$\displaystyle\Big{\|}\|v_{1}(t,x_{1},x^{\prime})+v_{2}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\Big{\|}^{\mathfrak{a}}_{L^{\mathfrak{b}}_{t}(I)}$
$\displaystyle\leq\Big{(}\big{\|}\|v_{1}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}(I)}+\big{\|}\|v_{2}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}(I)}\Big{)}^{\frac{\mathfrak{a}}{\mathfrak{b}}}$
$\displaystyle\leq\big{\|}\|v_{1}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{a}}_{L^{\mathfrak{b}}_{t}(I)}+\big{\|}\|v_{2}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{a}}_{L^{\mathfrak{b}}_{t}(I)}.$

Integrating in $x_{1}$ yields

$\|v_{1}+v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{\mathfrak{a}}\leq\|v_{1}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{\mathfrak{a}}+\|v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{\mathfrak{a}}\,.$

After raising both sides to the power $\frac{r}{\mathfrak{a}}$ and using again
that $(a+b)^{s}\leq a^{s}+b^{s}$, for $s=\frac{r}{\mathfrak{a}}\leq 1$, it
follows

$\|v_{1}+v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{r}\leq\|v_{1}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{r}+\|v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{r},$

as desired.

Second, we assume that $\mathfrak{a}\geq\mathfrak{b}$, and in particular that
$r\leq\mathfrak{b}$. Then, (3) and the triangle inequality yield

$\|v_{1}+v_{2}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{r}\begin{aligned}
&\leq\Big{(}\int_{\mathbb{R}}\big{(}\big{\|}\|v_{1}(x_{1})\|_{L^{\mathfrak{c}}_{x^{\prime}}}\|^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}+\big{\|}\|v_{2}(x_{1})\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\big{)}^{\frac{\mathfrak{a}}{\mathfrak{b}}}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x_{1}\Big{)}^{\frac{r}{\mathfrak{a}}}\\\
&=\Big{\|}\big{\|}\|v_{1}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}+\big{\|}\|v_{2}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\Big{\|}_{L^{\frac{\mathfrak{a}}{\mathfrak{b}}}_{x_{1}}}^{\frac{r}{\mathfrak{b}}}\\\
&\leq\Big{(}\Big{\|}\big{\|}\|v_{1}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\Big{\|}_{L^{\frac{\mathfrak{a}}{\mathfrak{b}}}_{x_{1}}}+\Big{\|}\big{\|}\|v_{2}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\Big{\|}_{L^{\frac{\mathfrak{a}}{\mathfrak{b}}}_{x_{1}}}\Big{)}^{\frac{r}{\mathfrak{b}}}\\\
&=\Big{(}\|v_{1}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{\mathfrak{b}}+\|v_{2}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{\mathfrak{b}}\Big{)}^{\frac{r}{\mathfrak{b}}}\\\
&\leq\|v_{1}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{r}+\|v_{2}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{r},\end{aligned}$

as desired.

Having shown that (3) holds, the desired bound (3) follows for
$r=\frac{2}{1-\varepsilon_{0}}$, which indeed satisfies
$r\leq\min\\{\mathfrak{a},\mathfrak{b}\\}$ for any directional norm in the
definition of $X_{N}(I)$ or $Y_{N}(I)$. The proof of the desired results is
finished. ∎

##  4\. The multilinear estimates

We start this section by proving a bilinear estimates in terms of directional
space-time norms. Then, we derive crucial trilinear estimates that allow us to
control the cubic nonlinearity in (1).

######  Lemma 4.1.

Fix any $0<\varepsilon\lesssim 1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon}1$. For any two functions
$h_{+},h_{-}\colon\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{C}$ and any
$N_{+},N_{-}\in 2^{\mathbb{N}}$ with $N_{+}\geq N_{-}$ it holds that

$\displaystyle\|h_{+}h_{-}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim_{\varepsilon_{0}}N_{+}^{\varepsilon}\Big{(}\frac{N_{+}}{N_{-}}\Big{)}^{-\frac{1}{2}}\times\begin{dcases}N_{-}^{\mathfrak{s}_{c}}\|h_{+}\|_{X_{N_{+}}(\mathbb{R})}\|h_{-}\|_{X_{N_{-}}(\mathbb{R})},\\\
\|h_{+}\|_{Y_{N_{+}}(\mathbb{R})}\|h_{-}\|_{Y_{N_{-}}(\mathbb{R})},\\\
\|h_{+}\|_{X_{N_{+}}(\mathbb{R})}\|h_{-}\|_{Y_{N_{-}}(\mathbb{R})},\\\
\|h_{+}\|_{Y_{N_{+}}(\mathbb{R})}\|h_{-}\|_{X_{N_{-}}(\mathbb{R})}\,,\end{dcases}$

where $\mathfrak{s}_{c}\coloneqq\frac{d-2}{2}$.

###### Proof.

Let $\mathop{\kern 0.0pt\mathrm{U}}\mathopen{}_{e_{l}}$ be as in Lemma 2.5,
and recall that $\sum_{l}\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}=\mathop{\kern
0.0pt\mathrm{Id}}\mathopen{}$. The triangle inequality implies

$\|h_{+}\,h_{-}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\leq\sum_{l=1}^{d}\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\,,$

and therefore it suffices to prove (4.1) for $h_{+}$ replaced by
$\mathop{\kern 0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}$.

By Fubini’s Theorem and Hölder’s inequality we get that

$\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\begin{aligned}
&=\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L_{e_{l}}^{(2,2,2)}(\mathbb{R})}\\\
&\lesssim\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}})}(\mathbb{R})}\|h_{-}\|_{L_{e_{l}}^{(\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}},\frac{2}{\varepsilon_{0}})}(\mathbb{R})}\,.\end{aligned}$

From the definitions (3) and (3) of the norms $X_{N}$ and $Y_{N}$ we directly
obtain the first three bounds in (4.1).

To prove the last estimate in (4.1), we use Hölder’s inequality to obtain the
bound

$\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L_{e_{l}}^{(2,2,2)}(\mathbb{R})}\lesssim\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{1}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2\mathfrak{c}_{0}}{\mathfrak{c}_{0}-2(1-\varepsilon_{0})})}(\mathbb{R})}\|h_{-}\|_{L_{e_{l}}^{(\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}},\frac{\mathfrak{c}_{0}}{1-\varepsilon_{0}})}(\mathbb{R})}\,.$

Then from the definition (3) of the norm $X_{N}$ it follows that

$\|h_{-}\|_{L_{e_{l}}^{(\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}},\frac{\mathfrak{c}_{0}}{1-\varepsilon_{0}})}(\mathbb{R})}\lesssim
N_{-}^{\frac{1}{2}}\|h_{-}\|_{X_{N_{-}}(\mathbb{R})}\,.$

To control thus we can use interpolation to control $\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}$ we use interpolation. Indeed, since
$\mathfrak{c}_{0}=2\frac{d-1}{d-2}\geq 2$, it holds that

$\frac{2}{1-\varepsilon_{0}}\leq\frac{2\mathfrak{c}_{0}}{\mathfrak{c}_{0}-2(1-\varepsilon_{0})}\leq\frac{2}{\varepsilon_{0}}$

for any $0<\varepsilon_{0}<2^{-100}$. Thus, interpolation with an
appropriately chosen parameter $\theta\in[0,1]$, followed by referencing the
definition (3) of the norm $Y_{N}$, gives that

$\displaystyle\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2\mathfrak{c}_{0}}{\mathfrak{c}_{0}-2(1-\varepsilon_{0})})}(\mathbb{R})}$
$\displaystyle\leq\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}})}(\mathbb{R})}^{1-\theta}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}})}(\mathbb{R})}^{\theta}$
$\displaystyle\lesssim N_{+}^{\frac{1-\theta}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{Y_{N_{+}}(\mathbb{R})}^{1-\theta}N_{+}^{\frac{\theta}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{Y_{N_{+}}(\mathbb{R})}^{\theta}$
$\displaystyle=N_{+}^{-\frac{1}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{Y_{N_{+}}(\mathbb{R})}$

and the proof is finished. ∎

The above lemma is the crucial ingredient for showing that product of three
functions taken from the spaces $Y^{S}$ or $X^{\mathfrak{s}}$ lies in the
space $X^{*,\sigma_{*}}$, which we use to control the non-homogeneity $h$ in
(3). The specific regularity of the product depends on whether the functions
in play belong to $Y^{S}$ or to $X^{\mathfrak{s}}$ and on the respective
regularity exponents $S$ and $\mathfrak{s}$. Since the three-fold product of
functions is estimated in the norm $X^{*,\sigma_{*}}$, defined by duality in
(3), the proof of Proposition 4.2 naturally reduces to showing bounds on a
$4$-linear integral form, formulated in Lemma 4.3. The $4$-linear estimate
(4.3) of Lemma 4.3 in turn can be reduced using the Cauchy-Schwarz inequality
to the bilinear $L^{2}$ estimate of Lemma 4.1.

