbb{R}^{n+1})}\lesssim\|f\|_{L^{p}(\mathbb{B}(0,1))}$

holds for $p>\frac{2(n+1)}{n}$. In the blue area, as far as the authors
knowledge, there is no result in this part.

In this paper, we focus on the problem for $n=2$. In this case, Wang [35]
proved the Fourier restriction estimate (1.10) for $p>3+\frac{3}{13}$. It
follows that (1.9) is true for $p>3+\frac{3}{13}$ and $p\leq r\leq\infty$ ($A$
of Figure 4). Combining the results of square function estimate obtained by
Lee, Rogers and Seeger [24], the red region in Figure 4 is verified for (1.9).
In [25], Lee, Rogers, and Seeger set up (1.9) for $p>3+\frac{1}{5}$ and $r\geq
4$ ($B$ of Figure 4) which allow us to have the yellow region. In this paper,
we obtain the brown region in Figure 4 by setting up the following theorem.

$\frac{1}{p}$$\frac{1}{r}$O$1/2$$(\frac{n-1}{2n},\frac{1}{2})$$\frac{n}{2n+1}$$(\frac{n}{2(n+1)},\frac{n}{2(n+1)})$
Figure 3.

$\frac{1}{p}$$\frac{1}{r}$O$1/2$$(\frac{1}{4},\frac{1}{2})$B$(\frac{1}{3},\frac{1}{3})$CA$\frac{2}{5}$
Figure 4. $A=(\frac{13}{42},\frac{13}{42})$, $B=(\frac{5}{16},\frac{1}{4})$,
$C=(\frac{7}{22},\frac{2}{11})$

######  Theorem 1.10.

When $n=2$, the local space-time estimate (1.9) holds for $p>3+1/7$ and $r\geq
5+1/2.$

######  Remark 1.11.

In higher dimensional case, the same argument as in Section 6 allows us to
verify (1.9) for some $(p,r)$. But they are still in the region implied by the
Fourier restriction estimate obtained by Hickman and Zahl [19].

This paper is organized as follows. In Section 2, Theorem 1.3 is proved by
using the polynomial partitioning of Guth [15] and the bilinear refined
Strichartz estimate of Du, Guth and Li [9]. In Section 3, Theorem 3.1, a
general form of Theorem 1.4, is reduced to a $2$-broad norm $BL_{A}^{p,q}$
estimate. For later use, a $BL_{A}^{2,\infty}$ estimate with input function
tangent to an $m$-variety is also studied. In Section 4, we set up the
$BL_{A}^{p,q}$ estimate by two algorithms introduced by Hickman and Rogers
[18]. In section 5, Theorem 1.6 is proved by using Theorem 1.3. In section 6,
Theorem 1.10 is proved. In Appendix A, we give a different proof of the
epsilon removal for the bilinear space-time estimate from Lee, Rogers and
Vargas [23]. In Appendix B, we formulate a conjecture on the bilinear space-
time estimate which implies Conjecture 1.2.

Notation list:

  * •

$A\lesssim B$: there exists a positive constant $C$ such that $A\leq CB$,
where $C$ may change from line to line. $A\sim B$ means $A\lesssim B$ and
$B\lesssim A$.

  * •

$A\lesssim_{\epsilon}B$: the implicant constant $C$ depends on a given
$0<\epsilon\ll 1$.

  * •

$A(R)\leq\text{RapDec}(R)B$: for any $N\geq 1$, there exists a constant
$C_{N}$ such that

$A(R)\leq C_{N}R^{-N}B\ \ \text{for all}\ R\geq 1.$

  * •

For a finite set $A$, $\sharp A$ denotes its cardinality.

  * •

$\epsilon>0$ is a very small number and $\delta\ll\epsilon$ (for example
$\delta=\epsilon^{2}$).

  * •

$R\geq 1$ is an arbitrary very large number. $r$ denotes some number $1\leq
r\leq R$. $K$ is also very large but far less than $R$.

  * •

$\mathbb{B}^{n}(Ne_{1},1):=\\{\xi\in\mathbb{R}^{n}:|\xi-Ne_{1}|\leq 1\\}$,
here $N\geq 1$ and $e_{1}=(1,0,\dots,0)$.

  * •

Let $z_{0}=(x_{0},t_{0})\in\mathbb{R}^{n+1}$ and $r\geq 1$,

$P_{r}(z_{0}):=\\{(x,t)\in\mathbb{R}^{n}\times[-r+t_{0},r+t_{0}]:x-4\pi
tNe_{1}\in Q_{r}(x_{0})\\}.$

Here $Q_{r}(x_{0})\subset\mathbb{R}^{n}$ denotes a cube centered at $x_{0}$
with side length $r$. The parameter $r$ is the size of $P_{r}(z_{0})$. We
abbreviate it as $P_{r}$ if the center is not important.

  * •

$T_{\theta,\nu}\in\mathbb{T}[r]$ means that a tube

$T_{\theta,\nu}:=\Big{\\{}(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\Big{|}\ 0\leq
t\leq r,\ |x-4\pi tc(\theta)-c(\nu)|\leq{r}^{\frac{1}{2}+\delta}\Big{\\}},$

here $c(\theta)\in{r}^{-\frac{1}{2}}\mathbb{Z}^{n}$ and
$c(\nu)\in{r}^{\frac{1}{2}}\mathbb{Z}^{n}$.

  * •

$Z$ or $Z(P)$ denotes the variety of a polynomial $P$. $T_{z}Z$ denotes the
tangent space of $Z$ at point $z\in Z$.

##  2\. High frequency input endpoint estimate for $n=2$

We devote this section to prove Theorem 1.3 by an induction argument.
Moreover, we set up the following theorem.

######  Theorem 2.1.

Let $p>3$ and $\ M\geq 1$, $f\in L^{2}(\mathbb{R}^{2})$ with
$\text{supp}\widehat{f}\subset\mathbb{B}^{2}(\xi_{0},\frac{1}{M})\subset\mathbb{B}^{2}(Ne_{1},1)$
for some $N\geq 1$. For any $\epsilon>0$, $R\geq 1$, and $r>1/\epsilon^{4}$,
there exists a constant $C_{p,\epsilon}$ such that

(2.1) $\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}\leq
C_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}.$

Since the constant $C_{p,\epsilon}$ does not depend on $r$, by taking
$r\to\infty$ and $M=1$, we have

$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{\infty}(P_{R})}\leq
C_{p,\epsilon}N^{\frac{1}{p}}R^{\epsilon}\|f\|_{L^{2}}.$

By interpolating with the trivial estimate

$\|e^{it\Delta}f\|_{L_{x}^{2}L_{t}^{\infty}(P_{R})}\lesssim(NR^{2})^{1/2}\|f\|_{L^{2}},$

we arrive that

$\|e^{it\Delta}f\|_{L_{x}^{3}L_{t}^{\infty}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{3}}R^{\epsilon}\|f\|_{L^{2}}.$

###  2.1. Wave packet decomposition

Wave packet decomposition is a fundamental tool in the proofs. In this
subsection, we collect several basic concepts and properties of the wave
packets without proof. We refer the readers to [32] for the details.

Let $\phi$ be a Schwartz function satisfying
$\text{supp}\,\widehat{\phi}\subset\mathbb{B}^{n}(0,3/2)$ and

$\sum_{k\in\mathbb{Z}^{n}}\widehat{\phi}(\xi-k)=1,\ \ \ \ \text{for
all}\quad\xi\in{\mathbb{R}^{n}}.$

Given $\lambda>0$, let

(2.2)
$\widehat{\phi_{\theta}}(\xi):={\lambda}^{\frac{n}{2}}\widehat{\phi}({\lambda}^{\frac{1}{2}}(\xi-c(\theta))),\
\widehat{\phi_{\theta,\nu}}(\xi):=e^{-2\pi
ic(\nu)\cdot\xi}\widehat{\phi_{\theta}}(\xi),$

where $c(\theta)\in{\lambda}^{-\frac{1}{2}}\mathbb{Z}^{n}$ and
$c(\nu)\in{\lambda}^{\frac{1}{2}}\mathbb{Z}^{n}$. $\theta$ denotes a
${\lambda}^{-\frac{1}{2}}$-ball in the frequency space and $\nu$ denotes a
${\lambda}^{\frac{1}{2}}$-ball in the physical space. Usually
$\phi_{\theta,\nu}$ is called as a wave packet with scale $\lambda$. Let
$\Omega:=\text{supp}\widehat{f}\subset\mathbb{R}^{n}$. A wave packet
decomposition of $f$ at scale $\lambda$ is

(2.3)
$f=\sum_{\theta,\nu}c_{\theta,\nu}\phi_{\theta,\nu}:=\sum_{\theta,\nu}f_{\theta,\nu},\
\ \theta\cap\Omega\neq\varnothing.$

Then

(2.4)
$e^{it\Delta}f=\sum_{\theta,\nu}c_{\theta,\nu}e^{it\Delta}\phi_{\theta,\nu}:=\sum_{\theta,\nu}e^{it\Delta}f_{\theta,\nu}.$

These decompositions satisfy the following properties.

  * •

The Fourier transform of $f_{\theta,\nu}$ is supported on $\theta$, and the
Fourier transform of $e^{it\Delta}f_{\theta,\nu}$ is supported on the
paraboloid

$\big{\\{}(\xi,\tau);\tau=2\pi|\xi|^{2},\xi\in\theta\big{\\}}.$

  * •

The functions $\phi_{\theta,\nu}$ are approximately orthogonal, and

(2.5)
$\sum_{\theta,\nu}|c_{\theta,\nu}|^{2}\approx\sum_{\theta,\nu}\|f_{\theta,\nu}\|_{L^{2}}^{2}\approx\|f\|^{2}_{L^{2}}.$

  * •

By the stationary phase method,

(2.6)
$|e^{it\Delta}\phi_{\theta,\nu}|\leq{\lambda}^{-\frac{n}{4}}\chi_{T_{\theta,\nu}}(x,t)+\text{RapDec}(\lambda),\
0\leq t\leq\lambda.$

Here

$T_{\theta,\nu}:=\Big{\\{}(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\Big{|}\ 0\leq
t\leq\lambda,\ |x-4\pi
tc(\theta)-c(\nu)|\leq{\lambda}^{\frac{1}{2}+\delta}\Big{\\}},$

where $\delta\ll\epsilon$. $T_{\theta,\nu}$ is a tube of length $\lambda$, of
radius ${\lambda}^{\frac{1}{2}+\delta}$ with the direction $(4\pi c(\theta),\
1)$. The collection of such tubes is denoted as $\mathbb{T}[\lambda]$ which is
also used to denote the set of the $(\theta,\nu)$ pairs.

Since the Fourier transform of $e^{it\Delta}f$ with
$\widehat{f}\subset\mathbb{B}^{n}(Ne_{1},1)\,(N\geq 1)$ is supported on a
rectangular box $1\times...\times 1\times N$. By the locally constant
property, $e^{it\Delta}f$ is roughly constant on a rectangular box
$1\times...\times 1\times 1/N$. Similarly, for each ball
$\tau\subset\mathbb{B}^{n}(Ne_{1},1)$ with radius $K^{-1}$,
$e^{it\Delta}f_{\tau}$ (with $\text{supp}\widehat{f}_{\tau}\subset\tau$) is
locally constant on a rectangular box $K\times...\times K\times K/N$.

###  2.2. The induction argument and the polynomial partitioning

Theorem 2.1 will be proved by induction on the physical radius $R$ and the
frequency radius $\frac{1}{M}$.

When $R\lesssim 1$, it’s easy to see that

$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{1})}\leq
C_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}\|f\|_{L^{2}}.$

When $R\gg 1$. If $M>R^{10}$, Theorem 2.1 is trivial. If
$R^{\frac{1}{2}-\delta}\leq M\leq R^{10}$, we adopt the wave packet
decomposition (2.3) at scale $R$. Since supp$\widehat{f}\subset
B(\xi_{0},\frac{1}{M})$, we have

$\sharp\Big{\\{}\theta:\exists\nu\ s.t.\
(\theta,\nu)\in\mathbb{T}\Big{\\}}\leq\frac{(1/M)^{2}}{1/R}\lesssim
R^{O(\delta)}.$

Then, for $p>3$ and $r>\epsilon^{-4}$,

$\displaystyle||e^{it\Delta}f||_{L_{x}^{p}L_{t}^{r}(P_{R})}$
$\displaystyle=\Big{\|}\Big{|}\sum_{\theta,\nu}c_{\theta,\nu}e^{it\Delta}\phi_{\theta,\nu}\Big{|}^{2}\Big{\|}_{L_{x}^{\frac{p}{2}}L_{t}^{\frac{r}{2}}(P_{R})}^{\frac{1}{2}}$
$\displaystyle\lesssim
R^{O(\delta)}\sup_{\theta_{0}}\Big{\|}\sum_{\nu}\Big{|}c_{\theta_{0},\nu}e^{it\Delta}\phi_{\theta_{0},\nu}\Big{|}^{2}\Big{\|}_{L_{x}^{\frac{p}{2}}L_{t}^{\frac{r}{2}}(P_{R})}^{\frac{1}{2}}$
$\displaystyle\lesssim
R^{O(\delta)}\sup_{\theta_{0}}R^{-\frac{1}{2}}\Big{(}\sum_{\nu}|c_{\theta_{0},\nu}|^{2}\|\chi_{T_{\theta_{0},\nu}}\|_{L_{x}^{\frac{p}{2}}L_{t}^{\frac{r}{2}}(P_{R})}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim
R^{O(\delta)}R^{-\frac{1}{2}}(N^{-1}R^{\frac{1}{2}})^{\frac{1}{r}}(R^{\frac{3}{2}}N)^{\frac{1}{p}}\sup_{\theta_{0}}\Big{(}\sum_{\nu}|c_{\theta_{0},\nu}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim
R^{O(\delta)}N^{\frac{1}{p}-\frac{1}{r}}R^{\frac{3}{2p}-\frac{1}{2}+\frac{1}{2r}}||f||_{L^{2}}$
(2.7) $\displaystyle\lesssim
N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}.$

Therefore it is sufficient to consider the case $R\gg 1$ and $M\ll
R^{\frac{1}{2}}$.

Induction Assumption:

(3.1) holds for $1\leq\tilde{R}\leq R/2$ or $\tilde{M}\geq 2M.$

We now introduce the polynomial partitioning. For a polynomial $P$, it is
called nonsingular if $\nabla P(z)\neq 0$ for each point $z\in Z(P)$, where
$Z(P):=\\{(x,t)\in{{{\mathbb{R}}}^{n}}\times\mathbb{R}:P(x,t)=0\\}$ denotes a
variety of the polynomial $P$. The following polynomial partitioning theorem
on mixed norm is obtained by Du, Guth, and Li.

######  Theorem 2.2.

[9] If $F\in
L^{1}_{x}L^{r}_{t}({{{\mathbb{R}}}^{n}}\times\mathbb{R})\backslash\\{0\\},\
1\leq r<\infty$, then for each $D$, there exists a nonzero polynomial $P$ of
degree at most $D$ such that $({{{\mathbb{R}}}^{n}}\times\mathbb{R})\backslash
Z(P)$ is a union of $\sim_{n}D^{n+1}$ disjoint open sets $O_{i}$ and for each
$i$, we have

$\|F\|_{L^{1}_{x}L^{r}_{t}({{{\mathbb{R}}}^{n}}\times\mathbb{R})}\leq
c_{n}D^{n+1}\|\chi_{O_{i}}F\|_{L^{1}_{x}L^{r}_{t}({{{\mathbb{R}}}^{n}}\times\mathbb{R})}.$

Moreover, the polynomial $P$ is a product of distinct nonsingular polynomials.
The open set $O_{i}$ is called as a cell.