######  Proposition 4.2.

Fix any

$0<S_{1}\leq S_{2}\leq S_{3}<\mathfrak{s}_{c}<\mathfrak{s}\,,$

and choose any $0<\varepsilon\lesssim_{S_{j},\mathfrak{s}}\\!1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S_{j},\mathfrak{s}}\\!1$. Then for any
$z_{j}$ and $v_{j}$, $j\in\\{1,2,3\\}$, the following estimates hold:

$\displaystyle\big{\|}z_{1}z_{2}z_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|z_{1}\|_{Y^{S_{1}+\varepsilon}(\mathbb{R})}\|z_{2}\|_{Y^{S_{2}+\varepsilon}(\mathbb{R})}\|z_{3}\|_{Y^{S_{3}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with }\sigma_{*}\coloneqq
S_{1}+\min\Big{(}S_{2},\frac{1}{2}\Big{)}+\min\Big{(}S_{3},\frac{1}{2}\Big{)}\,.$
$\displaystyle\big{\|}z_{1}z_{2}v_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|z_{1}\|_{Y^{S_{1}+\varepsilon}(\mathbb{R})}\|z_{2}\|_{Y^{S_{2}+\varepsilon}(\mathbb{R})}\|v_{3}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with }\sigma_{*}\coloneqq
S_{1}+\min\Big{(}S_{2},\frac{1}{2}\Big{)}+\frac{1}{2}\,.$
$\displaystyle\big{\|}z_{1}v_{2}v_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|z_{1}\|_{Y^{S_{1}+\varepsilon}(\mathbb{R})}\|v_{2}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}\|v_{3}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with
}\sigma_{*}\coloneqq\min\big{(}S_{1}+\mathfrak{s},\,S_{1}+1\big{)}\,.$
$\displaystyle\big{\|}v_{1}v_{2}v_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|v_{1}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}\|v_{2}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}\|v_{3}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with
}\sigma_{*}\coloneqq\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c})\,.$

The implicit constants are allowed to depend on $\varepsilon$,
$\varepsilon_{0}$, $\mathfrak{s}$, and $S_{j}$.

We deduce Proposition 4.2 by duality from Lemma 4.3 below, which is proved at
the end of the section.

Before proceeding, we introduce some notation. For $\sigma_{*}$ as in each
case of Proposition 4.2 (detailed below) we choose $v_{*}\in
X^{-\sigma_{*}}(\mathbb{R})$. Next, for any function
$h\in\\{v_{*},z_{j},v_{j}\\}$, $j=1,2,3$, and $N\in 2^{\mathbb{N}}$ we define
quantities

$Z_{N}(h)\coloneqq\begin{cases}\|v_{*}\|_{X_{N}(\mathbb{R})}&\text{if
}h=v_{*}\,,\\\ \|v_{j}\|_{X_{N}(\mathbb{R})}&\text{if }h=v_{j}\,,\\\
\|z_{j}\|_{Y_{N}(\mathbb{R})}&\text{if
}h=z_{j}\,,\end{cases}\qquad\alpha(h)\coloneqq\begin{cases}-\sigma_{*}&\text{if
}h=v_{*}\,,\\\ \mathfrak{s}&\text{if }h=v_{j}\,,\\\ S_{j}&\text{if
}h=z_{j}\,,\end{cases}$

and we denote by $\mathscr{S}(h_{1},\cdots,h_{4})$ the set of permutations of
a quadruple $(h_{1},\cdots,h_{4})$.

######  Lemma 4.3.

Fix exponents as in (4.2), and choose any $0<\varepsilon\lesssim 1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S_{j},\mathfrak{s}}1$. Then for any
$N_{j}\in 2^{\mathbb{N}}$, $j\in\\{1,2,3,4\\}$, we have that

$\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern 0.0pt\mathrm{d}}\mathopen{}x=0$

unless $N_{j}\lesssim\sum_{j^{\prime}\neq j}N_{j^{\prime}}$ for all
$j\in\\{1,2,3,4\\}$. Furthermore,

$\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\lesssim(N_{1}N_{2}N_{3}N_{4})^{\varepsilon}\prod_{j=1}^{4}N_{j}^{\alpha(h_{j})}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}$

provided that one of the following conditions is satisfied:

Case $zzz$:

$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},z_{1},z_{2},z_{3})$ and $\sigma_{*}$
as in (4.2).

Case $zzv$:

$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},z_{1},z_{2},v_{3})$ and $\sigma_{*}$
as in (4.2).

Case $zvv$:

$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},z_{1},v_{2},v_{3})$ and $\sigma_{*}$
as in (4.2).

Case $vvv$:

$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},v_{1},v_{2},v_{3})$ and $\sigma_{*}$
as in (4.2).

###### Proof of Proposition 4.2.

Choose any $h_{j}\in\\{z_{j},v_{j}\\}$, $j=\\{1,2,3\\}$. The definition of
$X^{*,\sigma_{*}}$ and $X_{N}$ in (3), $\mathop{\kern
0.0pt\mathrm{Id}}\mathopen{}=\sum_{N\in 2^{\mathbb{N}}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}$, the triangle inequality, and
$\Big{(}\sum|a_{j}|^{2}\Big{)}^{\nicefrac{{1}}{{2}}}\leq\sum|a_{j}|$ imply
that

$\big{\|}h_{1}h_{2}h_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\begin{aligned}
&=\Big{(}\sum_{N\in 2^{\mathbb{N}}}N^{2\sigma_{*}}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\big{(}h_{1}h_{2}h_{3}\big{)}\big{\|}_{X^{*}_{N}(\mathbb{R})}^{2}\Big{)}^{\nicefrac{{1}}{{2}}}\\\
&\leq\sum_{N,N_{1},N_{2},N_{3}\in
2^{\mathbb{N}}}N^{\sigma_{*}}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\big{(}P_{N_{1}}h_{1}P_{N_{2}}h_{2}P_{N_{3}}h_{3}\big{)}\big{\|}_{X^{*}_{N}(\mathbb{R})}\\\
&\leq\sum_{N,N_{1},N_{2},N_{3}\in
2^{\mathbb{N}}}N^{\sigma_{*}}\sup_{v_{*}}\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}v_{*}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\,,\end{aligned}$

where the upper bound in the last expression is taken over all $v_{*}\in
C^{\infty}_{c}(\mathbb{R}\times\mathbb{R}^{d})$ with $\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}v_{*}\|_{X_{N}}\leq 1$. Note that the
assumptions of Lemma 4.3 on $S_{j}$, $\mathfrak{s}$, $\varepsilon$,
$\varepsilon_{0}$, and $\sigma_{*}$ coincide with those of Proposition 4.2,
and therefore (4.3) with $\varepsilon$ replaced by $\frac{\varepsilon}{8}$
yields