Therefor, there exists a nonzero polynomial $P$ with degree
$D=R^{\epsilon^{4}}$ such that
$(\mathbb{R}^{2}\times\mathbb{R})\backslash\\{Z(P)\\}$ is a union of $\sim
D^{3}$ disjoint cells $O_{i}$ and

(2.8) $\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\lesssim
D^{3}\|\chi_{O_{i}}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p},\quad\forall\quad
1\leq i\lesssim D^{3}.$

A wall is the set $W:=N_{R^{1/2+\delta}}Z(P)\cap P_{R}$ with
$\delta=\epsilon^{2}$, where $N_{R^{1/2+\delta}}Z(P)$ stands for the
$R^{1/2+\delta}$-neighborhood of the variety $Z(P)$. For each cell $O_{i}$, we
define

(2.9) $O_{i}^{{}^{\prime}}:=[O_{i}\cap P_{R}]\backslash W,\ \
\mathbb{T}_{i}:=\\{(\theta,\nu)\in\mathbb{T}[R]:T_{\theta,\nu}\cap
O_{i}^{{}^{\prime}}\neq\emptyset\\}\ \ \text{and}\ \
f_{i}:=\sum_{(\theta,\nu)\in\mathbb{T}_{i}}f_{\theta,\nu}.$

For each shrunken cell $O^{\prime}_{i}$, we have

$\chi_{Q^{\prime}_{i}}e^{it\Delta}f\sim e^{it\Delta}f_{i}.$

With these notations,

(2.10)
$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\lesssim\sum_{i}\|\chi_{O_{i}^{{}^{\prime}}}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}+\|\chi_{W}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}.$

The estimate of (2.10) will be catalogued into two cases:

Cellular case:

there exist $O(D^{3})$ many cells $O_{i}^{{}^{\prime}}$ such that for each
$i$,

(2.11) $\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\lesssim
D^{3}\|\chi_{O_{i}^{{}^{\prime}}}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}.$

Wall case:

(2.12)
$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\lesssim\|\chi_{W}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}.$

For the cellular case, by the geometrical fact that each
$(\theta,\nu)\in\mathbb{T}[R]$ intersects at most $D+1$ cells
$O_{i}^{{}^{\prime}}$ , then

$\sum_{i}\|f_{i}\|_{L^{2}}^{2}\lesssim\sum_{i}\sum_{(\theta,\nu)\in\mathbb{T}_{i}}\|f_{\theta,\nu}\|_{L^{2}}^{2}\lesssim
D\sum_{\theta,\nu}\|f_{\theta,\nu}\|_{L^{2}}^{2}\sim D\|f\|_{L^{2}}^{2}.$

Then there exists $i_{0}$ satisfying (2.11) and

$\|f_{i_{0}}\|_{L^{2}}\lesssim D^{-1}\|f\|_{L^{2}}.$

We break $P_{R}$ into $O(1)$ parallelepipeds $P_{j}$ of size $R/2$, each has
the same direction with $P_{R}$. By the induction assumption, we have

$\displaystyle\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}$
$\displaystyle\lesssim
D^{3}\|\chi_{O_{i_{0}}^{{}^{\prime}}}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\lesssim
D^{3}\|e^{it\Delta}f_{i_{0}}\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}$
$\displaystyle\lesssim
D^{3}\max_{j}\|e^{it\Delta}f_{i_{0}}\|_{L_{x}^{p}L_{t}^{r}(P_{j})}^{p}\lesssim
D^{3}\Big{(}C_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}(R/2)^{\epsilon}\|f_{i_{0}}\|_{L^{2}}\Big{)}^{p}$
$\displaystyle\lesssim D^{3-p}2^{-\epsilon
p}\Big{(}C_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}\Big{)}^{p}.$

Since $D=R^{\epsilon^{4}}$ and $p>3$, with $R$ large enough, we can close the
induction.

To deal with the wall case, the set $P_{R}$ is broken into $R^{3\delta}$
parallelepipeds $P_{R^{1-\delta}}(z_{j})$ of size $R^{1-\delta}$. Denote
$P_{j}=P_{R^{1-\delta}}(z_{j})$. For any $T_{\theta,\nu}$ with
$(\theta,\nu)\in\mathbb{T}[R]$, there are two more conceptions: tangent tube
and transverse tube.

Tangent tube:

$T_{\theta,\nu}$ is $R^{-\frac{1}{2}+\delta}$ tangent to the variety $Z(P)$ on
a parallelepiped $P_{j}$ if $T_{\theta,\nu}\cap P_{j}\cap W\neq\emptyset$ and

$\angle(G_{0}(\theta),T_{z}[Z(P)])\leq R^{-\frac{1}{2}+\delta}$

for any non-singular point $z\in 10T_{\theta,\nu}\cap 2P_{j}\cap Z(P)$.

Transverse tube:

$T_{\theta,\nu}$ is $R^{-\frac{1}{2}+\delta}$ transverse to the variety $Z(P)$
on a parallelepiped $P_{j}$ if $T_{\theta,\nu}\cap P_{j}\cap W\neq\emptyset$
and

$\angle(G_{0}(\theta),T_{z}[Z(P)])>R^{-\frac{1}{2}+\delta}$

for some non-singular point $z\in 10T_{\theta,\nu}\cap 2P_{j}\cap Z(P)$.

Here $G_{0}(\theta)=(4\pi c(\theta),1)$ is the direction of the tube
$T_{\theta,\nu}$. $T_{z}[Z(P)]$ stands for the tangent space of $Z(P)$ at the
point $z$. Let $\mathbb{T}_{j,tang}$ represent the set of the tubes tangent to
$Z(P)$ in $P_{j}$, and $\mathbb{T}_{j,trans}$ denote the set of the tubes
transverse to $Z(P)$ in $P_{j}$. Set

$f_{j,tang}:=\sum_{(\theta,\nu)\in\mathbb{T}_{j,tang}}f_{\theta,\nu},\quad
f_{j,trans}:=\sum_{(\theta,\nu)\in\mathbb{T}_{j,trans}}f_{\theta,\nu}.$

We have, for $(x,t)\in P_{j}\cap W$,

$e^{it\Delta}f(x)\sim e^{it\Delta}f_{j,tang}(x)+e^{it\Delta}f_{j,trans}(x).$

Let parameter $K\gg 1$. We decompose $\mathbb{B}^{n}(\xi_{0},\frac{1}{M})$
into balls $\tau$ of radius $(KM)^{-1}$ and

$f=\sum_{\tau}f_{\tau},\,\quad\text{supp}\widehat{f_{\tau}}\subset\tau.$

We introduce in the following functions

(2.13)
$f_{\tau,j,tang}=\sum_{\theta\in\tau}\sum_{(\theta,v)\in\mathbb{T}_{j,trans}}f_{\theta,v},\quad
f_{\tau,j,trans}=\sum_{\theta\in\tau}\sum_{(\theta,v)\in\mathbb{T}_{j,tang}}f_{\theta,v}.$

Now we consider two cases.

  * •

If

(2.14)
$K^{\epsilon^{4}}\max_{\tau}|e^{it\Delta}f_{\tau}(x)|>|e^{it\Delta}f(x)|\ \
\text{for}\ \ (x,t)\in W.$

By the induction assumption,

$\displaystyle\|\max\limits_{\tau}|e^{it\Delta}f_{\tau}|\|_{L_{x}^{p}L_{t}^{r}(P_{R})}$
$\displaystyle\leq\Big{\|}\Big{(}\sum_{\tau}|e^{it\Delta}f_{\tau}|^{p}\Big{)}^{\frac{1}{p}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(P_{R})}$
$\displaystyle\leq\Big{(}\sum_{\tau}||e^{it\Delta}f_{\tau}||_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle\lesssim\Big{(}\sum_{\tau}(C_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}(KM)^{-\epsilon^{2}}R^{\epsilon}\|f_{\tau}\|_{L^{2}})^{p}\Big{)}^{\frac{1}{p}}$
$\displaystyle\lesssim
C_{p,\epsilon}K^{-\epsilon^{2}}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}.$

From this estimate and (2.14), it follows that

$\|\chi_{W}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}\lesssim
C_{p,\epsilon}K^{p(\epsilon^{4}-\epsilon^{2})}(N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}})^{p}.$

By taking $K$ sufficiently large, we can close the induction.

  * •

If

$K^{\epsilon^{4}}\max_{\tau}|e^{it\Delta}f_{\tau}(x)|\leq|e^{it\Delta}f(x)|\ \
\text{for}\ \ (x,t)\in W.$

By the broad-narrow catalogue as in [9, Section 6], we have

(2.15)
$\displaystyle\|\chi_{W}e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(P_{R})}^{p}$
$\displaystyle\lesssim$ $\displaystyle\sum_{j}\|\max_{I}\chi_{P_{j}\cap
W}e^{it\Delta}f_{I,j,trans}\|_{L_{x}^{p}L_{t}^{r}(P_{j})}^{p}+K^{10p}\sum_{j}\|\chi_{P_{j}\cap
W}\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x}^{p}L_{t}^{r}(P_{j})}^{p}$
$\displaystyle=:$ $\displaystyle\text{Linear transverse term}+\text{Bilinear
tangent term}.$

Here $I$ denotes a subset of the collection of $1/KM$ balls $\tau$,

$f_{I,j,trans}(x):=\sum_{\tau\in I}f_{\tau,j,trans}$

and

$\text{Bil}(e^{it\Delta}f_{j,tang}):=\max_{\\{(\tau_{1},\tau_{2}):dist(\tau_{1},\tau_{2})\geq\frac{1}{KM}\\}}|e^{it\Delta}f_{\tau_{1},j,tang}e^{it\Delta}f_{\tau_{2},j,tang}|^{\frac{1}{2}}.$

For the linear transverse term, Guth [15, Lemma 3.5] proved that for a given
$(\theta,\nu)\in\mathbb{T}$

$\sharp\Big{\\{}j:(\theta,\nu)\in\mathbb{T}_{j,trans}\Big{\\}}\leq
R^{O(\epsilon^{4})}.$

By the induction assumption,

$\displaystyle\sum_{j}\|\max_{I}\chi_{P_{j}\cap
W}e^{it\Delta}f_{I,j,trans}\|_{L_{x}^{p}L_{t}^{r}(P_{j})}^{p}$
$\displaystyle\lesssim\sum_{j}\sum_{I\subset\mathcal{T}}\|\chi_{P_{j}\cap
W}e^{it\Delta}f_{I,j,trans}\|_{L_{x}^{p}L_{t}^{r}(P_{j})}^{p}$
$\displaystyle\lesssim\sum_{j}2^{K^{2}}\Big{(}C_{\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{(1-\delta)\epsilon}\|f_{j,trans}\|_{L^{2}}\Big{)}^{p}$
$\displaystyle\lesssim 2^{K^{2}}R^{O(\epsilon^{4}-\delta\epsilon
p)}\Big{(}C_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}\Big{)}^{p}.$

Here we used the fact $\sharp\mathcal{T}\leq 2^{K^{2}}$ since $\mathcal{T}$ is
the possible collection of $I$ . We can close the induction since
$\delta=\epsilon^{2}$ and $K\ll R$.

The estimate of the bilinear tangent term is complicated. We will set up the
following estimate in the next subsection.

######  Proposition 2.3 (Bilinear tangent estimate).

Let $N,\,M\geq 1$. Suppose that $\xi_{0}\in\mathbb{B}^{2}(Ne_{1},1)$ and that
the Fourier supports of $f_{i}\,(i=1,2)$ are in
$\mathbb{B}^{2}(\xi_{0},\frac{1}{M})$ with distance at least $1/KM$ with the
same $K=K(\epsilon)$ as above. Further more assume that each $f_{i}$ is
concentrated in $\mathbb{T}_{Z}(R)$, where

$\mathbb{T}_{Z}(R):=\\{(\theta,\nu)\in\mathbb{T}[R]:T_{\theta,\nu}\ \text{is}\
R^{-\frac{1}{2}+\delta}\ \text{-tangent to}\ Z(P)\\}.$

Then

(2.16)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{3}L_{t}^{\infty}(P_{R})}\lesssim_{K,\epsilon}N^{\frac{1}{3}}R^{\epsilon}\|f_{1}\|_{L^{2}}^{\frac{1}{2}}\|f_{2}\|_{L^{2}}^{\frac{1}{2}}.$

In this proposition, we say a tube $T_{\theta,\nu}$ is
$R^{-\frac{1}{2}+\delta}$ tangent to $Z(P)$ if

$T_{\theta,\nu}\subset N_{R^{\frac{1}{2}+\delta}}Z(P)\cap P_{R}$

and

$\angle(G_{0}(\theta),T_{z}[Z(P)])\leq R^{-\frac{1}{2}+\delta}$

for any non-singular point $z\in
N_{2R^{\frac{1}{2}+\delta}}(T_{\theta,\nu})\cap 2P_{R}\cap Z(P)$. A function
$f$ is concentrated in $\boldsymbol{\Pi}\subset\mathbb{T}[R]$ if

(2.17)
$\sum_{(\theta,\nu)\notin\boldsymbol{\Pi}}\|f_{\theta,\nu}\|_{L^{2}}\lesssim\text{RapDec}(R)\|f\|_{L^{2}}.$

In order to apply Proposition 2.3 to deal with
$\text{Bil}(e^{it\Delta}f_{j,tang})$, wave packets decomposition at different
scales will be considered.

Let $R^{\frac{1}{2}}\leq r\leq R$ and $P_{r}(z_{j})$ with
$z_{j}=(x_{j},t_{j})\in P_{R}$. Define

$e^{it\Delta}f(x)=e^{i\tilde{t}\Delta}\tilde{f}(\tilde{x}),$

where $(x,t)=(\tilde{x},\tilde{t})+(x_{j},t_{j})$, $(\tilde{x},\tilde{t})\in
P_{r}$ and

$\widehat{\tilde{f}}(\xi)=e^{2\pi i(x_{j}\cdot\xi-2\pi
t_{j}|\xi|^{2})}\widehat{f}(\xi).$

$\tilde{f}$ has the following wave packets decomposition

$\tilde{f}=\sum_{\tilde{\theta},\tilde{\nu}\in\mathbb{T}[r]}\tilde{f}_{\tilde{\theta},\tilde{\nu}}+\text{RapDec}(R)\|f\|_{L^{2}}.$

This fact was proved by Guth.

######  Lemma 2.4.

[16] If $f$ is concentrated in $\boldsymbol{\Pi}\subset\mathbb{T}(R)$, then
$\tilde{f}$ is concentrated in $\tilde{\boldsymbol{\Pi}}\subset\mathbb{T}(r)$
with the following property. For every
$(\tilde{\theta},\tilde{\nu})\in\tilde{\boldsymbol{\Pi}}$, there exist a
$(\theta,\nu)\in\boldsymbol{\Pi}$ such that

$\text{dist}_{H}(T_{\tilde{\theta},\tilde{\nu}}+z_{j},T_{\theta,\nu}\cap
P_{r}(z_{j}))\lesssim R^{\frac{1}{2}+\delta}\ \ \text{and}\ \
\angle(G_{0}(\theta),G_{0}(\tilde{\theta}))\lesssim r^{-\frac{1}{2}},$

where $\text{dist}_{H}$ denotes the Hausdorff distance.

Let $g=f_{j,tang}$ and $g_{i}=f_{\tau_{i},j,tang}\ (i=1,2)$. From the
definition of $(\theta,\nu)\in\mathbb{T}_{j,tang}$ and Lemma 2.4, we obtain
$\tilde{g}$ is concentrated in $\mathbb{T}_{Z}(R^{1-\delta})$ in $P_{j}$. By
Proposition 2.3, we have

$\displaystyle\|\chi_{P_{j}\cap
W}\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x}^{3}L_{t}^{\infty}(P_{j})}$
$\displaystyle\lesssim\||e^{it\Delta}\tilde{g}_{1}e^{it\Delta}\tilde{g}_{2}|^{\frac{1}{2}}\|_{L_{x}^{3}L_{t}^{\infty}(P_{R^{1-\delta}})}$
$\displaystyle\lesssim_{K,\epsilon}N^{\frac{1}{3}}R^{\epsilon}\|f_{\tau_{1},j,tang}\|_{L^{2}}^{\frac{1}{2}}\|f_{\tau_{2},j,tang}\|_{L^{2}}^{\frac{1}{2}}.$

Interpolating the bilinear estimate with

$\|\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x,t}^{\infty}(P_{j})}\lesssim_{\epsilon}M^{-\frac{1}{2}}\|f_{\tau_{1},j,tang}\|_{L^{2}}^{\frac{1}{2}}\|f_{\tau_{2},j,tang}\|_{L^{2}}^{\frac{1}{2}},$

we have

(2.18) $\|\chi_{P_{j}\cap
W}\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x}^{p}L_{t}^{\infty}(P_{R})}\lesssim_{p,\epsilon}N^{\frac{1}{p}}M^{-\frac{1}{2}(1-\frac{3}{p})}R^{\frac{\epsilon}{2}}\|f_{\tau_{1},j,tang}\|_{L^{2}}^{\frac{1}{2}}\|f_{\tau_{2},j,tang}\|_{L^{2}}^{\frac{1}{2}}$

for $p>3$. By H$\ddot{\text{o}}$lder’s inequality, for any $r>1/\epsilon^{4}$,
we have

$\displaystyle\|\chi_{P_{j}\cap
W}\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x}^{p}L_{t}^{r}(P_{j})}$
$\displaystyle\leq(R/N)^{\frac{1}{r}}\|\chi_{P_{j}\cap
W}\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x}^{p}L_{t}^{\infty}(P_{R})}$
$\displaystyle\lesssim_{p,\epsilon}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}R^{\frac{\epsilon}{2}}\|f_{\tau_{1},j,tang}\|_{L^{2}}^{\frac{1}{2}}\|f_{\tau_{2},j,tang}\|_{L^{2}}^{\frac{1}{2}}.$

Then the bilinear term in (2.15) can be controlled by

$K^{10p}\sum_{j}\|\chi_{P_{j}\cap
W}\text{Bil}(e^{it\Delta}f_{j,tang})\|_{L_{x}^{p}L_{t}^{r}(P_{j})}^{p}\lesssim_{p,\epsilon}R^{O(\delta)+\frac{\epsilon
p}{2}}\Big{(}N^{\frac{1}{p}-\frac{1}{r}}M^{-\epsilon^{2}}\|f\|_{L^{2}}\Big{)}^{p}.$

It is enough to close the induction.

###  2.3. Bilinear tangent estimate

We now give the proof of Proposition 2.3. The key point is to convert the
$L^{p}_{x}L_{t}^{\infty}$ estimate to $L_{x,t}^{p}$ estimate through the
dyadic pigeonhole principle and the locally constant property.

For simplicity, assume $\|f_{i}\|_{L^{2}}=1,\ i=1,2$. Given a dyadic number
$H$, set

$A_{H}=\Big{\\{}x:(x,t)\in P_{R},\sup_{0<t\leq
R}|e^{it\Delta}f_{1}(x)e^{it\Delta}f_{2}(x)|^{\frac{1}{2}}\sim H\Big{\\}}.$

It is easy to see that $H\lesssim 1$. The case of $H\leq R^{-100}$ is trivial.
It is sufficient to consider the case $R^{-100}\leq H\leq 1$. By the
pigeonhole principle, there exists a dyadic number $H$ such that

(2.19)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{3}L_{t}^{\infty}(P_{R})}\lesssim(\log
R)H|A_{H}|^{\frac{1}{3}}.$

Since
${\rm{\,supp\,}}\widehat{f_{i}}\subset\mathbb{B}^{2}(\xi_{0},\frac{1}{M})\subset\mathbb{B}^{2}(Ne_{1},1)\,(N\gg
1)$, the Fourier transform of $e^{it\Delta}f_{i}$ is supported on a
rectangular box of dimensions

$\frac{1}{M}\times\frac{1}{M}\times\frac{N}{M}.$

By the locally constant property,
$|e^{it\Delta}f_{1}(x)e^{it\Delta}f_{2}(x)|^{\frac{1}{2}}$ is a essentially
constant on rectangular box of dimensions $M\times M\times M/N$ (A rigorous
discussion can be found in [9, Pages 635-636]). We can choose $X\subset P_{R}$
the union of the rectangular boxes of dimensions $M\times M\times M/N$
satisfying:

  * •

Each vertical thin tube of dimension $M\times M\times R$ contains at most one
rectangular box $M\times M\times M/N$ in $X$;

  * •

The projection of $X$ on $\mathbb{R}^{2}$ covers $A_{H}$.

By the construction of $X$,

(2.20) $|X|\sim\frac{M}{N}|A_{H}|.$

Decompose $P_{R}$ into parallelepipeds $P_{j}$ with size $R^{\frac{1}{2}}$.
For any $P_{j}\cap X\neq\emptyset$, making use of the Galilean transform
$\mathcal{G}:(x,t)\rightarrow(\tilde{x},\tilde{t})$:

$\tilde{t}=t\ \ \text{and}\ \ \ \tilde{x}=x-4\pi tNe_{1},$

we obtain

(2.21)
$\||e^{it\Delta}f_{1}(x)e^{it\Delta}f_{2}(x)|^{\frac{1}{2}}\|_{L^{6}(P_{j})}=\||e^{i\tilde{t}\Delta}g_{1}(\tilde{x})e^{i\tilde{t}\Delta}g_{2}(\tilde{x})|^{\frac{1}{2}}\|_{L^{6}(Q_{j})}.$

Here
${\rm{\,supp\,}}\widehat{g_{i}}\subset\mathbb{B}^{2}(\tilde{\xi}_{0},\frac{1}{M}),\,\tilde{\xi}_{0}\in\mathbb{B}^{2}(0,1)$,
dist$(\text{supp}\widehat{g}_{1},\text{supp}\widehat{g}_{2})\gtrsim\frac{1}{KM}$,
and $\|f_{i}\|_{L^{2}}=\|g_{i}\|_{L^{2}}=1$. The image
$\mathcal{G}(P_{j})=Q_{j}$ has side length $R^{\frac{1}{2}}$ (see Figure 5).
The case

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L^{6}(P_{j})}\leq
R^{-C}$

can be ignored for some large constant $C$. By the pigeonhole principle, we
can catalogue the $P_{j}$ according to

(2.22)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L^{6}(P_{j})}\sim\gamma,$

with a dyadic parameter $R^{-C}<\gamma<R^{C}$. Let $Y=\bigcup_{j=1}^{J}P_{j}$,
we have

(2.23) $|X|\lesssim(\log R)|X\cap Y|\lesssim(\log R)JMR.$

By (2.19), (2.20), (2.23) and the locally constant property, we have

$\displaystyle||e^{it\Delta}f_{1}e^{it\Delta}f_{2}||_{L_{x}^{3}L_{t}^{\infty}(P_{R})}$
$\displaystyle\lesssim R^{\epsilon}H|A_{H}|^{\frac{1}{3}}\lesssim
R^{\epsilon}N^{\frac{1}{3}}(M^{-1}JR)^{\frac{1}{6}}H|X|^{\frac{1}{6}}$
$\displaystyle\lesssim
R^{\epsilon}N^{\frac{1}{3}}(M^{-1}JR)^{\frac{1}{6}}\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L^{6}(Y)}$
(2.24) $\displaystyle\lesssim
R^{\epsilon}N^{\frac{1}{3}}(M^{-1}JR)^{\frac{1}{6}}\||e^{i\tilde{t}\Delta}g_{1}(\tilde{x})e^{i\tilde{t}\Delta}g_{2}(\tilde{x})|^{\frac{1}{2}}\|_{L^{6}(\tilde{Y})},$

where $\tilde{Y}=\bigcup_{j=1}^{J}Q_{j}\subset Q_{R}$.

$t:R^{\frac{1}{2}}$$x:R^{\frac{1}{2}}$$(4\pi
Ne_{1},1)$$\tilde{x}:R^{\frac{1}{2}}$$\tilde{t}$ Figure 5. $P_{j}\rightarrow
Q_{j}$

We now recall a proposition proved by Du, Guth, and Li [9, Proposition 8.2].

######  Proposition 2.5.

[9] Let $M\geq 1$. Suppose the Fourier supports of $g_{i}\,(i=1,2)$ are in
$\mathbb{B}^{2}(\xi_{0},\frac{1}{M})\subset\mathbb{B}^{2}(0,1)$ with the
distance at least $1/KM$ for some $K=K(\epsilon)$ large enough. $g_{i}$ is
concentrated in $\mathbb{T}_{Z}(R)$. Suppose that $Q_{1},Q_{2},...,Q_{J}$ are
lattice $R^{1/2}$-cubes in $Q_{R}$ so that

$\||e^{it\Delta}g_{1}e^{it\Delta}g_{2}|^{\frac{1}{2}}\|_{L^{6}(Q_{j})}\
\text{is essentially constant in}\ j.$

Let $\tilde{Y}:=\bigcup_{j=1}^{J}Q_{j}$. Then

(2.25)
$\||e^{it\Delta}g_{1}e^{it\Delta}g_{2}|^{\frac{1}{2}}\|_{L^{6}(\tilde{Y})}\lesssim_{K,\epsilon}M^{\frac{1}{6}}(JR)^{-\frac{1}{6}}R^{\epsilon}\|g_{1}\|_{L^{2}}^{\frac{1}{2}}\|g_{2}\|_{L^{2}}^{\frac{1}{2}}.$

The constant $K$ in Propositions 2.5 arises in a linear-bilinear argument. We
assume it is the same constant as in Proposition 2.2. It is easy to verify
that the tangent-to-variety condition is preserved under the Galilean
transform. We use (2.25) to bound the last term in (2.3), and finish the proof
of Proposition 2.3.

##  3\. The critical problem for $n\geq 3$

As Theorem 2.1, we are going to prove the following theorem.

######  Theorem 3.1.

Let $p\geq\tilde{p}_{n}$ and $f\in L^{2}(\mathbb{R}^{n})$ with
$\text{supp}\widehat{f}\subset\mathbb{B}^{n}(\xi_{0},\frac{1}{M})\subset\mathbb{B}^{n}(Ne_{1},1)$
for some $N\geq 1$ and $M\geq 1$. For any $\epsilon>0$ and $R\geq 1$, there
exists a constant $C_{p,\epsilon}$ such that

(3.1) $||e^{it\Delta}f||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}\leq
C_{p,\epsilon}N^{\frac{1}{p}}M^{-\epsilon^{2}}R^{\epsilon}||f||_{L^{2}}.$

In this section, we will introduce a $2$-broad norm $BL_{A}^{p,\infty}$ and
collect several properties of it. Then we reduce Theorem 3.1 to Theorem 3.4,
whose proof will be postponed to Section 4. Another aim is an
$L^{2}_{x}L^{\infty}_{t}$ estimate which is also an ingredient of the
induction argument.

###  3.1.  $2$-broad norm $BL_{A}^{p,\infty}$

Let parameter $K\gg 1$. We decompose $\mathbb{B}^{n}(\xi_{0},\frac{1}{M})$
into balls $\tau$ of radius $(KM)^{-1}$ and

$f=\sum_{\tau}f_{\tau},\,\quad\text{supp}\widehat{f_{\tau}}\subset\tau.$

Let

$G(\tau):=\\{G(\xi)\,|\,\xi\in\tau\\}\quad\text{with}\quad
G(\xi)=\frac{(4\pi\xi,1)}{|(4\pi\xi,1)|}.$

Given a linear subspace $V\subset\mathbb{R}^{n+1}$, we denote $\tau\in V$ if
and only if

$\angle(G(\tau),V)<(KM)^{-1},$

where

$\angle(G(\tau),V)=\min\\{\angle(v,v^{\prime}):\ \text{non-zero vectors}\
v,v^{\prime}\ \text{satisfy}\ v\in V\ \text{and}\ v^{\prime}\in G(\tau)\\}.$

Otherwise, we write $\tau\notin V$. The cube $Q_{R}$ is decomposed into sub-
cubes $Q_{K}$ of side length $K$. And the time interval $[0,R]$ is also
decomposed into intervals $\ I_{K/N}$ of length $K/N$. Denote

$D_{K}:=\\{x:(x,t)\in\mathcal{G}^{-1}(B_{K}\times I_{K/N})\\},$

then we obtain a finite overlap decomposition

$P_{R}=\bigcup D_{K}\times I_{K/N}.$

Given a positive integer $A$, we define

$\mu_{e^{it\Delta}f}(D_{K}\times
I_{K/N}):=\min_{V_{1},...V_{A}}\Big{(}\max_{\tau\notin V_{a}\ \text{for all}\
a}\int_{D_{K}\times I_{K/N}}|e^{it\Delta}f_{\tau}|^{p}dxdt\Big{)},$

where $V_{1},...,V_{A}$ are lines of $\mathbb{R}^{n+1}$. For any subset
$U\subset P_{R}$, the $2$-broad norm $BL_{A}^{p,\infty}(U)$ is defined by

$\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(U)}^{p}:=\sum_{D_{K}}\max_{I_{K/N}}\frac{|U\cap(D_{K}\times
I_{K/N})|}{|D_{K}\times I_{K/N}|}\mu_{e^{it\Delta}f}(D_{K}\times I_{K/N}).$

The $BL_{A}^{p,\infty}$-norm is not continuous, thus it can not be used to run
the polynomial partitioning argument. For this purpose, we recall the
$BL_{A}^{p,q}$-norm defined as

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(U)}^{p}:=\sum_{D_{K}}\Big{[}\sum_{I_{K/N}}\Big{(}\frac{|U\bigcap(D_{K}\times
I_{K/N})|}{|D_{K}\times I_{K/N}|}\mu_{e^{it\Delta}f}(D_{K}\times
I_{K/N})\Big{)}^{\frac{q}{p}}\Big{]}^{\frac{p}{q}}.$

The $BL_{A}^{p,q}$ norm has the following properties.

######  Proposition 3.2.

[8, 16] Let $1\leq p\leq q<\infty$. The following properties hold:

  * (i)

Let $U_{1},U_{2}\subset\mathbb{R}^{n+1}$ and $A\in\mathbb{N}$. Then

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(U_{1}\cup
U_{2})}^{p}\leq\|e^{it\Delta}f\|_{BL_{A}^{p,q}(U_{1})}^{p}+\|e^{it\Delta}f\|_{BL_{A}^{p,q}(U_{2})}^{p}.$

  * (ii)

Let $U\subset\mathbb{R}^{n+1}$ and $A\in\mathbb{N}$. Then

$\|e^{it\Delta}(f_{1}+f_{2})\|_{BL_{2A}^{p,q}(U)}\leq\|e^{it\Delta}f_{1}\|_{BL_{A}^{p,q}(U)}+\|e^{it\Delta}f_{2}\|_{BL_{A}^{p,q}(U)}$

  * (iii)

Let $U\subset\mathbb{R}^{n+1}$, $1\leq p,p_{0},p_{1}<\infty$ and
$A\in\mathbb{N}$. Suppose that $0\leq\alpha\leq 1$ satisfies

$\frac{1}{p}=\frac{1-\alpha}{p_{0}}+\frac{\alpha}{p_{1}}.$

Then

$\|e^{it\Delta}f\|_{BL_{2A}^{p,q}(U)}\lesssim\|e^{it\Delta}f\|_{BL_{A}^{p_{0},q}(U)}^{1-\alpha}\|e^{it\Delta}f\|_{BL_{A}^{p_{1},q}(U)}^{\alpha}.$

Moreover, $BL_{A}^{p,q}$ estimate can be reduced to $L_{x}^{p}L_{t}^{\infty}$
estimate.

######  Lemma 3.3.

Let $p\geq 2$, $\epsilon>0$, and $f\in L^{2}(\mathbb{R}^{n})$ with
$\text{supp}\widehat{f}\subset\mathbb{B}^{n}(\xi_{0},\frac{1}{M})\subset\mathbb{B}^{n}(Ne_{1},1)$.
If

$||e^{it\Delta}f||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}\lesssim_{p,\epsilon}N^{\frac{1}{p}}W||f||_{L^{2}}$

holds for some $W=W(M,R)>0$, then

$\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}\lesssim_{p,\epsilon}K^{O(1)}W\|f\|_{L^{2}}$

and

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim_{p,\epsilon}K^{O(1)}W\|f\|_{L^{2}}$

hold for any $q>1/\epsilon^{4}$.

###### Proof.

By the definition of $BL_{A}^{p,\infty}$, we have

$\displaystyle\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}^{p}$
$\displaystyle=\sum_{D_{K}}\max_{I_{K/N}}\mu_{e^{it\Delta}f}(D_{K}\times
I_{K/N})$
$\displaystyle\leq\sum_{D_{K}}\max_{I_{K/N}}\sum_{\tau}\|e^{it\Delta}f_{\tau}\|_{L^{p}(D_{K}\times
I_{K/N})}^{p}$ $\displaystyle\leq
N^{-1}K^{O(1)}\sum_{\tau}||e^{it\Delta}f_{\tau}||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}^{p}$
$\displaystyle\lesssim_{p,\epsilon}N^{-1}K^{O(1)}\sum_{\tau}\Big{(}N^{\frac{1}{p}}M^{-\epsilon^{2}}R^{\epsilon}||f_{\tau}||_{L^{2}}\Big{)}^{p}$
$\displaystyle\lesssim_{p,\epsilon}K^{O(1)}\Big{(}W||f||_{L^{2}}\Big{)}^{p}.$

Since $q>1/\epsilon^{4}$ and the number of $I_{K/N}$ is at most $R/K$, we have

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim_{p,\epsilon}K^{O(1)}W\|f\|_{L^{2}}.$

∎

###  3.2. A reduction argument

In this subsection, we will show that Theorem 3.1 can be reduced to the
following theorem.

######  Theorem 3.4.

Let $p\geq\tilde{p}_{n}$ and $f\in L^{2}(\mathbb{R}^{n})$ with
$\text{supp}\widehat{f}\subset\mathbb{B}^{n}(\xi_{0},\frac{1}{M})\subset\mathbb{B}^{n}(Ne_{1},1)$
for some $N\geq 1$ and $M\geq 1$. For any $\epsilon>0$ and $R\geq 1$, we have

$\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}\lesssim_{p,\epsilon}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}.$

###### Theorem 3.4 $\Rightarrow$ Theorem 3.1.

When $M\geq R^{\frac{1}{2}-\delta}$, we take the wave packet decomposition at
scale $R$. Since supp$\widehat{f}\subset B(\xi_{0},\frac{1}{M})$, we have

$\sharp\Big{\\{}\theta:\exists\nu\ s.t.\
(\theta,\nu)\in\mathbb{T}\Big{\\}}\leq\frac{(1/M)^{n}}{(1/R)^{n/2}}\lesssim
R^{O(\delta)}.$

Then, for $p>\frac{2(n+1)}{n}$, by the same argument as (2.2), we have

$||e^{it\Delta}f||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}\lesssim
N^{\frac{1}{p}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}.$

When $M<R^{\frac{1}{2}-\delta}$. We decompose

$||e^{it\Delta}f||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}^{p}=\sum_{D_{K}}\int_{D_{K}}\max_{I_{K/N}}\sup_{t\in
I_{K/N}}|e^{it\Delta}f(x)|^{p}dx.$

For each $D_{K}\times I_{K/N}$, there exist lines $V_{1},...,V_{A}$ which
achieve the minimum of $\mu_{e^{it\Delta}f}(D_{K}\times I_{K/N})$. Then, for
$(x,t)\in D_{K}\times I_{K/N}$, we have

$|e^{it\Delta}f(x)|^{p}\lesssim K^{O(1)}\max_{\tau\notin V_{a}\ \text{for
all}\ a}|e^{it\Delta}f(x)|^{p}+\sum_{a=1}^{A}\Big{|}\sum_{\tau\in
V_{a}}e^{it\Delta}f(x)\Big{|}^{p}$

and

$\displaystyle||e^{it\Delta}f||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}^{p}$
$\displaystyle\lesssim
K^{O(1)}\sum_{D_{K}}\int_{D_{K}}\max_{I_{K/N}}\sup_{t\in
I_{K/N}}\max_{\tau\notin V_{a}\ \text{for all}\
a}|e^{it\Delta}f_{\tau}(x)|^{p}dx$ $\displaystyle\ \ \ \ \
+\sum_{a=1}^{A}\sum_{D_{K}}\int_{D_{K}}\sup_{t\in[0,R]}|\sum_{\tau\in
V_{a}}e^{it\Delta}f_{\tau}(x)|^{p}dx$ $\displaystyle=:I+II.$

For $I$, from the local constant property and Theorem 3.4, it follows that

$\displaystyle I$ $\displaystyle\lesssim
NK^{O(1)}\sum_{D_{K}}\max_{I_{K/N}}\max_{\tau\notin V_{a}\ \text{for all}\
a}\int_{D_{K}\times I_{K/N}}|e^{it\Delta}f_{\tau}(x)|^{p}dx$
$\displaystyle\lesssim
NK^{O(1)}\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}^{p}$
$\displaystyle\lesssim
K^{O(1)}\Big{(}N^{\frac{1}{p}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}\Big{)}^{p}.$

For $II$, notice that $V_{a}\subset\mathbb{R}^{n}$ is a line, we have

$\sharp\\{\tau:\tau\in V_{a}\\}\lesssim 1.$

By induction on the frequency radius,

$\displaystyle II$
$\displaystyle\lesssim\sum_{a=1}^{A}\sum_{\tau}||e^{it\Delta}f_{\tau}||_{L_{x}^{p}L_{t}^{\infty}(P_{R})}^{p}$
$\displaystyle\lesssim\sum_{a=1}^{A}\sum_{\tau}[N^{\frac{1}{p}}(KM)^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}]^{p}$
$\displaystyle\lesssim
AK^{-p\epsilon^{2}}\Big{(}N^{\frac{1}{p}}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}\Big{)}^{p}.$

We can close the induction by choosing $K$ large enough and finish the proof.
∎

###  3.3. An maximal estimate

To end this section, we set up a $BL_{A}^{2,q}$ estimate which will be used in
the interpolation argument in subsection 4.3.