$\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}v_{*}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\begin{aligned}
&\lesssim(N_{1}N_{2}N_{3}N)^{\frac{\varepsilon}{8}}\prod_{j=1}^{3}N_{j}^{\alpha(h_{j})}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}\\\
&=(N_{1}N_{2}N_{3})^{-\frac{3\varepsilon}{8}}N^{\frac{\varepsilon}{8}}\prod_{j=1}^{3}N_{j}^{\alpha(h_{j})+\varepsilon/2}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}\,.\end{aligned}$

We may assume that $N\lesssim\max_{j}N_{j}$ since otherwise, by (4.3), the
left-hand side vanishes. We have that
$(N_{1}N_{2}N_{3})^{-\frac{3\varepsilon}{8}}N^{\frac{\varepsilon}{8}}\lesssim
N^{-\frac{\varepsilon}{4}}$, and consequently

$\displaystyle\sum_{N,N_{1},N_{2},N_{3}\in
2^{\mathbb{N}}}(N_{1}N_{2}N_{3})^{-\frac{3\varepsilon}{8}}N^{\frac{\varepsilon}{8}}\prod_{j=1}^{3}N_{j}^{\alpha(h_{j})+\varepsilon}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}$ $\displaystyle\hskip
100.00015pt\lesssim\sum_{N\in
2^{\mathbb{N}}}N^{-\frac{\varepsilon}{4}}\prod_{j=1}^{3}\Big{(}\sum_{N_{j}\in
2^{\mathbb{N}}}N_{j}^{\alpha(h_{j})+\varepsilon/2}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}\Big{)}\,.$

Using the Cauchy-Schwarz inequality and the definitions of $Z_{N_{j}}(h_{j})$,
$\alpha(h_{j})$, $X^{\mathfrak{s}}$, and $Y^{S}$ we have

$\displaystyle\sum_{N_{j}\in
2^{N}}N^{\alpha(h_{j})+\varepsilon/2}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})$
$\displaystyle\lesssim\Big{(}\sum_{N\in
2^{\mathbb{N}}}N_{j}^{-\varepsilon}\Big{)}^{1/2}\Big{(}\sum_{N\in
2^{N}}N_{j}^{2\alpha(h)+2\varepsilon}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}^{2}\Big{)}^{1/2}$
$\displaystyle\lesssim_{\varepsilon}\begin{cases}\|z_{j}\|_{Y^{S_{j}+\varepsilon}(\mathbb{R})}&\text{if
}h=z_{j}\,,\\\ \|v_{j}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}&\text{if
}h=v_{j}\,.\end{cases}$

∎

###### Proof of Lemma 4.3.

Note that the right-hand sides of (4.3) and (4.3), as well as the subsequent
conditions are the same if we exchange $P_{N_{j}}h_{j}$ with
$P_{N_{j^{\prime}}}h_{j^{\prime}}$ $j,j^{\prime}\in\\{1,2,3,4\\}$, and
therefore we assume without loss of generality, that $N_{1}\geq N_{2}\geq
N_{3}\geq N_{4}$.

First, we prove an orthogonality observation (4.3). For a contradiction
suppose that $N_{1}\lesssim N_{2}+N_{3}+N_{4}$ does not hold, and in
particular, suppose that $N_{1}\geq 2^{5}N_{2}$. By (2) we have $\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1})\big{)}\subset B_{N_{1}}\setminus
B_{2^{-2}N_{1}}$, since $N_{1}\geq 2^{5}N_{2}\geq 2^{5}>1$. On the other hand,
since $\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(h_{2}h_{3}h_{4})=\widehat{h_{2}}*\widehat{h_{3}}*\widehat{h_{4}}$,
it holds that

$\mathop{\kern 0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(h_{2}h_{3}h_{4})\big{)}\subset\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{2}}\big{)}+\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{3}}\big{)}+\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{4}}\big{)},$

and since $N_{2}\geq N_{3}\geq N_{4}$, $N_{1}\geq 2^{5}N_{2}$, and
$\mathop{\kern 0.0pt\mathrm{spt}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}h\big{)}\Big{)}\subset B_{N}$ by (2), we have

$\mathop{\kern 0.0pt\mathrm{spt}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\big{)}\Big{)}\subset
B_{N_{2}}+B_{N_{3}}+B_{N_{4}}\subset B_{2^{3}N_{2}}\subset B_{2^{-2}N_{1}}\,,$

Thus (4.3) holds by Plancherel’s identity.

We henceforth suppose that $N_{1}\approx N_{2}$ and we proceed to prove bound
(4.3). To estimate the left-hand side of (4.3) we apply the Cauchy-Schwarz
inequality pairing one function with high frequency ($N_{1}$ or $N_{2}$) with
a function with a lower frequency ($N_{3}$ or $N_{4}$). This gives the bound

$\displaystyle\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern 0.0pt\mathrm{d}}\mathopen{}x\Big{|}$
$\displaystyle\hskip 50.00008pt\leq\min\Big{(}\begin{aligned} &\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})},\\\
&\qquad\|\mathop{\kern 0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{4}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\Big{)}.\end{aligned}$

By definitions in (4) and Lemma 4.1 for any $l<l^{\prime}\in\\{1,2,3,4\\}$ it
holds that

$\displaystyle\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{l}}h_{l}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{l^{\prime}}}h_{l^{\prime}}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim
N_{l}^{\varepsilon-\nicefrac{{1}}{{2}}-\alpha(h_{l})}N_{l^{\prime}}^{\nicefrac{{1}}{{2}}-\alpha(h_{l^{\prime}})+\beta(h_{l},h_{l^{\prime}})}$
$\displaystyle\hskip
50.00008pt\times\prod_{j\in\\{l,l^{\prime}\\}}N_{j}^{\alpha(h_{j})}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})\,,$

where

$\beta(h_{l},h_{l^{\prime}})\coloneqq\begin{cases}\mathfrak{s}_{c}&\text{if
}h_{l},h_{l^{\prime}}\in\\{v_{*},v_{1},v_{2},v_{3}\\}\,,\\\
0&\text{otherwise}.\end{cases}$

Hence, since $N_{1}\approx N_{2}$, we obtain

$\displaystyle\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\leq
N_{1}^{2\varepsilon-1-\alpha(h_{1})-\alpha(h_{2})}N_{3}^{\nicefrac{{1}}{{2}}-\alpha(h_{3})}N_{4}^{\nicefrac{{1}}{{2}}-\alpha(h_{3})}$
$\displaystyle\hskip
50.00008pt\times\min\Big{(}N_{3}^{\beta(h_{2},h_{3})}N_{4}^{\beta(h_{1},h_{4})},N_{3}^{\beta(h_{1},h_{3})}N_{4}^{\beta(h_{2},h_{4})}\Big{)}\prod_{j=1}^{4}N_{j}^{\alpha(h_{j})}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})\,.$

By homogeneity of the required bound (4.3) we can assume, without loss of
generality, that $N_{j}^{\alpha(h_{j})}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})=1$, $j\in\\{1,2,3,4\\}$. Thus,
recalling that $N_{1}\approx N_{2}$, we reduce (4.3) to proving

(4.11)
$N_{3}^{\nicefrac{{1}}{{2}}-\alpha(h_{3})-\varepsilon}N_{4}^{\nicefrac{{1}}{{2}}-\alpha(h_{4})-\varepsilon}\min\Big{(}N_{3}^{\beta(h_{2},h_{3})}N_{4}^{\beta(h_{1},h_{4})},N_{3}^{\beta(h_{1},h_{3})}N_{4}^{\beta(h_{2},h_{4})}\Big{)}\\\
\lesssim N_{1}^{1+\alpha(h_{1})+\alpha(h_{2})}$

We first claim that $\alpha(h_{1})+\alpha(h_{2})+1\geq 0$. Indeed,
$\alpha(h)<0$ only if $h=v_{*}$, in which case $\alpha(h)=-\sigma_{*}$.
Assuming, without loss of generality, that $h_{1}=v_{*}$, in each of the cases
zzz, zzv, or zvv, we replace the minimum in (4.2), in (4.2), or in (4.2) by
$\frac{1}{2}$ or by $S+1$ to obtain that

$\sigma_{*}\leq S_{1}+1\leq S_{j}+1\leq\mathfrak{s}+1,\qquad j\in\\{1,2,3\\};$

the claim follows. Finally,in case vvv we have $h_{2}=v$, and therefore
$\alpha(h_{2})=\mathfrak{s}$. Consequently,

$\sigma_{*}=\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c})\leq\mathfrak{s}+1,$

and the claim follows.

Thus, since $N_{1}\geq N_{3}$ it is sufficient to show (4.11) for
$N_{1}=N_{3}$, that is, to show

$\displaystyle\min\Big{(}N_{3}^{\beta(h_{2},h_{3})}N_{4}^{\beta(h_{1},h_{4})},N_{3}^{\beta(h_{1},h_{3})}N_{4}^{\beta(h_{2},h_{4})}\Big{)}$
$\displaystyle\lesssim
N_{3}^{\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})}N_{4}^{-\nicefrac{{1}}{{2}}+\alpha(h_{4})}$

After standard algebraic manipulations, (4) follows if we show that for any
$N_{3}\geq N_{4}$ either

$N_{4}^{\beta(h_{1},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})}\lesssim
N_{3}^{\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{2},h_{3})}$

or

$N_{4}^{\beta(h_{2},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})}\lesssim
N_{3}^{\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{1},h_{3})}\,.$

Since $N_{3}\geq N_{4}$ are arbitrary, it suffices to show that either

$\max\\{0,\beta(h_{1},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})\\}\leq\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{2},h_{3})$

or

$\max\\{0,\beta(h_{2},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})\\}\leq\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{1},h_{3})$

holds. The expressions above can be rewritten in more symmetric form as

$\beta(h_{2},h_{3})+\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},\beta(h_{1},h_{4})\\}\leq\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})$

or

$\beta(h_{1},h_{3})+\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},\beta(h_{2},h_{4})\\}\leq\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})$

Next, we discuss each case separately.

Case $zzz$. Then,

$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+S_{1}+S_{2}+S_{3}$

and $\beta(h_{j},h_{k})=0$ for any $j\neq k$, and therefore (4) is equivalent
to

$\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},0\\}\leq-\sigma_{*}+S_{1}+S_{2}+S_{3}\,.$

Clearly the left hand side is largest if $\alpha(h_{4})=S_{3}$, and
consequently (4) holds if

$\sigma_{*}\leq
S_{1}+S_{2}+S_{3}-\max\\{S_{3}-\nicefrac{{1}}{{2}},0\\}=S_{1}+S_{2}+\min\\{S_{3},\nicefrac{{1}}{{2}}\\}\,,$

which holds by (4.2).

Case $zzv$. Then,

$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+S_{1}+S_{2}+\mathfrak{s}$

and either $\beta(h_{2},h_{3})=0$ or $\beta(h_{1},h_{3})=0$. Suppose
$\beta(h_{2},h_{3})=0$, the case $\beta(h_{1},h_{3})=0$ follows analogously by
proving (4) instead of (4). Then, (4) is equivalent to

$\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},\beta(h_{1},h_{4})\\}\leq-\sigma_{*}+S_{1}+S_{2}+\mathfrak{s}\,.$

Since $\alpha(h_{4})\leq\mathfrak{s}$ and
$\beta(h_{1},h_{4})\leq\mathfrak{s}_{c}$, then (4) follows if we show that

$\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\mathfrak{s}_{c}\\}\leq-\sigma_{*}+S_{1}+S_{2}+\mathfrak{s}\,,$

which is equivalent to

$\sigma_{*}\leq
S_{1}+S_{2}+\mathfrak{s}+\min\\{\nicefrac{{1}}{{2}},\mathfrak{s}-\mathfrak{s}_{c}\\}\,,$

that follows from (4.2).

Case $zvv$. Then,

$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+S_{1}+\mathfrak{s}+\mathfrak{s}\,.$

We discuss three cases. If $h_{4}=v$, then $z_{1}\in\\{h_{1},h_{2},h_{3}\\}$
and consequently either $\beta(h_{2},h_{3})=0$ or $\beta(h_{1},h_{3})=0$.
Suppose $\beta(h_{2},h_{3})=0$, the case $\beta(h_{1},h_{3})=0$ follows
analogously by proving (4) instead of (4). Then, (4) is equivalent to

$\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\beta(h_{1},h_{4})\\}\leq-\sigma_{*}+S_{1}+2\mathfrak{s}\,.$

Since $\beta(h_{1},h_{4})\leq\mathfrak{s}_{c}$, then (4) follows if we show
that

$\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\mathfrak{s}_{c}\\}\leq-\sigma_{*}+S_{1}+2\mathfrak{s}\,,$

which is equivalent to

$\sigma_{*}\leq
S_{1}+\mathfrak{s}+\min\\{\nicefrac{{1}}{{2}},\mathfrak{s}-\mathfrak{s}_{c}\\}\,,$

that follows from (4.2).

If $h_{4}=z_{1}$, then $\beta(h_{1},h_{4})=0$,
$\beta(h_{2},h_{3})\leq\mathfrak{s}_{c}$, and $\alpha(h_{4})=S_{1}$ implies
that (4) holds if

$\mathfrak{s}_{c}+\max\\{S_{1}-\nicefrac{{1}}{{2}},0\\}\leq-\sigma_{*}+S_{1}+2\mathfrak{s}\,.$

After standard algebraic manipulations, the latter is equivalent to

$\sigma_{*}\leq
2\mathfrak{s}-\mathfrak{s}_{c}+\min\\{\nicefrac{{1}}{{2}},S_{1}\\}\,,$

which follows from (4.2).

If $h_{4}=v_{*}$, then $\alpha(h_{4})-\nicefrac{{1}}{{2}}<0$, and either
$\beta(h_{2},h_{3})=0$ or $\beta(h_{1},h_{3})=0$. As above, we suppose
$\beta(h_{2},h_{3})=0$ and we note that (4) follows if we show

$\mathfrak{s}_{c}\leq-\sigma_{*}+S_{1}+2\mathfrak{s},$

or equivalently

$\sigma_{*}\leq S_{1}+2\mathfrak{s}-\mathfrak{s}_{c},$

which follows from (4.2).

Case $vvv$. Then,

$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+3\mathfrak{s}$

and $\beta(h_{j},h_{k})\leq\mathfrak{s}_{c}$ for each $j\neq k$. Then, (4)
follows if we show

$\mathfrak{s}_{c}+\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\mathfrak{s}_{c}\\}\leq-\sigma_{*}+3\mathfrak{s}\,.$

The latter is equivalent to

$\sigma_{*}\leq
2(\mathfrak{s}-\mathfrak{s}_{c})+\min\\{\nicefrac{{1}}{{2}}+\mathfrak{s}_{c},\mathfrak{s}\\}\,,$

and (4) follows from (4.2). ∎

We conclude this section by a crucial estimate that justifies the regularity
computation encoded by the defintion (1.3) of $\mu(k,S)$

######  Lemma 4.4 (Inductive property of $\mu(k,S)$).

For any $S>0$ and any $k_{1},k_{2},k_{3}\in\mathbb{N}\setminus\\{0\\}$, with
$k_{1}\leq k_{2}\leq k_{3}$, it holds that

$\mu(k_{1}+k_{2}+k_{3},S)\leq\mu(k_{1},S)+\min\big{(}\mu(k_{2},S),\nicefrac{{1}}{{2}}\big{)}+\min\big{(}\mu(k_{3},S),\nicefrac{{1}}{{2}}\big{)}.$

Furthermore, setting $k\coloneqq k_{1}+k_{2}+k_{3}$
$0<\varepsilon\lesssim_{k}1$, and any
$0<\varepsilon_{0}\lesssim_{k,S,\varepsilon}1$. Then for any functions
$z_{j}$, $j\in\\{1,2,3\\}$, it holds that

$\|z_{1}z_{2}z_{3}\|_{X^{*,\mu(k,S)}}\lesssim\prod_{j=1}^{3}\|z_{j}\|_{Y^{\mu(k_{j},S)+\varepsilon}}\,,$

where the implicit constant depends on $k$, $\varepsilon$, $\varepsilon_{0}$,
and $S$.