Let $1\leq m\leq n+1$. Let $Z$ denote the $m$ dimension variety
$Z(P_{1},...,P_{n+1-m})$ which consists of the common zeros of polynomials
$P_{1},...,P_{n+1-m}$ on $\mathbb{R}^{n+1}$. The variety $Z$ is a transverse
complete intersection in the sense,

$\nabla P_{1}(x)\wedge...\wedge\nabla P_{n+1-m}(x)\neq 0\ \ \text{for all}\
x\in Z.$

Given a small constant $\delta_{m}>0$ depending on the dimension $m$. For any
$(\theta,\nu)\in\mathbb{T}[R]$, we say that a tube $T_{\theta,\nu}$ is
$R^{-1/2+\delta_{m}}$-tangent to $Z$ on $P_{R}$ if

$T_{\theta,\nu}\subset N_{R^{1/2+\delta_{m}}}Z\cap P_{R}$

and

$\angle(G(\theta),T_{z}Z)\lesssim R^{-1/2+\delta_{m}}$

for any $z\in N_{2R^{1/2+\delta_{m}}}(T_{\theta,\nu})\cap 2P_{R}\cap Z$. Here
$T_{z}Z$ denotes the $m$-dimension tangent space of the variety $Z$ at $z$. As
before

$\mathbb{T}_{Z}(R):=\\{(\theta,\nu)\in\mathbb{T}[R]:T_{\theta,\nu}\ \text{is}\
R^{-\frac{1}{2}+\delta_{m}}\ \text{-tangent to}\ Z\\}.$

The main result in this part is the following theorem.

######  Theorem 3.5.

Let $n\geq 3$ and $2\leq m\leq n+1$. Suppose that $f\in L^{2}(\mathbb{R}^{n})$
is Fourier supported in $\mathbb{B}^{n}(Ne_{1},1)$ and concentrated in
$\mathbb{T}_{Z}(R)$. For any $\epsilon>0$,

(3.2)
$\|e^{it\Delta}f\|_{BL_{A}^{2,q}(P_{R})}\lesssim_{\epsilon}K^{O(1)}R^{\frac{m}{2(m+1)}+\epsilon}\|f\|_{L^{2}}$

hold for any $q>1/\epsilon^{4}$.

From Lemma 3.3, it can be reduce to set up the following maximal estimate.

######  Proposition 3.6.

Let $2\leq m\leq n+1$. Suppose that $f\in L^{2}(\mathbb{R}^{n})$ is Fourier
supported in $\mathbb{B}^{n}(Ne_{1},1)$ and concentrated in
$\mathbb{T}_{Z}(R)$. For any $\epsilon>0$ and $R\geq 1$, then

(3.3)
$\|e^{it\Delta}f\|_{L^{2}_{x}L^{\infty}_{t}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{2}}R^{\frac{m}{2(m+1)}+\epsilon}\|f\|_{L^{2}}.$

The proof is similar to Proposition 2.3.

###### Proof.

No loss of generality, we assume $\|f\|_{L^{2}}=1$. In the same way, we
introduce in the notation

$A_{H}=\big{\\{}x:(x,t)\in P_{R},\sup_{0<t\leq R}|e^{it\Delta}f(x)|\sim
H\big{\\}}$

and assume $R^{-100}\leq H\leq 1$. By the dyadic pigeonhole principle, we have

(3.4) $\|e^{it\Delta}f\|_{L_{x}^{2}L_{t}^{\infty}(P_{R})}\lesssim(\log
R)H|A_{H}|^{\frac{1}{2}}.$

The Fourier transform of $e^{it\Delta}f$ is supported on a rectangular box of
dimensions

$1\times\cdots\times 1\times N.$

Then it is essentially constant on a rectangular box of dimensions
$1\times\cdots\times 1\times 1/N$. One can choose a collection $X\subset
P_{R}$ is the union of the rectangular box of dimensions $1\times...\times
1\times 1/N$ satisfying:

  * •

Each vertical thin tube of dimensions $1\times...\times 1\times R$ contains at
most one rectangular box $1\times...\times 1\times 1/N$ in $X$;

  * •

The projection of set $X$ on $\mathbb{R}^{n}$ covers $A_{H}$.

By the definition of $X$, we have

(3.5) $|A_{H}|\sim N|X|.$

The parallelepiped $P_{R}$ is decomposed into
$R^{\frac{1}{2}}$-parallelepipeds $P_{j}$ and intersect $X$. By pigeonhole
principle, we choose $P_{j}$s satisfying:

  * •

$\|e^{it\Delta}f\|_{L^{\frac{2(m+1)}{(m-1)}}(P_{j})}\sim\gamma\quad\text{for
some dyadic number}\,R^{-C}\leq\gamma\leq R^{C}$;

  * •

Each horizontal $R^{\frac{1}{2}}$-slab contains $\sim\sigma$ many $P_{j}$s.

Let $Y=\bigcup_{j=1}^{J}P_{j}$, we obtain

(3.6) $|X|\lesssim(\log R)^{2}|X\cap Y|\lesssim(\log R)^{2}JR^{\frac{n}{2}}$

and

$J\lesssim\sigma R^{\frac{1}{2}}.$

Using the locally constant property and the Galilean transform, we can obtain

$\displaystyle\|e^{it\Delta}f\|_{L^{2}_{x}L^{\infty}_{t}(P_{R})}$
$\displaystyle\lesssim_{\epsilon}R^{\epsilon}N^{\frac{1}{2}}|X|^{\frac{1}{2}-\frac{m-1}{2(m+1)}}H|X\cap
Y|^{\frac{m-1}{2(m+1)}}$
$\displaystyle\lesssim_{\epsilon}R^{\epsilon}N^{\frac{1}{2}}(\sigma
R^{\frac{n+1}{2}})^{\frac{1}{m+1}}\|e^{it\Delta}f(x)\|_{L^{\frac{2(m+1)}{m-1}}(Y)}$
(3.7) $\displaystyle\lesssim_{\epsilon}R^{\epsilon}N^{\frac{1}{2}}(\sigma
R^{\frac{n+1}{2}})^{\frac{1}{m+1}}\|e^{i\tilde{t}\Delta}g(\tilde{x})\|_{L^{\frac{2(m+1)}{m-1}}(\tilde{Y})}.$

Here $g$ is the image of $f$ under the Galilean transform satisfying
${\rm{\,supp\,}}\widehat{g}\subset\mathbb{B}^{n}(0,1)$ and $\|g\|_{L^{2}}=1$.
And $\tilde{Y}=\bigcup_{j=1}^{J}Q_{j}\subset\mathbb{B}^{n+1}(0,R)$ is the
image of $Y$. Since the tangent-to-variety condition is preserved under the
Galilean transform, we finish the proof of Proposition 3.6 by recalling the
follow linear refined Strichartz estimate. ∎

######  Theorem 3.7.

[10] (Linear refined Strichartz for $m$-variety in dimension $n+1$). Let
$2\leq m\leq n+1$ and $p_{m}=2(m+1)/(m-1)$. Suppose that
$Z=Z(P_{1},...,P_{n+1-m})$ is a transverse complete intersection where
$DegP_{i}\leq R^{\delta_{i}}$. Here $\delta_{i}\ll\delta$ is a small
parameter. Suppose that $g\in L^{2}(\mathbb{R}^{n})$ is Fourier supported in
$\mathbb{B}^{n}(0,1)$ and concentrated in $\mathbb{T}_{Z}(R)$ on
$\mathbb{B}^{n+1}(0,R)$. Suppose that $Q_{1},\ Q_{2},...$ are lattice
$R^{1/2}$-cubes in $\mathbb{B}^{n+1}(0,R)$, so that

$\|e^{it\Delta}g\|_{L^{p_{m}}(Q_{j})}\ \ \text{is essentially constant in}\
j.$

Suppose that these cubes are arranged in horizontal slabs of the form
$\mathbb{R}\times...\times\mathbb{R}\times\\{t_{0},t_{0}\\}$, and that each
such slab contains $\sim\sigma$ cubes $Q_{j}$. Let $Y$ denote
$\bigcup_{j}Q_{j}$. Then

$\|e^{it\Delta}g\|_{L^{p_{m}}(Y)}\leq
C_{\epsilon}R^{\epsilon}R^{-(n+1-m)/2(m+1)}\sigma^{-\frac{1}{m+1}}\|g\|_{L^{2}}.$

##  4\. Algorithm

We now give the proof of Theorem 3.4. If $M\geq R^{\frac{1}{2}-\delta}$, by
Lemma 3.3, we have

$\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}\lesssim_{p,\epsilon}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}$

holds for $p>\frac{2(n+1)}{n}$. So we can assume $M<R^{\frac{1}{2}-\delta}$.
To show that, for supp$\widehat{f}\subset\mathbb{B}^{n}(\xi_{0},\frac{1}{M})$,

$\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}\lesssim_{p,\epsilon}M^{-\epsilon^{2}}R^{\epsilon}\|f\|_{L^{2}}$

holds for $p\geq\tilde{p}_{n}$. We only need to prove that, for
supp$\widehat{f}\subset\mathbb{B}^{n}(Ne_{1},1)$,

(4.1)
$\|e^{it\Delta}f\|_{BL_{A}^{p,\infty}(P_{R})}\lesssim_{p,\epsilon}R^{\epsilon}\|f\|_{L^{2}}$

holds for $p\geq\tilde{p}_{n}$ since

$R^{\frac{\epsilon}{2}}\leq R^{\epsilon}R^{-\frac{\epsilon^{2}}{2}}\leq
R^{\epsilon}M^{-\epsilon^{2}}.$

For (4.1), it is suffice to prove that

(4.2)
$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim_{p,\epsilon}R^{\epsilon}\|f\|_{L^{2}}$

holds for $p\geq\tilde{p}_{n}$ and any $q>1/\epsilon^{4}$.

We will adopt two algorithms which were introduced by Hickman and Rogers [18]
and developed by Guth [16] in the study of Fourier restriction conjecture. The
algorithms conclude the scale induction and the dimensional induction. We
refer the readers [16, 18, 7] for more details.

######  Proposition 4.1.

[16] Let $1\leq r^{\frac{1}{2}}\leq\rho\leq r$ and $P_{r}\cap
N_{{r}^{1/2+\delta_{m}}}Z\neq\emptyset$. Here $m$ denotes the dimension of the
variety $Z$ and $\delta_{m}$ is a small number depends $m$. $f$ is
concentrated in $\mathbb{T}_{Z}(r)$. Then there is a set of translates
$\mathcal{B}\subset\mathbb{B}^{n+1}(0,{r}^{1/2+\delta_{m}})$ such that

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Z)}^{p}\lesssim\log^{2}r\sum_{b\in\mathcal{B}}\|e^{it\Delta}\tilde{f}_{b}\|_{BL_{A}^{p,q}(P_{\rho}\cap
N_{{\rho}^{1/2+\delta_{m}}}(Z+b))}^{p}+r^{-M}\|f\|_{L^{2}}^{p}$

where $\tilde{f}_{b}$ is concentrated in $\mathbb{T}_{Z+b}(\rho)$ with
$Z+b=\\{z+b:z\in Z\\}$. Moreover,

$\sum_{b\in\mathcal{B}}\|\tilde{f}_{b}\|_{L^{2}}^{2}\lesssim\|f\|_{L^{2}}^{2}$

and for any $b\in\mathcal{B}$,

$\|\tilde{f}_{b}\|_{L^{2}}^{2}\lesssim
r^{O(\delta_{m})}\big{(}\frac{r}{\rho}\big{)}^{-\frac{n+1-m}{2}}\|f\|_{L^{2}}^{2}.$

######  Theorem 4.2.

[16, 7] Fix $r\gg 1$ and $1\leq p,q<\infty$. Suppose that $Z$ is an
$m$-dimensional transverse complete intersection of degree at most $D_{Z}$.
Suppose that $f$ is concentrated in $\mathbb{T}_{Z}(r)$. Then there exists a
constant $D=D(\epsilon,D_{Z})$ such that at least one of the following cases
holds.
Cellular case: there exist $O(D^{m})$ shrunken cells $O_{i}$ with size at most
$r/2$ such that for each $i$

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Z)}^{p}\lesssim
D^{m}\|e^{it\Delta}f\|_{BL_{A}^{p,q}(O_{i})}^{p}.$

Algebraic case: there exists an $m-1$-dimensional transverse complete
intersection $Y$ of degree at most $O(D)$ such that

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Z)}^{p}\lesssim\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Y)}^{p}.$

###  4.1. Sketch of the first algorithm

Suppose function $f$ satisfing the conditions of Theorem 4.2, we consider the
cellular case and the algebraic case, respectively.

If the cellular case dominates,

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Z)}^{p}\lesssim
D^{m}\|e^{it\Delta}f\|_{BL_{A}^{p,q}(O_{i})}^{p}.$

For each $i$, set

$f_{O_{i}}=\sum_{(\theta,\nu)\in\mathbb{T}_{Z}(r),T_{\theta,\nu}\cap
O_{i}\neq\emptyset}f_{\theta,\nu}.$

We have

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Z)}^{p}\lesssim\sum_{O_{i}}\|e^{it\Delta}f_{O_{i}}\|_{BL_{A}^{p,q}(O_{i})}^{p}+\text{RapDec}(r)\|f\|_{L^{2}}^{p},$
$\sum_{O_{i}}\|f_{O_{i}}\|_{L^{2}}^{2}\lesssim D\|f\|_{L^{2}}^{2}$

and

$\|f_{O_{i}}\|_{L^{2}}^{2}\lesssim D^{-(m-1)}\|f\|_{L^{2}}^{2},\ \text{for
each}\ O_{i}.$

So far the estimates are reduced from the $r$-size set $P_{r}$ to the
$r/2$-size set $Q_{i}$. One can continue the argument to $f_{O_{i}}$ because
$f_{O_{i}}$ concentrated in $\mathbb{T}_{Z}(r/2)$ (see Lemma 2.4).

If the algebraic case dominates, there exists an $m-1$-dimensional transverse
complete intersection $Y$ of degree at most $O(D)$ such that

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Z)}^{p}\lesssim\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Y)}^{p}.$

Even the dimension of the variety $Y$ is already $m-1$, but the function $f$
may not tangent to $Y$. Let $\mathcal{P}$ be a cover of $P_{r}\cap
N_{r^{1/2+\delta_{m}}}Y$ by finitely-overlapping $P_{r_{1}}$ with some
suitable chosen $r_{1}<r$. For each $P_{r_{1}}\in\mathcal{P}$, let

$\mathbb{T}_{P_{r_{1}}}=\\{(\theta,\nu)\in\mathbb{T}_{Z}(r):T_{\theta,\nu}\cap
P_{r_{1}}\cap N_{r^{1/2+\delta_{m}}}Y\neq\emptyset\\}.$

A tube $T_{\theta,\nu}$ is $r_{1}^{1/2+\delta_{m-1}}$-tangent to $Y$ on
$P_{r_{1}}$ means

$T_{\theta,\nu}\cap 2P_{r_{1}}\subset
N_{2r_{1}^{1/2+\delta_{m-1}}}Y=N_{2r^{1/2+\delta_{m}}}Y$

and for any $y\in Y\cap 2P_{r_{1}}\cap
N_{r_{1}^{1/2+\delta_{m-1}}}(T_{\theta,\nu})$,

$\angle(G(\theta),T_{y}Y)\lesssim r_{1}^{-1/2+\delta_{m-1}}.$

Here we used a fact that
$r_{1}^{\frac{1}{2}+\delta_{m-1}}=r^{\frac{1}{2}+\delta_{m}}.$ Which can be
obtained by suitable choose of $\delta_{m-1}$. Denote

$\mathbb{T}_{P_{r_{1}},tang}=\\{(\theta,\nu)\in\mathbb{T}_{P_{r_{1}}}:T_{\theta,\nu}\
\text{is}\ r_{1}^{1/2+\delta_{m-1}}-\text{tangency to}\ Y\text{on}\
P_{r_{1}}\\}$

and
$\mathbb{T}_{P_{r_{1}},trans}=\mathbb{T}_{P_{r_{1}}}\setminus\mathbb{T}_{P_{r_{1}},tang}$,
we have

$\displaystyle\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r}\cap
N_{r^{1/2+\delta_{m}}}Y)}^{p}$
$\displaystyle\lesssim\sum_{P_{r_{1}}\in\mathcal{P}}\|e^{it\Delta}f_{P_{r_{1}},tang}\|_{BL_{A/2}^{p,q}(P_{r_{1}})}^{p}+\sum_{P_{r_{1}}\in\mathcal{P}}\|e^{it\Delta}f_{P_{r_{1}},trans}\|_{BL_{A/2}^{p,q}(P_{r_{1}})}^{p}+\text{RapDec}(r)\|f\|_{L^{2}}^{p},$

where

$f_{P_{r_{1}},tang}=\sum_{(\theta,\nu)\in\mathbb{T}_{P_{r_{1}},tang}}f_{\theta,\nu}\
\ \text{and}\ \
f_{P_{r_{1}},trans}=\sum_{(\theta,\nu)\in\mathbb{T}_{P_{r_{1}},trans}}f_{\theta,\nu}.$

By the pigeonhole principle, if
$\sum_{P_{r_{1}}\in\mathcal{P}}\|e^{it\Delta}f_{P_{r_{1}},tang}\|_{BL_{A/2}^{p,q}(P_{r_{1}})}^{p}$
dominates, by Lemma 2.4, we have

$\sum_{P_{r_{1}}\in\mathcal{P}}\|e^{it\Delta}f_{P_{r_{1}},tang}\|_{BL_{A/2}^{p,q}(P_{r_{1}})}^{p}\lesssim\sum_{O_{j}}\|e^{it\Delta}f_{O_{j}}\|_{BL_{A/2}^{p,q}(O_{j})}^{p},$

where $f_{O_{j}}$ is concentrated in $\mathbb{T}_{Y}(r_{1})$ and the size of
$O_{j}$ is $r_{1}$. We obtain the dimensional reducing aim and with the lost
of some constant independent of $D$, $r$, and $m$.

Otherwise, we face the most difficult case

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r})}^{p}\lesssim\sum_{P_{r_{1}}\in\mathcal{P}}\|e^{it\Delta}f_{P_{r_{1}},trans}\|_{BL_{A/2}^{p,q}(P_{r_{1}})}^{p}.$

By Propositions 4.1, we have the following estimates:

$\sum_{P_{r_{1}}\in\mathcal{P}}\|e^{it\Delta}f_{P_{r_{1}},trans}\|_{BL_{A/2}^{p,q}(P_{r_{1}})}^{p}\lesssim\log^{2}r\sum_{O_{j}}\|e^{it\Delta}f_{O_{j}}\|_{BL_{A/2}^{p,q}(O_{j})}^{p},$
$\sum_{O_{j}}\|f_{O_{j}}\|_{L^{2}}^{2}\lesssim\|f\|_{L^{2}}^{2}$

and for each $O_{j}$,

$\|f_{O_{j}}\|_{L^{2}}^{2}\lesssim
r^{O(\delta_{m})}\Big{(}\frac{r}{r_{1}}\Big{)}^{-\frac{n+1-m}{2}}\|f\|_{L^{2}}^{2},$

where $f_{O_{j}}$ is concentrated in $\mathbb{T}_{Z}(r_{1})$ with
$b\in\mathbb{B}^{n+1}(0,{r}^{1/2+\delta_{m}})$ and the size of $O_{j}$ at most
$r_{1}$. Next, for each $f_{O_{j}}$, we proceed to apply Theorem 4.2.

###  4.2. The first algorithm [alg 1]

Let $p\geq 2$, $0<\epsilon\ll 1$ and a family of small parameters

$\epsilon^{C}\leq\delta\ll\delta_{n}\ll\delta_{n-1}\ll\dots\ll\delta_{1}\ll\delta_{0}\ll\epsilon.$

Input

  * •

A parallelepiped $P_{r}$ with size $r\gg 1$.

  * •

An admissible large integer $A\in\mathbb{N}$.

  * •

A function $f$ with supp$\widehat{f}\subset\mathbb{B}^{n}(Ne_{1},1)$ and
concentrated on wave packets which are $r^{-\frac{1}{2}+\delta_{m}}$-tangent
to $Z$ in $P_{r}$ with a transverse complete intersection $Z$ of dimension
$m\geq 2$.

Output The $j$th stage of [alg 1] outputs:

  * •

A choice of spatial scale $\rho_{j}\geq 1$.

  * •

Certain integers $\sharp a(j),\ \sharp c(j)\in\mathbb{N}_{0}$ satisfying
$\sharp a(j)+\sharp c(j)=j$, where the integers $\sharp a(j),\ \sharp c(j)$
indicate the number of occurrences of the algebraic cases and the cellular
cases, respectively.

  * •

A family of subsets $\mathcal{O}_{j}$ of $\mathbb{R}^{n+1}$ which will be
referred to as cells. Each cell $O_{j}\in\mathcal{O}_{j}$ has size at most
$\rho_{j}$.

  * •

A collection of functions $\\{f_{O_{j}}\\}_{O_{j}\in\mathcal{O}_{j}}$. Each
$f_{O_{j}}$ is concentrated on wave packets of scale $\rho_{j}$ which are
$\rho_{j}^{-\frac{1}{2}+\delta_{m}}$ -tangent to some translation of $Z$ on
$O_{j}$.

  * •

A large integer $d\in\mathbb{N}$ which depends only on the admissible
parameters and deg$Z$.

With these outputs, we denote

$C_{j,\delta}^{I}(d,r)=d^{\sharp c(j)\delta}(\log r)^{2\sharp
a(j)(1+\delta)},$ $C_{j,\delta}^{II}(d)=d^{\sharp c(j)\delta+(n+1)\sharp
a(j)(1+\delta)},$ $C_{j,\delta}^{III}(d,r)=d^{\sharp c(j)\delta+\sharp
a(j)\delta}r^{\tilde{C}\sharp a(j)\delta_{m}}.$

It is clear that

(4.3)
$C_{j,\delta}^{I}(d,r),\,C_{j,\delta}^{II}(d),\,C_{j,\delta}^{III}(d,r)\lesssim_{d,\delta}r^{\delta_{0}}d^{\sharp
c(j)\delta}.$

Let $A_{j}=2^{-\sharp a(j)}A\in\mathbb{N}$. We obtain the decomposition which
satisfy the following properties.
Property 1. Most of the mass of $\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r})}^{p}$
is concentrated on the $O_{j}\in\mathcal{O}_{j}$, i.e.

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{r})}^{p}\leq
C_{j,\delta}^{I}(d,r)\sum_{O_{j}\in\mathcal{O}_{j}}\|e^{it\Delta}f_{O_{j}}\|_{BL_{A_{j}}^{p,q}(O_{j})}^{p}+\text{RapDec}(r)\|f\|_{L^{2}}^{p}.$

Property 2. The functions $f_{O_{j}}$ satisfy

$\sum_{O_{j}\in\mathcal{O}_{j}}\|f_{O_{j}}\|_{L^{2}}^{2}\leq
C_{j,\delta}^{II}(d)d^{\sharp c(j)}\|f\|_{L^{2}}^{2}.$

Property 3. Each individual $f_{O_{j}}$ satisfies

$\|f_{O_{j}}\|_{L^{2}}^{2}\leq
C_{j,\delta}^{III}(d,r)\Big{(}\frac{r}{\rho_{j}}\Big{)}^{-\frac{n+1-m}{2}}d^{-\sharp
c(j)(m-1)}\|f\|_{L^{2}}^{2}.$

Stopping conditions Suppose that we are at the $j$th stage of [alg 1]. We stop
our algorithm if one of [tiny] and [tang] holds true:

[tiny] The algorithm terminates if $\rho_{j}\leq r^{\tilde{\delta}_{m-1}}$.

[tang] The algorithm terminates if

$\sum_{O_{j}\in\mathcal{O}_{j}}\|e^{it\Delta}f_{O_{j}}\|_{BL_{A_{j}}^{p,q}(O_{j})}^{p}\leq
C_{tang}\sum_{S\in\mathcal{S}}\|e^{it\Delta}f_{S}\|_{BL_{A_{j}/2}^{p,q}(P_{\tilde{\rho}})}^{p}$

and

$\sum_{S\in\mathcal{S}}\|f_{S}\|_{L^{2}}^{2}\leq
C_{tang}r^{(n+1)\tilde{\delta}_{m-1}}\sum_{O_{j}\in\mathcal{O}_{j}}\|f_{O_{j}}\|_{L^{2}}^{2}$

hold for $(1-\tilde{\delta}_{m-1})(1/2+\delta_{m-1})=(1/2+\delta_{m})$,
$\tilde{\rho}=\rho_{j}^{1-\tilde{\delta}_{m-1}}$, and for some fixed
dimensional constant $C_{tang}$. Here $\mathcal{S}$ is a collection of
$(m-1)$-dimensional transverse complete intersections in $\mathbb{R}^{n+1}$
all of degree $O(d)$. $f_{S}$ is a function which is
$\tilde{\rho}^{-1/2+\delta_{m-1}}$-tangent to $S\in\mathcal{S}$ on
$P_{\tilde{\rho}}$.

By repeatedly applying the algorithm [alg 1], we have the following algorithm
[alg 2]

###  4.3. The second algorithm [alg 2]

Let $p_{n+1},\dots,p_{k}$ denote Lebesgue exponents to be fixed later and
satisfy

$2\leq p:=p_{n+1}\leq...\leq p_{k+1}\leq p_{k}.$

Let $0\leq\alpha_{l},\,\beta_{l}\leq 1$ be defined by

$\frac{1}{p_{l}}=\frac{1-\alpha_{l-1}}{2}+\frac{\alpha_{l-1}}{p_{l-1}}\
\text{and}\
\beta_{l}=\prod_{i=l}^{n}\alpha_{i}=\Big{(}\frac{1}{2}-\frac{1}{p_{l}}\Big{)}^{-1}\Big{(}\frac{1}{2}-\frac{1}{p_{n}}\Big{)}\
\ \text{for}\ k+1\leq l\leq n,$

and $\alpha_{n+1}=\beta_{n+1}=1$.
Input Fix $1\ll A\ll R$. Function $f$ satisfies
$supp\widehat{f}\subset\mathbb{B}^{n}(Ne_{1},1)$ and the non-degeneracy
hypothesis

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\geq CR^{\epsilon}\|f\|_{L^{2}}.$

Output The $(n+2-l)$-th step of the recursion will produce:

  * •

An $(n+2-l)$-tuple of scales $\vec{r}=(r_{n+1},...,r_{l})$ satisfying
$R=r_{n+1}>r_{n}>...>r_{l}$; large parameters $\vec{D}=(D_{n+1},...,D_{l})$,
where $D_{i}(l\leq i\leq n+1)$ are defined by the upper bound of
$O(r_{i+1}^{\delta_{0}}d^{\delta\log r_{i+1}})$; integers
$\vec{A}=(A_{n+1},\dots,A_{l})$ satisfying $A=A_{n+1}>A_{n}>\dots>A_{l}$.

  * •

A family transverse complete intersections $\mathcal{S}_{l}$. Each
$S_{i}\in\mathcal{S}_{i}(l\leq i\leq n+1)$ satisfies dim$S_{i}=i$.

  * •

A function $f_{S_{l}}$ associated with each $S_{l}\in\mathcal{S}_{l}$ is
concentrated on scale $r_{l}$ wave packets which are
$r_{l}^{-1/2+\delta_{l}}$-tangent to $S_{l}$ in $P_{r_{l}}$.

These data satisfy the following properties.
Property 1.

(4.4) $\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim
M(\vec{r_{l}},\vec{D_{l}})R^{O(\delta_{0})}\|f\|_{L^{2}}^{1-\beta_{l}}\Big{(}\sum_{S_{l}\in\mathcal{S}_{l}}\|e^{it\Delta}f_{S_{l}}\|_{BL_{A_{l}}^{p_{l},q}(P_{r_{l}})}^{p_{l}}\Big{)}^{\frac{\beta_{l}}{p_{l}}},$

where

$M(\vec{r_{l}},\vec{D_{l}}):=\Big{(}\prod_{i=l}^{n}D_{i}\Big{)}^{(n+1-l)\delta}\Big{(}\prod_{i=l}^{n}r_{i}^{\frac{i}{2(i+1)}(\beta_{i+1}-\beta_{i})}D_{i}^{\frac{1}{2}(\beta_{i+1}-\beta_{l})}\Big{)}.$

Propety 2. For $l\leq n$,

$\sum_{S_{l}\in\mathcal{S}_{l}}\|f_{S_{l}}\|_{L^{2}}^{2}\lesssim
D_{l}^{1+\delta}R^{O(\delta_{0})}\sum_{S_{l+1}\in\mathcal{S}_{l+1}}\|f_{S_{l+1}}\|_{L^{2}}^{2}.$

Property 3. For $l\leq n$,

$\max_{S_{l}\in\mathcal{S}_{l}}\|f_{S_{l}}\|_{L^{2}}^{2}\lesssim\Big{(}\frac{r_{l+1}}{r_{l}}\Big{)}^{-\frac{n-l}{2}}D_{l}^{-l+\delta}R^{O(\delta_{0})}\max_{S_{l+1}\in\mathcal{S}_{l+1}}\|f_{S_{l+1}}\|_{L^{2}}^{2}.$

Stopping condition The recursive stage of [alg 2] terminates if he following
[tiny-dom] happens.

[tiny-dom] Suppose that the inequality

$\sum_{S_{l}\in\mathcal{S}_{l}}\|e^{it\Delta}f_{S_{l}}\|_{BL_{A_{l}}^{p_{l},q}(P_{r_{l}})}^{p_{l}}\leq
2\sum_{S_{l}\in\mathcal{S}_{l,tiny}}\|e^{it\Delta}f_{S_{l}}\|_{BL_{A_{l}/2}^{p_{l},q}(P_{r_{l}})}^{p_{l}}$

holds, where the right-hand summation is restricted to those
$S_{l}\in\mathcal{S}_{l}$ for which [alg 1] terminates owing to the stopping
condition [tiny].

We now prove Properties 1 of [alg 2] which relies on Theorem 3.5. Properties 2
and 3 can be obtained by repeatedly applying of Properties 2 and 3 in [alg 1].