###### Proof of Lemma 4.4.

Without loss of generality, suppose $k_{1}\leq k_{2}\leq k_{3}$. Let

$J\coloneqq\mu(k_{1},S)+\min\big{(}\mu(k_{2},S),\nicefrac{{1}}{{2}}\big{)}+\min\big{(}\mu(k_{3},S),\nicefrac{{1}}{{2}}\big{)}.$

Trivially, we have $\mu(|k_{j}|,S)\geq S$, $j\in\\{1,2,3\\}$. If
$\mu(k_{3},S)\geq\mu(k_{2},S)\geq\frac{1}{2}$ then

$J\geq S+\frac{1}{2}+\frac{1}{2}=S+1\geq\mu(k_{1}+k_{2}+k_{3},S)\,,$

and (4.4) follows. If $\mu(k_{3},S)\geq\frac{1}{2}\geq\mu(k_{2},S)$, then

$J\geq S+S+\frac{1}{2}=2S+\frac{1}{2}\geq\mu(k_{1}+k_{2}+k_{3},S)\,,$

and (4.4) follows. Finally, if $\frac{1}{2}\geq\mu(k_{3},S)\geq\mu(k_{2},S)$
then

$J=\mu(k_{1},S)+\mu(k_{2},S)+\mu(k_{3},S)$

and in addition $\mu(k_{j},S)=k_{j}S$, $j\in\\{1,2,3\\}$ since
$\min(S+1,2S+\frac{1}{2})>\frac{1}{2}$ and $k_{1}\leq k_{2}$. Then we deduce
that

$J=\big{(}k_{1}+k_{2}+k_{3}\big{)}S\geq\mu(k_{1}+k_{2}+k_{3},S),$

and (4.4) follows.

Finally, bound (4.4) follows from bound (4.2) with $S$ replaced by
$S+\varepsilon$. This completes the proof. ∎

##  5\. The deterministic fixed point

This section we prove Theorem 1.5.

Recall that solution $u$ to (1) is a fixed point of the map
$u\mapsto\mathbb{I}_{f}(|u|^{2}u)$ with $\mathbb{I}_{f}$ given by (1). Having
assumed that $u$ is a function of the form given by (1.5) for some
$M\in\mathbb{N}$, it follows by direct substitution that $u$ is a solution to
(1) if and only if the remainder term $u^{\\#}_{M}\coloneqq u-z_{\leq M}$,
with $z_{\leq M}\coloneqq\sum_{k\leq M}z_{k}$, is fixed point of the iteration
map
$u^{\\#}_{M}\mapsto\mathcal{J}_{z,M}(u^{\\#}_{M})\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$, given by (1.3).

The boundedness of the map $\mathbb{I}_{0}$ from
$X^{*,\mathfrak{s}_{*}}(\mathbb{R})$ to $X^{*,\mathfrak{s}_{*}}(\mathbb{R})$,
for some $\mathfrak{s}_{*}>\mathfrak{s}$, has already been established in
Proposition 3.1. Thus, the proof of Theorem 1.5 relies on estimates on
$\Phi_{z_{\leq M}}[v]$ and $\big{[}z,z,z\big{]}_{>M}$ in the
$X^{*,\mathfrak{s}_{*}}([0,T])$ norm for some $\mathfrak{s}_{*}>\mathfrak{s}$
and for small enough intervals $T>0$.

In Lemma 5.1 we establish bounds on the Lipschitz constant of the nonlinear
map $X^{\mathfrak{s}}(I)\ni v\mapsto\Phi_{z}[v]\in
X^{*,\mathfrak{s}_{*}}\big{(}I\big{)}$ for an interval $I\subset\mathbb{R}$.
This will allow us to prove Theorem 1.5 using the uniqueness of a fixed point
of $v\mapsto\mathcal{I}(v)$ by using the Banach fixed point theorem.

######  Lemma 5.1.

Fix $0<S<\mathfrak{s}_{c}<\mathfrak{s}<\mathfrak{s}_{c}+\nicefrac{{1}}{{2}}$.
Choose $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,\mathfrak{s}}1$. For any interval
$I\subseteq\mathbb{R}$, any $z\in Y^{S}(I)$ and $v_{1},v_{2}\in
X^{\mathfrak{s}}(I)$ and any

$\mathfrak{s}_{*}<\min\big{(}\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\nicefrac{{1}}{{2}},\,S+1,S+\mathfrak{s}\big{)}-\varepsilon\,,$

it holds that

$\big{\|}\Phi_{z}[v_{1}]\big{\|}_{X^{*,\mathfrak{s}_{*}}(I)}\lesssim\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}\|_{X^{\mathfrak{s}}(I)},$
$\Big{\|}\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]\Big{\|}_{X^{*,\mathfrak{s}_{*}}(I)}\lesssim\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|v_{2}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}-v_{2}\|_{X^{\mathfrak{s}}(I)}\,,$

and

$\Big{\|}\Phi_{z}[v]-\Phi_{\widetilde{z}}[v]\Big{\|}_{X^{*,\mathfrak{s}_{*}}(I)}\lesssim\big{(}\|v\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}+\|\widetilde{z}\|_{Y^{S}(I)}\big{)}^{2}\|z-\widetilde{z}\|_{Y^{S}(I)}.$

In particular, if
$\mathfrak{s}<\min\big{(}2S+\nicefrac{{1}}{{2}},\,S+1\big{)}$, then for any
sufficiently small $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$, (5.1) holds with
$\mathfrak{s}_{*}=\mathfrak{s}+\varepsilon$.

We remark that the implicit constants may depend on $S$, $\mathfrak{s}$,
$\mathfrak{s}_{*}$, $\varepsilon$, and $\varepsilon_{0}$ but not on $I$, $z$,
$v_{1}$ or $v_{2}$.

###### Proof of Lemma 5.1.

Without loss of generality, we assume that $I=\mathbb{R}$. The general
statement follows by replacing $z$ with $z\mathbbm{1}_{I}$ and using that
$\|z\|_{Y^{S}(I)}=\|z\mathbbm{1}_{I}\|_{Y^{S}(\mathbb{R})}$.

By (1.3) we have $\Phi_{z}[0]=0$, and therefore (5.1) follows from (5.1) after
setting $v_{2}=0$. Thus we concentrate on proving (5.1). Using the expression
(1.3) for $\Phi_{z}[v]$ it follows that

$\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]=G_{1}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}(v_{1}-v_{2})+G_{2}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}(\overline{v_{1}-v_{2}})\,,$

where $G_{j}[z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}]$,
$j\in\\{1,2\\}$, are homogeneous polynomials of degree 2 in the variables $z$,
$\overline{z}$, $v_{1}$, $\overline{v_{1}}$, $v_{2}$, $\overline{v_{2}}$.
Assume $\varepsilon<\min\big{(}S,\mathfrak{s}-\mathfrak{s}_{c}\big{)}$ and
observe that

$\displaystyle\mathfrak{s}_{*}<\min\big{(}2S+\nicefrac{{1}}{{2}},\,S+1\big{)}-\varepsilon,$
$\displaystyle\mathfrak{s}_{*}<\min\big{(}S+\mathfrak{s},\,S+1\big{)}-\varepsilon,$
$\displaystyle\mathfrak{s}_{*}<\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c})-\varepsilon.$

Then, (4.2), (4.2), and (4.2) with $\mathfrak{s}-\varepsilon/3$ and
$S_{j}-\varepsilon/3$ replacing, respectively, $\mathfrak{s}$ and $S_{j}$,
$j=1,2,3$ imply

$\displaystyle\Big{\|}\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]\Big{\|}_{X^{*,\mathfrak{s}_{*}}(\mathbb{R})}$
$\displaystyle\leq\begin{aligned}
&\Big{\|}G_{1}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}\big{(}v_{1}-v_{2}\big{)}\Big{\|}_{X^{*,\mathfrak{s}_{*}}(\mathbb{R})}\\\
&\qquad+\Big{\|}G_{2}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}\big{(}\overline{v_{1}-v_{2}}\big{)}\Big{\|}_{X^{*,\mathfrak{s}_{*}}(\mathbb{R})}.\end{aligned}$
$\displaystyle\lesssim\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}-v_{2}\|_{X^{\mathfrak{s}}(I)}\,,$

as desired.

Bound (5.1) follows analogously by representing

$\Phi_{z}[v]-\Phi_{\widetilde{z}}[v]=\Psi_{1}\big{[}z,\overline{z},\widetilde{z},\overline{\widetilde{z}},v,\overline{v}\big{]}(z-\widetilde{z})+\Psi_{2}\big{[}z,\overline{z},\widetilde{z},\overline{\widetilde{z}},v,\overline{v}\big{]}\overline{(z-\widetilde{z})}$

with
$\Psi_{1}\big{[}z,\overline{z},\widetilde{z},\overline{\widetilde{z}},v,\overline{v}\big{]}$,
$j\in\\{1,2\\}$, are homogeneous polynomials of degree 2 in the input
variables. ∎

Next, we define the time-delayed version of the iteration map
$v\mapsto\mathbb{I}_{0}\big{(}\Phi_{z}[v]+h\big{)}$ with non-trivial intial
data. Define for any $t_{0}>0$ the map

$\mathcal{K}(v):=e^{i(t-t_{0})\Delta}v_{0}\mp
i\int_{t_{0}}^{t}e^{i(t-s)\Delta}(\Phi_{z}[v]+h)\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}s$

In the following lemma we show that $v\mapsto\mathcal{K}(v)$ is a contraction
on $X^{\mathfrak{s}}([t_{0},t_{1}))$ if the time interval $[t_{0},t_{1})$ is
short enough, or if $v_{0}$,$z$, and $h$ are small in appropriate norms.

######  Lemma 5.2.

Let $I$ with $I\ni t_{0}$ be a time interval. Fix
$0<S<\mathfrak{s}_{c}<\mathfrak{s}<\mathfrak{s}_{c}+\nicefrac{{1}}{{2}}$.
Choose $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,\mathfrak{s}}1$. Then there exist
constants $C=C(\varepsilon,\varepsilon_{0},S,\mathfrak{s})>0$ and
$c=c(\varepsilon,\varepsilon_{0},S,\mathfrak{s})>0$ such that for any $z\in
Y^{S}(I)$, $h\in X^{\mathfrak{s}+\varepsilon}(I)$, and $v_{1},v_{2}\in
X^{\mathfrak{s}}(I)$ the following bounds hold:

$\Big{\|}\mathcal{K}(v_{1})\Big{\|}_{X^{\mathfrak{s}}(I)}\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\begin{aligned}
\Big{(}&\|v_{0}\|_{H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})}+\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}\\\
&\quad+\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}\|_{X^{\mathfrak{s}}(I)}\Big{)}\end{aligned}$

and

$\displaystyle\Big{\|}\mathcal{K}(v_{1})-\mathcal{K}(v_{2})\Big{\|}_{X^{\mathfrak{s}}(I)}$
$\displaystyle\qquad\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|v_{2}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\big{\|}v_{1}-v_{2}\big{\|}_{X^{\mathfrak{s}}(I)}\,.$

As a consequence:

  * •

For any $v_{0}\in H^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})$, $z\in
Y^{S}(I)$, $h\in X^{*,\mathfrak{s}+\varepsilon}(I)$, and $\delta_{0}>0$ we
define

$\displaystyle
T_{\delta_{0}}\big{(}\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})},\|z\|_{Y^{S}(I)},\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}\big{)}$
$\displaystyle\quad\coloneqq\frac{1}{2^{100}(C\delta_{0})^{\nicefrac{{1}}{{c}}}}\big{\langle}\delta_{0}+\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})}+\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(\mathbb{R})}+\|z\|_{Y^{S}(\mathbb{R})}\big{\rangle}^{-\nicefrac{{3}}{{c}}}\,.$

Then for any interval $J=[t_{0},t_{1}]\subseteq I$ with

$|J|<T_{\delta_{0}}\big{(}\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})},\|z\|_{Y^{S}(I)},\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}\big{)}$

the map $v\mapsto\mathcal{K}(v)$ is a contraction on the set

$\overline{\mathcal{B}}_{\delta_{0}}^{J}\coloneqq\Big{\\{}v\in
X^{\mathfrak{s}}(J)\colon\|v\|_{X^{\mathfrak{s}}(J)}\leq\delta_{0}\Big{\\}}\,,$

and thus it admits a unique fixed point in
$\overline{\mathcal{B}}_{\delta_{0}}^{J}$.

  * •

If $I=\mathbb{R}$, there exists $\delta_{0}>0$ such that if

$\|z\|_{Y^{S}(\mathbb{R})}\leq\frac{\delta_{0}}{2C},\qquad\|v_{0}\|_{H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})}\leq\frac{\delta_{0}}{2C},\qquad\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(\mathbb{R})}\leq\frac{\delta_{0}}{2C},$

then $v\mapsto\mathcal{K}(v)$ is a contraction on the set

$\overline{\mathcal{B}}_{\delta_{0}}^{\mathbb{R}}\coloneqq\Big{\\{}v\in
X^{\mathfrak{s}}(\mathbb{R})\colon\|v\|_{X^{\mathfrak{s}}(\mathbb{R})}\leq\delta_{0}\Big{\\}}\,,$

and thus admits a unique fixed point in
$\overline{\mathcal{B}}_{\delta_{0}}^{\mathbb{R}}$.

###### Proof of Lemma 5.2.

Assume that $\varepsilon>0$ is chosen small, and in particular that it
satisfies

$\varepsilon<\frac{1}{2}\min\big{(}S,2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\nicefrac{{1}}{{2}}-\mathfrak{s},S+1-\mathfrak{s}\big{)}.$

Fix $z\in Y^{S}(I)$, $h\in X^{\mathfrak{s}+\varepsilon}(I)$, and
$v_{1},v_{2}\in X^{\mathfrak{s}}(I)$. Then, by (3.1)

$\displaystyle\Big{\|}\mathcal{K}(v_{1})\Big{\|}_{X^{\mathfrak{s}}(I)}\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\Big{(}\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})}+\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}+\big{\|}\Phi_{z}[v_{1}]\big{\|}_{X^{*,\mathfrak{s}+\varepsilon}(I)}\Big{)}\,.$

Since
$\mathfrak{s}+\varepsilon<\min\big{(}\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c}),\,2S+\nicefrac{{1}}{{2}},S+1,S+\mathfrak{s}\big{)}-\varepsilon$
for $\varepsilon>0$ small enough, (5.2) follows from (5.1) with
$s_{*}=\mathfrak{s}+\varepsilon$. Next, (3.1) implies

$\Big{\|}\mathcal{K}(v_{1})-\mathcal{K}(v_{2})\Big{\|}_{X^{\mathfrak{s}}(I)}\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\big{\|}\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]\big{\|}_{X^{*,\mathfrak{s}+\varepsilon}(I)}.$

and (5.1) with $s_{*}=\mathfrak{s}+\varepsilon$ yields (5.2).