Assume that the condition [tiny-dom] does not happen. Necessarily,

$\sum_{S_{l}\in\mathcal{S}_{l}}\|e^{it\Delta}f_{S_{l}}\|_{BL_{A_{l}}^{p_{l},q}(P_{r_{l}})}^{p_{l}}\leq
2\sum_{S_{l}\in\mathcal{S}_{l,tang}}\|e^{it\Delta}f_{S_{l}}\|_{BL_{A_{l}/2}^{p_{l},q}(P_{r_{l}})}^{p_{l}},$

where the right-hand summation is restricted to those
$S_{l}\in\mathcal{S}_{l}$ for which [alg 1] terminates owing to [tang]. For
each $S_{l}\in\mathcal{S}_{l,tang}$, let $\mathcal{S}_{l-1}[S_{l}]$ denote a
collection of transverse complete intersections in $\mathbb{R}^{n+1}$ of
dimension $l-1$ which is produced by the stopping condition [tang] in [alg 1]
on $\|e^{it\Delta}f_{S_{l}}\|_{BL_{A}^{p_{l},q}(P_{r_{l}})}^{p_{l}}$. Then

$\|e^{it\Delta}f_{S_{l}}\|_{BL_{A_{l}}^{p_{l},q}(P_{r_{l}})}^{p_{l}}\lesssim
R^{O(\delta_{0})}D_{l-1}^{\delta}\sum_{S_{l-1}\in\mathcal{S}_{l-1}[S_{l}]}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{2A_{l-1}}^{p_{l},q}(P_{r_{l-1}})}^{p_{l}}.$

Let $\mathcal{S}_{l-1}$ denote the structured set

$\mathcal{S}_{l-1}=\\{S_{l-1}:S_{l}\in\mathcal{S}_{l,tang}\ \text{and}\
S_{l-1}\in\mathcal{S}_{l-1}[S_{l}]\\}.$

By induction on $l$, we have

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim
D_{l-1}^{\delta}M(\vec{r_{l}},\vec{D_{l}})R^{O(\delta_{0})}\|f\|_{L^{2}}^{1-\beta_{l}}\Big{(}\sum_{S_{l-1}\in\mathcal{S}_{l-1}}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{2A_{l-1}}^{p_{l},q}(P_{r_{l-1}})}^{p_{l}}\Big{)}^{\frac{\beta_{l}}{p_{l}}}.$

By Proposition 3.2 (iii) and Hölder’s inequality, we have

$\displaystyle\Big{(}\sum_{S_{l-1}\in\mathcal{S}_{l-1}}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{2A_{l-1}}^{p_{l},q}(P_{r_{l-1}})}^{p_{l}}\Big{)}^{\frac{1}{p_{l}}}$
$\displaystyle\leq\Big{\|}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{A_{l-1}}^{p_{l-1},q}(P_{r_{l-1}})}\Big{\|}_{l^{p_{l-1}}(\mathcal{S}_{l-1})}^{\alpha_{l-1}}\Big{\|}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{A_{l-1}}^{2,q}(P_{r_{l-1}})}\Big{\|}_{l^{2}(\mathcal{S}_{l-1})}^{1-\alpha_{l-1}}.$

Combining Theorem 3.5 and Property 2, it follows that

$\Big{\|}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{A_{l-1}}^{2,q}(P_{r_{l-1}})}\Big{\|}_{l^{2}(\mathcal{S}_{l-1})}\lesssim
r_{l-1}^{\frac{l-1}{2l}}(\prod_{i=l-1}^{n}D_{i}^{1+\delta})^{\frac{1}{2}}R^{O(\delta_{0})}\|f\|_{L^{2}}.$

Thus, we obtain

$\displaystyle\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}$ $\displaystyle\lesssim
M(\vec{r}_{l-1},\vec{D}_{l-1})R^{O(\delta_{0})}\|f\|_{L^{2}}^{1-\beta_{l-1}}\Big{(}\sum_{S_{l-1}\in\mathcal{S}_{l-1}}\|e^{it\Delta}f_{S_{l-1}}\|_{BL_{A_{l-1}}^{p_{l-1},q}(P_{r_{l-1}})}^{p_{l-1}}\Big{)}^{\frac{\beta_{l-1}}{p_{l-1}}}.$

The final stage The algorithm necessarily terminates at the $n$-th step. In
fact, if (4.4) would hold for $l=1$ and functions $f_{S_{1}}$ concentrated on
wave packets which are tangent to some transverse complete intersections of
dimension $1$. By the vanishing property of the $2$-broad norm which can be
easily proven, we have

(4.5)
$\|e^{it\Delta}f_{S_{1}}\|_{BL_{A}^{p_{1},q}(P_{r_{1}})}\lesssim\text{RapDec}(R)\|f\|_{L^{2}}.$

and hence
$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim\text{RapDec}(R)\|f\|_{L^{2}}$
by (4.4). This contradicts the non-degeneracy hypothesis.

In fact, from the definition of $BL_{A}^{p,q}$ and choosing the large number
$K\ll r_{1}^{1/2}$. Let $D_{K}\times I_{K/N}\subset
N_{r_{1}^{1/2+\delta_{1}}}(S_{1})\cap P_{r_{1}}$ and $V=T_{z_{0}}S_{1}$ with
$z_{0}\in S_{1}\cap N_{r_{1}^{1/2+\delta_{1}}}(D_{K}\times I_{K/N})$. Then
$\dim V=1$. For each $(\theta,\nu)\in\mathbb{T}_{S_{1}}(r_{1})$ and
$T_{\theta,\nu}\cap(D_{K}\times I_{K/N})\neq\emptyset$, we have

$\angle(G(\theta),V)\lesssim r_{1}^{-\frac{1}{2}+\delta_{1}}.$

It implies that for any $\theta\subset\tau$,

$\angle(G(\tau),V)\leq K^{-1}.$

So such $\tau$’s do not contribute to $\mu_{e^{it\Delta}f_{S_{1}}}(D_{K}\times
I_{K/N})$ and hence obtain $\eqref{e4.12}$.

Suppose the recursion process terminates at step $n+2-k$ for some $k\geq 2$.
For each $S_{k}\in\mathcal{S}_{k,tiny}$, let $\mathcal{O}[S_{k}]$ denote the
final collection of cells outputted by [alg 1]. Each $O\in\mathcal{O}[S_{k}]$
has the size at most $R^{\delta_{0}}$ by the stopping condition [tiny]. Let

$\mathcal{O}=\bigcup\\{\mathcal{O}[S_{k}]:S_{k}\in\mathcal{S}_{k,tiny}\\}.$

From Properties 1, 2 and 3 of [alg 1] and [alg 2], it follows that the
following three properties hold.

Property 1

(4.6) $\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim
M(\vec{r}_{k},\vec{D}_{k})R^{O(\delta_{0})}\|f\|_{L^{2}}^{1-\beta_{k}}\Big{(}\sum_{O\in\mathcal{O}}\|e^{it\Delta}f_{O}\|_{BL_{A_{k-1}}^{p_{k},q}(O)}^{p_{k}}\Big{)}^{\frac{\beta_{k}}{p_{k}}},$

where

$M(\vec{r}_{k},\vec{D}_{k}):=\Big{(}\prod_{i=k}^{n}D_{i}\Big{)}^{(n+1-l)\delta}\Big{(}\prod_{i=k}^{n}r_{i}^{\frac{i}{2(i+1)}(\beta_{i+1}-\beta_{i})}D_{i}^{\frac{1}{2}(\beta_{i+1}-\beta_{k})}\Big{)}.$

Property 2 For $k\leq n+1$,

(4.7)
$\sum_{O\in\mathcal{O}}\|f_{O}\|_{L^{2}}^{2}\lesssim\Big{(}\prod_{i=k-1}^{n}D_{i}^{1+\delta}\Big{)}R^{O(\delta_{0})}\|f\|_{L^{2}}^{2}.$

Property 3 For $k\leq n+1$,

(4.8)
$\max_{O\in\mathcal{O}}\|f_{O}\|_{L^{2}}^{2}\lesssim\prod_{i=k-1}^{n}r_{i}^{-\frac{1}{2}}D_{i}^{-i+\delta}R^{O(\delta_{0})}\|f\|_{L^{2}}^{2}.$

Since each $O\in\mathcal{O}$ has the size at most $R^{\delta_{0}}$, we have

$\|e^{it\Delta}f_{O}\|_{BL_{A}^{p_{l},q}(O)}\lesssim
R^{\delta_{0}}\|f_{O}\|_{L^{2}}.$

Combining this estimate and the above properties, one concludes

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim\prod_{i=k-1}^{n}r_{i}^{X_{i}}D_{i}^{Y_{i}+O(\delta)}R^{O(\delta_{0})}\|f\|_{L^{2}},$

where

$X_{i}:=\frac{i}{2(i+1)}(\beta_{i+1}-\beta_{i})-\frac{1}{2}(\frac{1}{2}-\frac{1}{p_{n+1}})$

and

$Y_{i}:=\frac{\beta_{i+1}}{2}-(i+1)(\frac{1}{2}-\frac{1}{p_{n+1}}).$

One easily verifies that

(4.9) $X_{i}\leq
0\iff\Big{(}\frac{1}{2}-\frac{1}{p_{i+1}}\Big{)}^{-1}-\Big{(}\frac{1}{2}-\frac{1}{p_{i}}\Big{)}^{-1}\leq\frac{i+1}{i}$

and

(4.10) $Y_{i-1}\leq 0\iff\Big{(}\frac{1}{2}-\frac{1}{p_{i}}\Big{)}^{-1}-2i\leq
0.$

We start with

$\Big{(}\frac{1}{2}-\frac{1}{p_{k}}\Big{)}^{-1}=2k.$

By (4.9) and (4.10), we have

$\Big{(}\frac{1}{2}-\frac{1}{p_{k+1}}\Big{)}^{-1}\leq\min\\{2(k+1),\
2k+\frac{k+1}{k}\\}.$

We choose $p_{k+1}$ such that

$\Big{(}\frac{1}{2}-\frac{1}{p_{k+1}}\Big{)}^{-1}=2k+\frac{k+1}{k}.$

Repeating the above argument, we obtain

$\Big{(}\frac{1}{2}-\frac{1}{p_{n+1}}\Big{)}^{-1}=2k+\frac{k+1}{k}+...+\frac{n+1}{n}$

and hence

$p_{n+1}=2+\frac{4}{n+k-1+s_{k}^{n}}.$

The worst case happens when $k=2$, i.e.,

$\|e^{it\Delta}f\|_{BL_{A}^{p,q}(P_{R})}\lesssim R^{\epsilon}\|f\|_{L^{2}},\
\text{when}\ p\geq\tilde{p}_{n}=p_{n+1}=2+\frac{4}{n+1+s_{2}^{n}}.$

And hence we complete the proof of Theorem 1.4.

##  5\. The proof of Theorem 1.6

Let $n\geq 2$, we need to find

$p_{n}\in\big{[}\frac{2(n+1)}{n},\frac{2(n+3)}{(n+1)}\big{)}\ \ \text{and}\ \
\frac{n+1}{p_{n}}+\frac{1}{r_{n}}=\frac{n}{2}$

such that

(5.1)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p_{n}}L_{t}^{r_{n}}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{p_{n}}-\frac{1}{r_{n}}}R^{\epsilon}||f_{1}||_{L^{2}}^{\frac{1}{2}}||f_{2}||_{L^{2}}^{\frac{1}{2}}.$

Here $\widehat{f_{1}},\ \widehat{f_{2}}$ are supported on subsets of
$\mathbb{B}^{n}(Ne_{1},1)$ with $N\geq 1$ and
$\text{dist}(\text{supp}\widehat{f_{1}},\text{supp}\widehat{f_{2}})\geq\frac{1}{2}$.

In fact, by (5.1) and Proposition A.1, we have

(5.2)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}\lesssim
N^{\frac{1}{p}-\frac{1}{r}}||f_{1}||_{L^{2}}^{\frac{1}{2}}||f_{2}||_{L^{2}}^{\frac{1}{2}}$

for $p>p_{n}$ and $r>r_{n}$. On the other hand, using Bernstein’s inequality
and the sharp bilinear estimate established by Tao [32], we have, for $r\geq
p>\frac{2(n+3)}{n+1}$,

$\displaystyle\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}$
$\displaystyle\lesssim
N^{\frac{1}{p}-\frac{1}{r}}\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x,t}^{p}(\mathbb{R}^{n+1})}$
(5.3) $\displaystyle\lesssim
N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|_{L^{2}}^{\frac{1}{2}}\|f_{2}\|_{L^{2}}^{\frac{1}{2}}.$

Interpolating (5.2) with (5), for each critical pair $(\tilde{p},\tilde{r})$
with $p_{n}<\tilde{p}\leq\frac{2(n+2)}{n}$, we can find a neighborhood
$\mathcal{N}(\tilde{p},\tilde{r})$ such that

(5.4)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}\lesssim
N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|_{L^{2}}^{\frac{1}{2}}\|f_{2}\|_{L^{2}}^{\frac{1}{2}}$

holds for each $(p,r)\in\mathcal{N}(\tilde{p},\tilde{r})$. By rescaling, we
obtain

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}\lesssim
2^{j(\frac{n+1}{p}+\frac{1}{r}-\frac{n}{2})}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}},$

where $\widehat{f_{1}},\ \widehat{f_{2}}$ are supported on the subsets of
$\mathbb{B}^{n}(\xi_{0},2^{-j})\subset A(1)$ and
$\text{dist}(\text{supp}\widehat{f_{1}},\text{supp}\widehat{f_{2}})\geq
2^{-j}$. Theorem 1.6 is then a direct result of the following crucial
proposition.

######  Proposition 5.1.

[23]. Let $2<\tilde{p}<\tilde{r}<\infty$ and
$\frac{n+1}{\tilde{p}}+\frac{1}{\tilde{r}}=\frac{n}{2}$. If

(5.5)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}\lesssim
2^{j(\frac{n+1}{p}+\frac{1}{r}-\frac{n}{2})}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}$

holds for $(p,r)$ in a neighborhood of $(\tilde{p},\tilde{r})$, where
$\widehat{f_{1}},\ \widehat{f_{2}}$ are supported on subsets of
$\mathbb{B}^{n}(\xi_{0},2^{-j})\subset A(1):=\\{\xi:|\xi|\sim 1\\}$ and
$\text{dist}(\text{supp}\widehat{f_{1}},\text{supp}\widehat{f_{2}})\geq
2^{-j}$. Then, (1.2) holds for $(\tilde{p},\tilde{r})$.

Our aim becomes to find out the index $p_{n}$ such that (5.1) holds. For
$n=2$, from Theorem 1.3, (5.1) holds for $p_{2}=3$. For $n\geq 3$, by Theorem
1.4, we have

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{\tilde{p}_{n}}L_{t}^{\infty}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{\tilde{p}_{n}}}R^{\epsilon}||f_{1}||_{L^{2}}^{\frac{1}{2}}||f_{2}||_{L^{2}}^{\frac{1}{2}}.$

Interpolating the estimate with the local bilinear estimate of Tao [32],

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{\frac{2(n+3)}{n+1}}L_{t}^{\frac{2(n+3)}{n+1}}(P_{R})}\lesssim_{\epsilon}R^{\epsilon}||f_{1}||_{L^{2}}^{\frac{1}{2}}||f_{2}||_{L^{2}}^{\frac{1}{2}},$

we obtain that (5.1) holds for

$p_{n}=\frac{2(n+1)[(n+2)\tilde{p}_{n}-2(n+3)]}{(n+1)n\tilde{p}_{n}-2(n^{2}+2n-1)}.$

##  6\. The proof of Theorem 1.10

Let $\widehat{f_{1}},\ \widehat{f_{2}}$ be supported on subsets of
$\mathbb{B}^{n}(0,1)$ with
$\text{dist}(\text{supp}\widehat{f_{1}},\text{supp}\widehat{f_{2}})\geq\frac{1}{2}$.
We make the wave packet decompositions at scale $R$,

$e^{it\Delta}f_{1}=\sum_{\theta_{1},\nu_{1}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\quad\text{and}\quad
e^{it\Delta}f_{2}=\sum_{\theta_{2},\nu_{2}}e^{it\Delta}f_{\theta_{2},\nu_{2}},$

where $f_{\theta_{i},\nu_{i}}(i=1,2)$ is the same as in (2.2). The $R$-cube
$Q_{R}$ can be decomposed into $O(R^{n\delta})$ many $R^{1-\delta}$-cubes
$Q\in\mathcal{Q}_{R^{1-\delta}}$.

The relation ”$\sim$” between the tubes $T_{\theta_{i},\nu_{i}}$ and the cubes
$Q$ was introduced by Tao [32]. Here we state several facts about it without
proof. The readers can refer to [32] for the details. Let
$\mathbb{T}_{i}:=\cup T_{\theta_{i},\nu_{i}},\,i=1,2$, and
$\mathbb{T}=\mathbb{T}_{1}\cup\mathbb{T}_{2}.$ Since only the tubes intersect
with $Q_{R}$ would have significant effects in the estimate, we assume that
$\\#\mathbb{T}\leq R^{C}$ for some constant $C$. We decompose the $Q_{R}$ into
$R^{\frac{1}{2}}$ cubes $q$ and denote the sets as
$\mathcal{Q}_{R^{\frac{1}{2}}}$. For each $q\in\mathcal{Q}_{R^{\frac{1}{2}}}$,
set

$\mathbb{T}_{i}(q)=\big{\\{}T_{\theta_{i},\nu_{i}}\in\mathbb{T}_{i};\,\,T_{\theta_{i},\nu_{i}}\cap
R^{\delta}q\neq\emptyset\big{\\}},\quad\text{for }\,\,i=1,2.$

For some dyadic numbers $1\leq\mu_{1},\mu_{2}\leq R^{C}$,

(6.1)
$q[\mu_{1},\mu_{2}]=\big{\\{}q\in\mathcal{Q}_{R^{\frac{1}{2}}};\,\,\mu_{i}\leq\\#\mathbb{T}_{i}(q)\leq
2\mu_{i},\,\,i=1,2\big{\\}},$

and for any fixed $T_{\theta_{i},\nu_{i}}$

(6.2) $\lambda(T_{\theta_{i},\nu_{i}},\mu_{1},\mu_{2})=\sharp\Big{\\{}q\in
q[\mu_{1},\mu_{2}];\,\,T_{\theta_{i},\nu_{i}}\cap
R^{\delta}q\neq\emptyset\Big{\\}},\quad\text{for }\,\,i=1,2.$

It is easy to see that
$1\leq\lambda(T_{\theta_{i},\nu_{i}},\mu_{1},\mu_{2})\leq R^{C}$. Then for any
dyadic number $1\leq\lambda_{1},\lambda_{2}\leq R^{C}$, we introduce in the
notation

(6.3)
$\mathbb{T}_{i}[\lambda_{i},\mu_{1},\mu_{2}]=\Big{\\{}T_{\theta_{i},\nu_{i}}\in\mathbb{T}_{i};\,\,\lambda_{i}\leq\lambda(T_{\theta_{i},\nu_{i}},\mu_{1},\mu_{2})\leq
2\lambda_{i}\Big{\\}}\quad\text{for }\,i=1,2.$

So far, we have four dyadic parameters
$\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}$, each has $O(\log R)$ many choices.
Then for a fixed triple $(\lambda_{1},\mu_{1},\mu_{2})$ and
$T_{\theta_{1},\nu_{1}}\in\mathbb{T}_{1}[\lambda_{1},\mu_{1},\mu_{2}]$, we say
that $T_{\theta_{1},\nu_{1}}\sim_{\lambda_{1},\mu_{1},\mu_{2}}Q$ if and only
if the cube $Q\in\mathcal{Q}_{R^{1-\delta}}$ maximizes the quantity

$\sharp\Big{\\{}q\in q[\mu_{1},\mu_{2}];\,\,T_{\theta_{1},\nu_{1}}\cap
R^{\delta}q\neq\emptyset,\,\,q\cap Q\neq\emptyset\Big{\\}}.$

By the definition, for any fixed triple $(\lambda_{1},\mu_{1},\mu_{2})$, there
is $O(1)$ cubes $Q\in\mathcal{Q}_{R^{1-\delta}}$ satisfying
$T_{\theta_{1},\nu_{1}}\sim_{\lambda_{1},\mu_{1},\mu_{2}}Q$. And we call

(6.4) $T_{\theta_{1},\nu_{1}}\sim Q\quad\text{if and only
if}\quad\exists(\lambda_{1},\mu_{1},\mu_{2})\quad\text{such that
}T_{\theta_{1},\nu_{1}}\sim_{\lambda_{1},\mu_{1},\mu_{2}}Q.$

The relation $T_{\theta_{2},\nu_{2}}\sim Q$ can be defined in the same way
(see Figure 6). By the definition, it is easy to see that

(6.5)
$\sharp\big{\\{}Q\in\mathcal{Q}_{R^{1-\delta}};\,\,T_{\theta_{i},\nu_{i}}\sim
Q\big{\\}}\lesssim_{\epsilon}R^{\epsilon}\quad\text{for }i=1,2.$

The relation $T_{\theta_{i},\nu_{i}}\sim Q$ describes the fact that the wave
packet mainly concentrated in the cube $Q$. Otherwise, we use the notation
$T_{\theta_{i},\nu_{i}}\nsim Q$. For this part, Tao proved the following
proposition.

$R$$R^{1-\delta}$$R^{1/2}$$Q_{1}$$Q_{2}$$T_{\theta_{1},\nu_{1}}$$T_{\theta_{1},\nu_{1}^{\prime}}$$T_{\theta_{2},\nu_{2}}$$q$$q$
Figure 6. $T_{\theta_{1},\nu_{1}}\sim Q_{2},\
T_{\theta_{1},\nu_{1}^{\prime}}\sim Q_{1},\ T_{\theta_{2},\nu_{2}}\sim Q_{1}$

######  Proposition 6.1.

[32] For any given $T_{\theta_{i},\nu_{i}}$, we have

(6.6) $\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim
Q}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim
Q}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x,t}^{4}(Q)}\lesssim_{\epsilon}R^{\epsilon+c\delta-\frac{n-1}{8}}\|f_{1}\|^{\frac{1}{2}}_{2}\|f_{2}\|^{\frac{1}{2}}_{2},$

and

(6.7) $\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim
Q}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\nsim
Q}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x,t}^{4}(Q)}\lesssim_{\epsilon}R^{\epsilon+c\delta-\frac{n-1}{8}}\|f_{1}\|^{\frac{1}{2}}_{2}\|f_{2}\|^{\frac{1}{2}}_{2}$

for some constant $c$ independent of $\delta,\epsilon$.

By the Galilean transform $\mathcal{G}$, we have
$\widehat{\tilde{f_{i}}}(\xi)=\widehat{f_{i}}(\xi-Ne_{1})$ supported on
subsets of $\mathbb{B}^{n}(Ne_{1},1)$ with
$\text{dist}(\text{supp}\widehat{\tilde{f_{1}}},\text{supp}\widehat{\tilde{f_{2}}})\geq\frac{1}{2}$.
The cube $Q_{R}$ becomes the parallelepiped $P_{R}$, and the cube
$Q\in\mathcal{Q}_{R^{1-\delta}}$ turns out to be the
$R^{1-\delta}$-parallelepiped $\tilde{P}\in\mathcal{P}$ . The image of the
tube $T_{\theta_{i},\nu_{i}}$ will be denote as
$\tilde{T}_{\theta_{i},\nu_{i}}$. We define
$\tilde{T}_{\theta_{i},\nu_{i}}\sim\tilde{P}$ if and only if
$T_{\theta_{i},\nu_{i}}\sim Q$. By Proposition 6.1, we have

######  Proposition 6.2.

For any given $\tilde{T}_{\theta_{i},\nu_{i}}$,

(6.8)
$\sharp\\{\tilde{P}\in\mathcal{P}:\tilde{T}_{\theta_{i},\nu_{i}}\sim\tilde{P}\\}\lesssim_{\epsilon}R^{\epsilon},$

(6.9)
$\Big{\|}\Big{|}\Big{(}\sum_{\tilde{T}_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{i\tilde{t}\Delta}\tilde{f}_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{\tilde{T}_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{i\tilde{t}\Delta}\tilde{f}_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x,t}^{4}(\tilde{P})}\lesssim_{\epsilon}R^{\epsilon+c\delta-\frac{n-1}{8}}\|\tilde{f}_{1}\|^{\frac{1}{2}}_{2}|\|\tilde{f}_{2}\|^{\frac{1}{2}}_{2},$

and

(6.10)
$\Big{\|}\Big{|}\Big{(}\sum_{\tilde{T}_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{i\tilde{t}\Delta}\tilde{f}_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{\tilde{T}_{\theta_{2},\nu_{2}}\nsim\tilde{P}}e^{i\tilde{t}\Delta}\tilde{f}_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x,t}^{4}(\tilde{P})}\lesssim_{\epsilon}R^{\epsilon+c\delta-\frac{n-1}{8}}\|\tilde{f}_{1}\|^{\frac{1}{2}}_{2}|\|\tilde{f}_{2}\|^{\frac{1}{2}}_{2}$

for some constant $c$ independent of $\delta,\epsilon$.

######  Lemma 6.3.

Let $p=3+\frac{1}{7}$, $r=5+\frac{1}{2}$ and $N\geq 1$. Suppose that
$\widehat{f_{1}},\ \widehat{f_{2}}$ are supported on subsets of
$\mathbb{B}^{n}(Ne_{1},1)$ with
$\text{dist}(\text{supp}\widehat{f_{1}},\text{supp}\widehat{f_{2}})\geq\frac{1}{2}$.
Then, for any $\epsilon>0$,

(6.11)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(P_{R})}\leq
R^{\epsilon}N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

###### Proof.

By Theorem 1.3 and H$\ddot{\text{o}}$lder’s inequality, we have

(6.12)
$||e^{it\Delta}f||_{L_{x}^{2}L_{t}^{\infty}(P_{R})}\leq(NR^{2})^{\frac{1}{2}-\frac{1}{3}}||e^{it\Delta}f||_{L_{x}^{3}L_{t}^{\infty}(P_{R})}\lesssim
N^{\frac{1}{2}}R^{\frac{1}{3}+\epsilon}||f||_{L^{2}}.$

Let $C_{R}$ be the minimum number such that

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(P_{R})}\leq
C_{R}N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

Assume that

(6.13) $C_{R}\lesssim R^{\alpha}$

for some $\alpha>0$ and all $R\geq 1$. We claim that, for any
$0<\delta\ll\epsilon\ll 1$,

(6.14)
$C_{R}\lesssim_{\delta,\epsilon}R^{\max\\{(1-\delta)\alpha,c\delta\\}+\epsilon},$

where the constant $c$ is independent of $\delta,\epsilon,\alpha,R$. By a
bootstrapping argument, we have

$C_{R}\lesssim_{\epsilon}R^{\epsilon}.$

We now give the proof of the claim. By Hölder’s inequality and the bilinear
Fourier restriction estimate, for $r>10/3$

$\displaystyle\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(P_{R})}$
$\displaystyle\leq(R^{2}N)^{\frac{1}{p}-\frac{1}{r}}\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{r}L_{t}^{r}(\mathbb{R}^{2+1})}$
$\displaystyle\lesssim(R^{2}N)^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

Hence (6.13) holds for $\alpha=2(\frac{1}{p}-\frac{1}{r})$.

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(P_{R})}$
can be decomposed into the local part:

(6.15)
$\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})},$

and the global parts:

(6.16)
$\displaystyle\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})},$
(6.17)
$\displaystyle\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})},$
(6.18)
$\displaystyle\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})}.$

By Cauchy-Schwarz, (6.8), and (6.13), the local part can be bounded by

$\displaystyle\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})}$
$\displaystyle\ \ \lesssim
N^{\frac{1}{p}-\frac{1}{r}}\sum_{\tilde{P}\in\mathcal{P}}R^{(1-\delta)\alpha}\Big{\|}\sum_{T_{\theta_{1},\nu_{1}}\sim\tilde{P}}f_{\theta_{1},\nu_{1}}\Big{\|}^{\frac{1}{2}}_{L^{2}}\Big{\|}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}f_{\theta_{2},\nu_{2}}\Big{\|}^{\frac{1}{2}}_{L^{2}}$
$\displaystyle\ \ \lesssim
N^{\frac{1}{p}-\frac{1}{r}}R^{(1-\delta)\alpha}\Big{(}\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\sum_{T_{\theta_{1},\nu_{1}}\sim\tilde{P}}f_{\theta_{1},\nu_{1}}\Big{\|}_{L^{2}}^{2}\Big{)}^{\frac{1}{4}}\Big{(}\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}f_{\theta_{2},\nu_{2}}\Big{\|}_{L^{2}}^{2}\Big{)}^{\frac{1}{4}}$
$\displaystyle\ \ \lesssim
N^{\frac{1}{p}-\frac{1}{r}}R^{(1-\delta)\alpha}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}}\sum_{T_{\theta_{1},\nu_{1}}\sim\tilde{P}}\|f_{\theta_{1},\nu_{1}}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{4}}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}\|f_{\theta_{2},\nu_{2}}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{4}}$
(6.19) $\displaystyle\ \
\lesssim_{\epsilon}N^{\frac{1}{p}-\frac{1}{r}}R^{(1-\delta)\alpha+\epsilon}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

For the global parts, we give the estimate of (6.16). (6.17) and (6.18) can be
bounded in the same way. From Cauchy-Schwarz and (6.12), it follows that

$\displaystyle\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{2}L_{t}^{\infty}(\tilde{P})}$
$\displaystyle\ \
\leq\Big{\|}\sum_{T_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{\|}^{\frac{1}{2}}_{L_{x}^{2}L_{t}^{\infty}(\tilde{P})}\Big{\|}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{\|}^{\frac{1}{2}}_{L_{x}^{2}L_{t}^{\infty}(\tilde{P})}$
(6.20) $\displaystyle\ \
\lesssim_{\epsilon}N^{\frac{1}{2}}R^{\frac{1}{3}+\epsilon}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

Interpolating (6) with (6.9) $(n=2)$, we obtain

$\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})}\lesssim_{\epsilon}N^{\frac{1}{p}-\frac{1}{r}}R^{c\delta+\epsilon}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}$

for $p=22/7$ and $r=22/4$. Notice that
$\sharp\\{\tilde{P}:\tilde{P}\in\mathcal{P}\\}\lesssim R^{(n+1)\delta}$,

$\sum_{\tilde{P}\in\mathcal{P}}\Big{\|}\Big{|}\Big{(}\sum_{T_{\theta_{1},\nu_{1}}\nsim\tilde{P}}e^{it\Delta}f_{\theta_{1},\nu_{1}}\Big{)}\Big{(}\sum_{T_{\theta_{2},\nu_{2}}\sim\tilde{P}}e^{it\Delta}f_{\theta_{2},\nu_{2}}\Big{)}\Big{|}^{\frac{1}{2}}\Big{\|}_{L_{x}^{p}L_{t}^{r}(\tilde{P})}\lesssim_{\epsilon}N^{\frac{1}{p}-\frac{1}{r}}R^{c\delta+\epsilon}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

Combining this estimate and (6.19), we finally obtain

$C_{R}\lesssim_{\delta,\epsilon}R^{\max\\{(1-\delta)\alpha,c\delta\\}+\epsilon}.$

∎

From (6.11) and Proposition A.1, we have

$\sup_{\rho>1}\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}(\mathbb{R}^{2}\times[0,\rho])}\lesssim
N^{\frac{1}{p_{0}}-\frac{1}{r_{0}}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}$

for $p_{0}>3+\frac{1}{7}$ and $r_{0}>5+\frac{1}{2}$. By the following
Proposition 6.4, we obtain that

$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{2}\times[0,1])}\lesssim\|f\|_{B_{\alpha,p}^{p}}$

holds for $\alpha=n(1-2/p)-2/r$, $p>3+\frac{1}{7}$ and $r\geq 5+\frac{1}{2}$.

######  Proposition 6.4.

[25] Let $2(n+1)/n\leq p_{0}\leq r_{0}\leq 2p_{0}/(p_{0}-2)$ and $N\geq 1$.
Suppose the Fourier supports of $f_{1},f_{2}$ are in
$\mathbb{B}^{n}(Ne_{1},1)$ with
$dist(supp\widehat{f_{1}},supp\widehat{f}_{2})\geq\frac{1}{2}$ and

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}(\mathbb{R}^{n}\times[0,\rho])}\lesssim
N^{\gamma}\|f_{1}\|_{L^{2}}^{\frac{1}{2}}\|f_{2}\|_{L^{2}}^{\frac{1}{2}}$

holds for some $\gamma<2n(1-\frac{2}{p_{0}})-\frac{4}{r_{0}}$ and any
$\rho>1$. Then, for any $p_{0}<p<\infty$ and $r_{0}\leq r<\infty$,

$\|e^{it\Delta}f\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n}\times[0,1])}\lesssim\|f\|_{B_{\alpha,p}^{p}}$

holds for $\alpha=n(1-2/p)-2/r$.

##  Appendix A Epsilon removal for bilinear space-time estimate

######  Proposition A.1.

Let $N\geq 1$. Suppose the Fourier supports of $f_{1},f_{2}$ are in
$\mathbb{B}^{n}(Ne_{1},1)$ with

$dist(supp\widehat{f_{1}},supp\widehat{f}_{2})\geq\frac{1}{2}$

and

(A.1)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}(P_{R})}\lesssim_{\epsilon}N^{\frac{1}{p_{0}}-\frac{1}{r_{0}}}R^{\epsilon}\|f_{1}\|_{L^{2}}^{\frac{1}{2}}\|f_{2}\|_{L^{2}}^{\frac{1}{2}}$

holds for $r_{0}\geq p_{0}\geq 2$. Then, for any $p>p_{0}$ and $r>r_{0}$,

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}(\mathbb{R}^{n+1})}\lesssim
N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|_{L^{2}}^{\frac{1}{2}}\|f_{2}\|_{L^{2}}^{\frac{1}{2}}.$

The above space-time $\epsilon$-removing argument was proven by Lee, Rogers
and Vargas [23]. Here, we give another proof by making use of an argument of
Bourgain and Guth [3].

######  Definition A.2 ($(M,R)$-sparse parallelepipeds).

A finite collection $\\{P_{R}(z_{\alpha})\\}_{\alpha=1}^{M}$ of
parallelepipeds in $\mathbb{R}^{n+1}$ is called $(M,R)$-sparse if the centers
$z_{\alpha}$ are $(MR)^{\gamma}$-separated, i.e.

$|z_{\alpha_{0}}-z_{\alpha_{1}}|\geq(MR)^{\gamma}\ \ \text{with}\ \
\alpha_{0}\neq\alpha_{1},$

where $\gamma$ is a fixed number.

The following cover lemma which was first proved by Tao [31, Lemma 3.3] and
refined by Cho, Koh and Lee [6, Lemma 2.6]. We transform their result into the
parallelepiped by Galilean transform.

######  Lemma A.3.

Let $E\subset\mathbb{R}^{n+1}$ be a finite union of finitely overlapping
parallelepipeds of size $c\sim 1$. Then for each $K\in\mathbb{N}$, there are
subsets $E_{1},E_{2},...,E_{K}$ of $E$ with

$E=\bigcup_{k=1}^{K}E_{k}$

such that each $E_{k}$ has $O(|E|^{1/K})$ number of
$(O(|E|),|E|^{O({\gamma}^{k-1})})$ sparse collections

$S_{1},S_{2},...,S_{O(E^{1/K})}$

of which the union $S_{1}\cup S_{2}\cup...\cup S_{O(E^{1/K})}$ is a covering
of $E_{k}$.