Assuming, without loss of generality, that $C>1>c>0$, direct computation
allows us to deduce from (5.2) that $\mathcal{K}$ maps the sets
$\overline{\mathcal{B}}_{\delta_{0}}^{J}$ and
$\overline{\mathcal{B}}_{\delta_{0}}^{\mathbb{R}}$ to themselves under the
corresponding assumptions on $\delta_{0}$, $z$, and $h$. Similarly, bound
(5.2) shows that $\mathcal{K}$ is a contraction on these sets. The existence
and uniqueness of the fixed point follows from the Banach contraction mapping
principle. ∎

Finally, let us record the bound on $[z,z,z]_{>M}$ appearing in the defintion
of the iteration map
$\mathcal{J}_{z,M}(v)\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$.

######  Lemma 5.3.

Fix $0<S<\mathfrak{s}_{c}$, $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$, and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,\mathfrak{s}}1$. Then it holds that

$\big{\|}[z,z,z]_{>M}\big{\|}_{X^{*,\mu(M+1,S)}(\mathbb{R})}\lesssim\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}$

and the the map

$\vec{z}_{M}=(z_{k})_{k\leq M}\mapsto[z,z,z]_{>M}$

is Lipschitz on bounded sets with a Lipschitz constant bounded by

$C\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{2}$

for some $C>0$ with respect to distance introduced in (1.5).

###### Proof of Lemma 5.3.

To prove (5.3) we note that, according to (1.3) it holds that

$[z,z,z]_{>M}=\sum_{\mathclap{\qquad\begin{subarray}{c}\hskip
3.5ptk_{1}+k_{2}+k_{3}\geq M+1\\\ k_{j}\leq
M\end{subarray}}}z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}\,,$

Since the sum above is finite, we estimate each summand separately in the
space $X^{*,\mu(M+1,S)}(\mathbb{R})$. Without loss of generality, assume that
$k_{1}\leq k_{2}\leq k_{3}$ and use (4.4) to obtain the desired claim.

To prove Lipschitz regularity it is sufficient to note that

$[z,z,z]_{>M}-[\widetilde{z},\widetilde{z},\widetilde{z}]_{>M}=\sum_{\mathclap{\qquad\begin{subarray}{c}\hskip
3.5ptk_{1}+k_{2}+k_{3}\geq M+1\\\ k_{j}\leq
M\end{subarray}}}\Big{(}(z_{k_{1}}-\widetilde{z}_{k_{1}})\overline{z_{k_{2}}}z_{k_{3}}+\begin{aligned}
&\widetilde{z}_{k_{1}}\overline{(z_{k_{2}}-\widetilde{z}_{k_{2}})}z_{k_{3}}\\\
&+\widetilde{z}_{k_{1}}\overline{\widetilde{z}_{k_{2}}}(z_{k_{3}}-\widetilde{z}_{k_{3}})\Big{)}\,,\end{aligned}$

and apply the same reasoning as above. ∎

We are now ready to prove Theorem 1.5.

###### Proof of Theorem 1.5.

We suppose that $\varepsilon>0$ is small enough, and in particular it
satisfies

$\varepsilon<\frac{1}{3}\min\Big{(}S,2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\frac{1}{2}-\mathfrak{s},S+1-\mathfrak{s},\mu(M+1,S)-\mathfrak{s}\Big{)}.$

Let $C,c$ be constants from Lemma 5.2.

According to the discussion at the beginning of this section, $u$ is a
solution to (1) on $[0,T)$ if and only if $u^{\\#}_{M}$ is a fixed point of
the iteration map
$u^{\\#}_{M}\mapsto\mathcal{J}_{z,M}(u^{\\#}_{M})\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$ given by (1.3). Henceforth, we use $v$ as a
shorthand for the remainder term $u^{\\#}_{M}$.

Local existence of solutions. Let $T=T_{\delta_{0}=1}$, where $T_{\delta_{0}}$
is as in (• ‣ 5.2). Note that

$\mathcal{J}_{z,M}(v)\coloneqq\mathcal{K}(v)$

with $v_{0}=0$, $z=z_{\leq M}$, $h=[z,z,z]_{>M}$, and $t_{0}=0$. Thus
existence of fixed point in $X^{\mathfrak{s}}([0,T))$ follows from Lemma 5.2.
The bound from below on the time of existence follows from Lemma 5.2, from the
bounds (5.3) from Lemma 5.3, which gives

$\big{\|}[z,z,z]_{>M}\big{\|}_{X^{*,\mu(M+1,S)}(\mathbb{R})}\lesssim\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}\,,$

and from the fact that

$\|z_{\leq M}\|_{Y^{S}(I)}\leq
M\max_{k\in\\{1,\ldots,M\\}}\|z_{k}\|_{Y^{\mu(k,S)}(I)}.$

The bound on
$\|v\|_{C^{0}\big{(}[0,T);H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})\big{)}}$
follows from the next part of the proof.

Global existence of solutions. Analogously to the proof above, note that

$\mathcal{J}_{z,M}(v)\coloneqq\mathcal{K}(v)$

with $v_{0}=0$, $z=z_{\leq M}$, $h=[z,z,z]_{>M}$, and $t_{0}=0$. Thus
existence of fixed point in $X^{\mathfrak{s}}([0,+\infty))$ again follows from
Lemma 5.2. Indeed, the bounds above show that $\|z_{\leq M}\|_{Y^{S}(I)}$ and
$\big{\|}[z,z,z]_{>M}\big{\|}_{X^{*,\mu(M+1,S)}(\mathbb{R})}$ can be made
arbitrarily small as long as $\delta_{0}$ in (Global existence of solutions: )
is chosen small enough. Note that the value of $\delta_{0}$ here is chosen
differently, smaller, than the value of $\delta_{0}$ appearing in Lemma 5.2.

Time-continuity of the solution We show that if $v\in X^{\mathfrak{s}}(I)$ for
any interval $I\subset[0,T_{0})$, then $\mathcal{J}_{z,M}(v)\in
C^{0}\big{(}I,H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})\big{)}$.

Indeed, from Proposition 3.3, we have

$\displaystyle\|\mathcal{J}_{z,M}(v)\|_{C^{0}(I,H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d}))}\lesssim$
$\displaystyle\begin{aligned} &\big{\|}\Phi_{z_{\leq
M}}[v]\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\\\
&+\big{\|}[z,z,z]_{>M}\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\,.\end{aligned}$
$\displaystyle\lesssim$ $\displaystyle\begin{aligned}
&\big{(}\|v\|_{X^{\mathfrak{s}}(I)}+M\max_{k\in\\{1,\ldots,M\\}}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(I)}\big{)}^{3},\end{aligned}$

where we used the bounds

$\displaystyle\big{\|}[z,z,z]_{>M}\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\lesssim\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}\,.$
$\displaystyle\big{\|}\Phi_{z_{\leq
M}}[v]\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\lesssim\big{(}\|v\|_{X^{\mathfrak{s}}(I)}+M\max_{k\in\\{1,\ldots,M\\}}\|z_{k}\|_{Y^{\mu(k,S)}(I)}\big{)}^{2}\|v\|_{X^{\mathfrak{s}}(I)}$

from (5.3) of Lemma 5.3 and from (5.1) of Lemma 5.1, respectively. The
assumptions of the latter are satisfied since because (5) implies that

$\mathfrak{s}+2\varepsilon<\min\big{(}S+\mathfrak{s},\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\frac{1}{2},\,S+1\big{)}-\varepsilon\,.$

This concludes the proof of our claim.