###### Proof.

We assume that $\|f_{i}\|_{L^{2}}=1\,(i=1,2)$ and denote
$F(x,t):=|e^{it\Delta}f_{1}(x)e^{it\Delta}f_{2}(x)|^{\frac{1}{2}}$. Given a
$(M,R)$-sparse set $E=\bigcup_{\alpha=1}^{M}P_{R}(z_{\alpha})$ with
$z_{\alpha}=(x_{\alpha},t_{\alpha})$. We first show that

(A.2) $\|F|_{E}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}}\leq
C_{\epsilon}N^{\frac{1}{p_{0}}-\frac{1}{r_{0}}}R^{\epsilon}.$

In fact, let $\phi(x,t;\xi)=x\cdot\xi-2\pi t|\xi|^{2}$ and $(x,t)\in
P_{R}(z_{\alpha})$, we write

(A.3) $e^{it\Delta}f(x)=\int e^{2\pi
i(\phi(x,t;\xi)-\phi(x_{\alpha},t_{\alpha};\xi))}(e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{f}(\xi))\omega(\xi)d\xi,$

where $\omega(\xi)$ is a smooth function localized on
${\rm{\,supp\,}}\widehat{f}$. Define $\widehat{g}(\xi):=e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{f}(\xi)$, then

(A.4) $\eqref{e4.2x}=\int\int e^{2\pi
i(\phi(x,t;\xi)-\phi(x_{\alpha},t_{\alpha};\xi)-\xi y)}\omega(\xi)d\xi
g(y)dy.$

Let $\eta$ be a bump function satisfy $0\leq\eta\leq 1$ and
${\rm{\,supp\,}}\eta\subset\mathbb{B}^{n}(0,1)$,
$\eta_{R_{1}}(y):=\eta(\frac{y}{R_{1}})$, $R_{1}:=100MR$ and
$P_{(0,R_{1})}g:=g\eta_{R_{1}}$. Since

$\displaystyle|\bigtriangledown_{\xi}[\phi(x,t;\xi)-\phi(x_{\alpha},t_{\alpha};\xi)]|$
$\displaystyle=|x-x_{\alpha}-4\pi(t-t_{\alpha})\xi|$
$\displaystyle\leq|x-x_{\alpha}-4\pi(t-t_{\alpha})Ne_{1}|+|4\pi(t-t_{\alpha})(\xi-
Ne_{1})|$ $\displaystyle\lesssim R,$

we have

$|\eqref{e4.3x}|\lesssim\Big{|}\int\int e^{2\pi
i(\phi(x,t;\xi)-\phi(x_{\alpha},t_{\alpha};\xi)-\xi y)}\omega(\xi)d\xi
P_{(0,R_{1})}g(y)dy\Big{|}+\text{RapDec}(MR).$

Define

$\widehat{f_{\alpha}}:=e^{-2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{P_{(0,R_{1})}g},$

then

$|e^{it\Delta}f(x)|\lesssim|e^{it\Delta}f_{\alpha}(x)|+\text{RapDec}(MR).$

By the above argument, $r_{0}\geq p_{0}\geq 2$ and (A.1), we have

$\displaystyle\|F|_{E}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}}^{p_{0}}$
$\displaystyle\leq\sum_{\alpha}\|F\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}(P_{R}(z_{\alpha}))}^{p_{0}}$
$\displaystyle\lesssim_{\epsilon}R^{p_{0}\epsilon}N^{p_{0}(\frac{1}{p_{0}}-\frac{1}{r_{0}})}\sum_{\alpha}(\|f_{1,\alpha}\|_{L^{2}}^{\frac{1}{2}}\|f_{2,\alpha}\|_{L^{2}}^{\frac{1}{2}})^{p_{0}}$
(A.5)
$\displaystyle\lesssim_{\epsilon}R^{p_{0}\epsilon}N^{p_{0}(\frac{1}{p_{0}}-\frac{1}{r_{0}})}\Big{(}\sum_{\alpha}\|f_{\alpha}\|_{L^{2}}^{2}\Big{)}^{\frac{p_{0}}{2}},$

where

$\sum_{\alpha}\|f_{\alpha}\|_{L^{2}}^{2}=\max\Big{\\{}\sum_{\alpha}\|f_{1,\alpha}\|_{L^{2}}^{2},\sum_{\alpha}\|f_{2,\alpha}\|_{L^{2}}^{2}\Big{\\}}.$

By duality and Plancherel’s theorem, there exist
$\\{\zeta_{\alpha}\\}_{\alpha}$ satisfying supp
$\zeta_{\alpha}\subset\mathbb{B}^{n}(0,R_{1})$ and
$\sum_{\alpha}\|\zeta_{\alpha}\|_{L^{2}}^{2}=1$ such that

$\displaystyle\Big{(}\sum_{\alpha}\|f_{\alpha}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle=\sum_{\alpha}<P_{(0,R_{1})}g,\zeta_{\alpha}>$
$\displaystyle=\sum_{\alpha}<e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{f},\widehat{\zeta_{\alpha}}>$ (A.6)
$\displaystyle=\sum_{\alpha}<\widehat{f},e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{\zeta_{\alpha}}>.$

From (A) and Cauchy-Schwarz inequality, it follows that

$\displaystyle\Big{(}\sum_{\alpha}\|f_{\alpha}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{\|}\sum_{\alpha}e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{\zeta_{\alpha}}\Big{\|}_{L^{2}}$
(A.7)
$\displaystyle\leq\Big{(}\sum_{\alpha}\|\zeta_{\alpha}\|_{L^{2}}^{2}\Big{)}^{\frac{1}{2}}+\Big{(}\sum_{\alpha,\beta:\alpha\neq\beta}|<e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{\zeta_{\alpha}},e^{2\pi
i\phi(x_{\beta},t_{\beta};\xi)}\widehat{\zeta_{\beta}}>|\Big{)}^{\frac{1}{2}}.$

For the off-diagonal terms $\alpha\neq\beta$,

(A.8) $\displaystyle|<e^{2\pi
i\phi(x_{\alpha},t_{\alpha};\xi)}\widehat{\zeta_{\alpha}},e^{2\pi
i\phi(x_{\beta},t_{\beta};\xi)}\widehat{\zeta_{\beta}}>|=\Big{|}\int e^{2\pi
i[\phi(x_{\alpha},t_{\alpha};\xi)-\phi(x_{\beta},t_{\beta};\xi)]}\widehat{\zeta_{\alpha}}(\xi)\overline{\widehat{\zeta_{\beta}}}(\xi)d\xi\Big{|}.$

Since $|z_{\alpha}-z_{\beta}|\geq(MR)^{\gamma}$ with
$z_{\alpha}=(x_{\alpha},t_{\alpha}),\ z_{\beta}=(x_{\beta},t_{\beta})$,
$|\phi(x_{\alpha},t_{\alpha};\xi)-\phi(x_{\beta},t_{\beta};\xi)|$ satisfies
either

$|\bigtriangledown_{\xi}[\phi(x_{\alpha},t_{\alpha};\xi)-\phi(x_{\beta},t_{\beta};\xi)]|\gtrsim(MR)^{\gamma}$

or

$|\text{det}\,D_{\xi}^{2}[\phi(x_{\alpha},t_{\alpha};\xi)-\phi(x_{\beta},t_{\beta};\xi)]|\gtrsim(MR)^{n\gamma}.$

By the method of stationary phase, we have

$\text{\eqref{eA.5}}\lesssim(MR)^{-n\gamma/2}\|\zeta_{\alpha}\|_{L^{1}}\|\zeta_{\beta}\|_{L^{1}}\lesssim
R_{1}^{n}(MR)^{-n\gamma/2}\|\zeta_{\alpha}\|_{L^{2}}\|\zeta_{\beta}\|_{L^{2}}.$

By choosing $\gamma>2$, we obtain $\eqref{eA.4}\lesssim 1$ which implies
(A.2).

Since $\widehat{f}_{1}$ and $\widehat{f}_{2}$ are supported on
$\mathbb{B}^{n}(Ne_{1},1)$, we have
$\|F\|_{\infty}\leq\|f_{1}\|_{L^{2}}\|f_{2}\|_{L^{2}}=1$. We decompose

$F=\sum_{k\geq 0}F|_{|F|\sim 2^{-k}}:=\sum_{k\geq 0}F_{k},$

where $\text{supp}F_{k}\subset E_{k}$ and $E_{k}$ is the union of
parallelepipeds of size $c$ ($c$-parallelepipeds). We denote by proj$(E_{k})$
the projection of $E_{k}$ along the $(4\pi Ne_{1},1)$ onto the $x$-plane. For
each grid point $x\in c\mathbb{Z}^{n}\cap\text{proj}(E_{k})$, we define
$E_{k,x}$ to be the union of $c$-parallelepipeds such that $x\in$proj$(E_{k})$
and

$E_{k}^{j}=\bigcup\\{E_{k,x}:\sharp(E_{k,x})\sim 2^{j}\\}.$

We have

$E_{k}=\bigcup_{j\geq 0}E_{k}^{j}.$

Let

$I=\frac{\log(1/\epsilon)}{2\log\gamma}+1.$

By Lemma A.3, we have

$E_{k}^{j}=\bigcup_{i=1}^{I}E_{k,i}^{j}\ \ \text{and}\ \
E_{k,i}^{j}\subset\bigcup_{|m|\lesssim|E_{k}^{j}|^{1/I}}\bigcup_{\alpha\lesssim|E_{k}^{j}|}P_{R_{m}}(z_{\alpha}),$

where $R_{m}\lesssim|E_{k}^{j}|^{C\gamma^{i-1}}$. From (A.2), we obtain that

$\|F_{k}|_{E_{k,i}^{j}}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}}\leq
C_{\epsilon}N^{\frac{1}{p_{0}}-\frac{1}{r_{0}}}|E_{k}^{j}|^{1/I}(|E_{k}^{j}|^{C{\gamma}^{i-1}})^{\epsilon}.$

Sum up $1\leq i\leq I$,

$\|F_{k}|_{E_{k}^{j}}\|_{L_{x}^{p_{0}}L_{t}^{r_{0}}}\leq
C_{\epsilon}N^{\frac{1}{p_{0}}-\frac{1}{r_{0}}}|E_{k}^{j}|^{1/I}(|E_{k}^{j}|^{C{\gamma}^{I-1}})^{\epsilon}.$

Combining this estimate and the definition of $E_{k}^{j}$, we have

$2^{-k}2^{j/r_{0}}|\text{proj}(E_{k}^{j})|^{1/p_{0}}\lesssim
N^{\frac{1}{p_{0}}-\frac{1}{r_{0}}}(2^{j}|\text{proj}(E_{k}^{j})|)^{\delta(\epsilon)},$

where $\delta(\epsilon)=1/I+C\epsilon{\gamma}^{I-1}$. Since $\lim_{\epsilon\to
0}\delta(\epsilon)=0$, then there exists $\epsilon>0$ such that

$\frac{1}{r}+\epsilon\leq\Big{(}\frac{1}{r_{0}}-\delta(\epsilon)\Big{)}(1-\epsilon)$

and

$\frac{1}{p_{1}}=\Big{(}\frac{1}{p_{0}}-\delta(\epsilon)\Big{)}(1-\epsilon).$

For the decomposition of $F$, we have

$\displaystyle\|F\|_{L_{x}^{p_{1}}L_{t}^{r}}$ $\displaystyle\leq\sum_{k\geq
0}\sum_{j\geq 0}\|F_{k}|_{E_{k}^{j}}\|_{L_{x}^{p_{1}}L_{t}^{r}}$
$\displaystyle\lesssim\sum_{k\geq 0}\sum_{j\geq
0}2^{-k}2^{j/r}|\text{proj}(E_{k}^{j})|^{1/p_{1}}$ $\displaystyle\lesssim
N^{\frac{1}{p_{1}}-\frac{1}{r}}\sum_{k\geq 0}\sum_{j\geq 0}2^{(-k-j)\epsilon}$
(A.9) $\displaystyle\lesssim CN^{\frac{1}{p_{1}}-\frac{1}{r}}.$

On the other hand, for $1\leq r\leq\infty$,

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{[}\frac{1}{2}]\|_{L_{x}^{\infty}L_{t}^{r}(\mathbb{R}^{n+1})}\lesssim
N^{-\frac{1}{r}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

It was proved by [30]. By interpolation it with (A), we obtain for $p\geq
p_{1}$,

$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}\leq
C_{p,r}N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}}.$

Hence, we complete the proof of Proposition A.1.

∎

##  Appendix B A bilinear space-time estimate conjecture

We now formulate a bilinear space-time conjecture which implies Conjecture
1.2.

######  Conjecture B.1 (Bilinear space-time estimate).

Suppose that $\widehat{f_{1}},\ \widehat{f_{2}}$ are supported on subsets of
$\mathbb{B}^{n}(Ne_{1},1)$ with
$\text{dist}(\text{supp}\widehat{f_{1}},\text{supp}\widehat{f_{2}})\geq\frac{1}{2}$.

(B.1)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}|^{\frac{1}{2}}\|_{L_{x}^{p}L_{t}^{r}(\mathbb{R}^{n+1})}\leq
C_{p,r}N^{\frac{1}{p}-\frac{1}{r}}\|f_{1}\|^{\frac{1}{2}}_{L^{2}}\|f_{2}\|^{\frac{1}{2}}_{L^{2}},$

holds if and only if $\frac{n+2}{p}+\frac{1}{r}\leq\frac{n+1}{2},2\leq
p,r<\infty$.

######  Remark B.2.

  * (I)

By the same argument of Subsection 5 of this paper, we have Conjecture B.1
$\implies$ Conjecture 1.2.

  * (II)

Conjecture B.1 is also closely related to Conjecture 1.9. By Proposition 6.4,
we have

$\text{Conjecture \ref{c1.2}}\ \implies\ \text{Conjecture \ref{c1.4} holds
for}\ p>\frac{2(n+1)}{n}\ \text{and}\ r\geq 2(n+1).$

The above conjecture is also deeply related to the Fourier restrictive
conjecture of paraboloid in $\mathbb{R}^{n+1}$ (see [25, Theorem 1.1]).
However, it is very difficult to prove the restrictive conjecture with the
endpoint $\frac{2(n+1)}{n}$. Conjecture B.1 provides a new viewpoint to
improve Conjecture 1.9.

We now give a necessary condition for (B.1) by modifying an example from
Foschi-Klainerman [14].

######  Proposition B.3.

Let $p,r\in[2,\infty)$. If (B.1) is true, then

$\frac{n+2}{p}+\frac{1}{r}\leq\frac{n+1}{2}.$

###### Proof.

Let $\epsilon>0$,

$\widehat{f_{1}}(\xi)=\chi_{\\{|\xi_{1}-(N+1/2)|<\epsilon\\}}(\xi_{1})\chi_{\\{|\xi_{2}-1/2|<\epsilon^{2}\\}}(\xi_{2})\chi_{\\{|\xi_{3}|<\epsilon\\}}(\xi_{3})...\chi_{\\{|\xi_{n}|<\epsilon\\}}(\xi_{n})$

and

$\widehat{f_{2}}(\eta)=\chi_{\\{|\eta_{1}-(N+1/2)|<\epsilon\\}}(\eta_{1})\chi_{\\{|\eta_{2}+1/2|<\epsilon^{2}\\}}(\eta_{2})\chi_{\\{|\eta_{3}|<\epsilon\\}}(\eta_{3})...\chi_{\\{|\eta_{n}|<\epsilon\\}}(\eta_{n}).$

Then

$\text{supp}\widehat{f_{i}}\subset\mathbb{B}^{n}(Ne_{1},1),\quad\|f_{i}\|_{L^{2}}\sim\epsilon^{\frac{n+1}{2}},\quad\text{for
}i=1,2.$

The distance between their Fourier supports is at least $\frac{1}{2}$. An easy
calculus shows that

$\displaystyle|\xi|^{2}$
$\displaystyle=|(\xi_{1}-(N+1/2))+(N+1/2)|^{2}+|(\xi_{2}-1/2)+1/2|^{2}+|\xi_{3}|^{2}+...+|\xi_{n}|^{2}$
$\displaystyle\sim N^{2}+2(N+1/2)(\xi_{1}-1)+O(\epsilon^{2})$

and

$|\eta|^{2}\sim N^{2}+2(N+1/2)(\eta_{1}-1)+O(\epsilon^{2}).$

Let $A\subset\mathbb{R}^{n+1}$ denotes the set

$x_{1}-4\pi(N+1/2)t=O(\epsilon^{-1}),\
x_{2}=O(\epsilon^{-2}),x_{i}=O(\epsilon^{-1}),3\leq i\leq
n,t=O(\epsilon^{-2}).$

For each point $(x,t)\in A$, we have

$\displaystyle|e^{it\Delta}f_{1}e^{it\Delta}f_{2}(x)|$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}e^{2\pi
i[x\cdot(\xi+\eta)-2\pi
t(|\xi|^{2}+|\eta|^{2})]}\widehat{f_{1}}(\xi)\widehat{f_{2}}(\eta)d\xi
d\eta\Big{|}$ $\displaystyle\sim\epsilon^{2(n+1)}.$

Hence

(B.2)
$\||e^{it\Delta}f_{1}e^{it\Delta}f_{2}(x)|^{\frac{1}{2}}\|_{L^{p}_{x}L^{r}_{t}(\mathbb{R}^{n}\times\mathbb{R}^{n})}\geq\epsilon^{n+1}\|\chi_{A}\|_{L^{p}_{x}L^{r}_{t}(\mathbb{R}^{n}\times\mathbb{R}^{n})}\geq\epsilon^{(n+1)-\frac{1}{r}-\frac{n+2}{p}}N^{\frac{1}{p}-\frac{1}{r}}.$

From the estimate (B.1), we have

$\epsilon^{(n+1)-\frac{1}{r}-\frac{n+2}{p}}N^{\frac{1}{p}-\frac{1}{r}}\lesssim
N^{\frac{1}{p}-\frac{1}{r}}\epsilon^{\frac{n+1}{2}}.$

It implies that $\frac{n+2}{p}+\frac{1}{r}\leq\frac{n+1}{2}$. ∎

### Acknowledgement

The authors would like to express their gratitude to Dr. Zhenbin Cao for many
valuable comments and suggestions on the early visions of this paper.

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