Uniqueness of the solution and Blow-up criterion. Next, we show the uniqueness
of the fixed point without any a-priori assumption on the smallness of its
$X^{\mathfrak{s}}$ norm (cf. Lemma 5.2). For any two fixed points $v_{j}\in
X^{\mathfrak{s}}([0,T_{j}])$, $j\in\\{1,2\\}$ of the map
$v\mapsto\mathcal{J}_{z,M}(v)$, set $T_{\cap}\coloneqq\min(T_{1},T_{2})$ and

$t_{0}\coloneqq\sup\big{(}t\in[0,T_{\cap})\colon v_{1}(s)=v_{2}(s)\text{ for
all }s\in[0,t]\big{)}$

and assume, by way of contradiction, that $t_{0}<T_{\cap}$. Note that $t_{0}$
is well defined since $v_{1}(0)=v_{2}(0)$, and therefore the supremum is taken
over non-empty set. Since $v_{1},v_{2}\in
C_{t}^{0}\big{(}[0,T_{\cap}),H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})\big{)}$,
thanks to (Time-continuity of solutions: ), it holds that
$v_{1}(t_{0})=v_{1}(t_{0})$. Thereby, from the group properties of the linear
Schrödinger evolution $t\mapsto e^{it\Delta}$ for all $t\in[t_{0},T_{\cap}]$
we have that

$v_{j}(t)=e^{i(t-t_{0})\Delta}v_{0}\mp
i\int_{t_{0}}^{t}e^{i(t-s)\Delta}(\Phi_{z_{\leq
M}}[v_{j}]+[z,z,z]_{>M})\,ds\qquad j=1,2\,.$

with $v_{0}=v_{1}(t_{0})=v_{1}(t_{0})$, that is $v_{j}$ are both fixed points
of the map $v\mapsto\mathcal{K}(v)$ with $z_{\leq M}$ in place of $z$, and
with $[z,z,z]_{>M}$ in place of $h$.

Thanks to (5), we can fix $R>0$ for which

$\|z_{\leq M}\|_{Y^{S}([0,T_{\cap}))}\leq
R,\quad\|[z,z,z]_{>M}\|_{X^{\mathfrak{s}}([0,T_{\cap}))}\leq
R,\quad\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}}\leq R.$

Estimate (5.2) shows that there exists $T$ with $t_{0}<t_{0}+T<T_{\cap}$ for
which

$\|v_{1}\|_{X^{\mathfrak{s}}([t_{0},t_{0}+T])}\leq 1\qquad\textrm{and
}\quad\|v_{2}\|_{X^{\mathfrak{s}}([t_{0},T])}\leq 1$

and the map $v\mapsto\mathcal{K}(v)$ has a unique fixed point on
$\overline{\mathcal{B}}_{1}^{[t_{0},t_{0}+T]}\big{\\{}v\colon\|v\|_{X^{\mathfrak{s}}([t_{0},t_{0}+T])}\leq
1\big{\\}}$. This contradicts our assumption that $t_{*}<T_{\cap}$ is the
upper bound of times $s$ for which $v_{1}(s)=v_{2}(s)$ and proves the desired
assertion.

The uniqueness above allows us to define $T_{\mathrm{max}}$ as the upper bound
of for the existence and uniqueness time of solutions. To prove the blow-up
criterion, let us assume that $T_{\mathrm{max}}<T_{0}$ and by way of
contradiction let us assume that we can fix $R>0$ and find
$t_{0}<T_{\mathrm{max}}$, arbitrarily close to $T_{\mathrm{max}}$ for which
(5) holds with $V_{0}=u^{\\#}_{M}(t_{0})$. Then Lemma 5.2 guarantees that we
can extend our solution to at least $\big{[}0,\min(t_{0}+T,T_{0})\big{)}$ by
finding a fixed point to the map $v\mapsto\mathcal{K}(v)$ on
$\big{[}t_{0},\min(t_{0}+T,T_{0})\big{]}$ and extending the remainder
$u^{\\#}_{M}$ past $t_{0}$. By subadditivity of norms, the $u^{\\#}_{M}$
extended this way satisfies

$\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}\big{(}[0,\min(t_{0}+T,T_{0}))\big{)}}+\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,\min(t_{0}+T,T_{0}));H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})\big{)}}<\infty.$

This contradicts our assumption that $T_{\mathrm{max}}<T_{0}$ as long as
$t_{0}$ is chosen close enough to $T_{\mathrm{max}}$.

Time-continuity and sattering of the multilinear data Clearly
$z_{1}=e^{it\Delta}f$ so the scattering for $z_{1}$ is immediate. It also
holds that

$z_{1}=\mathbb{I}_{f}(0)$

so (3.3) from Proposition 3.3 we obtain

$\|z_{1}\|_{C^{0}([0,T_{0});H^{S}_{x}(\mathbb{R}^{d}))}\leq\|f\|_{H^{S}_{x}(\mathbb{R}^{d})}.$

Next we focus on $z_{k}$ with $2\leq k\leq M$.

According to (1) it holds that that

$z_{k}=\mp i\sum_{\mathclap{\begin{subarray}{c}k_{1}+k_{2}+k_{3}=k+1\\\
k_{1},k_{2},k_{3}\leq
k\end{subarray}}}\mathbb{I}_{0}(z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}})\,,$

Using (3.3) from Proposition 3.3 we obtain that

$\|z_{k}\|_{C^{0}\big{(}[0,T_{0});H^{\mu(k,S)}_{x}(\mathbb{R}^{d})\big{)}}\leq\sum_{\mathclap{\begin{subarray}{c}k_{1}+k_{2}+k_{3}=k+1\\\
k_{1},k_{2},k_{3}\leq
k\end{subarray}}}\Big{\|}z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}h\Big{\|}_{X^{*,\mu(k,S)+\varepsilon}([0,T_{0}))}.$

Using (4.4) applied with $S+\varepsilon$ in place of $S$ we obtain

$\displaystyle\|z_{k}\|_{C^{0}\big{(}[0,T_{0});H^{\mu(k,S)}_{x}(\mathbb{R}^{d})\big{)}}$
$\displaystyle\lesssim\|z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}\|_{Y^{\mu(k,S)+\varepsilon}([0,T_{0}))}$
$\displaystyle\leq\prod_{j=1}^{3}\|z_{k_{j}}\|_{Y^{\mu(k_{j},S)+(M+1)\varepsilon}([0,T_{0}))}.$

This implies the desired claim as long as we replace $\varepsilon$ with
$\nicefrac{{\varepsilon}}{{(M+1)}}$ above.

To show that $z_{k}$, $2\leq k\leq M$ scatters set

$\mathfrak{g}_{k}\coloneqq\lim_{t\to+\infty}e^{-it\Delta}z_{k}=\mp
i\sum_{\mathclap{\begin{subarray}{c}k_{1}+k_{2}+k_{3}=k\\\
k_{1},k_{2},k_{3}<k\end{subarray}}}\lim_{t\to+\infty}e^{-it\Delta}\mathbb{I}_{0}(z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}})\,,$

where we used the inductive definition (1) of $z_{k}$. As before, using (4.4)
applied with $S+\varepsilon$ in place of $S$

$\|z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}\|_{Y^{\mu(k,S)+\varepsilon}([0,+\infty))}\lesssim\prod_{j=1}^{3}\|z_{k_{j}}\|_{Y^{\mu(k_{j},S)+(M+1)\varepsilon}([0,+\infty))}\,,$

while from (3.4) and (3.4) it follows that

$\lim_{t\to+\infty}\Big{\|}z_{k}(t)-e^{it\Delta}\mathfrak{g}_{k}\Big{\|}_{H^{\mu(k,S)}_{x}(\mathbb{R}^{d})}=\lim_{t\to+\infty}\Big{\|}e^{-it\Delta}z_{k}(t)-\mathfrak{g}_{k}\Big{\|}_{H^{\mu(k,S)}_{x}(\mathbb{R}^{d})}=0.$

This is the desired claim as long as we replace $\varepsilon$ with
$\nicefrac{{\varepsilon}}{{(M+1)}}$ above.

Scattering of global solutions. Now let us show that the remainder
$u^{\\#}_{M}$ scatters in $H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ under the
assumptions (Global existence of solutions: ), Since $u^{\\#}_{M}$ satisfies

$u^{\\#}_{M}=\mathbb{I}_{0}\big{(}\Phi_{\mathfrak{z}_{\leq
M}}[u^{\\#}_{M}]+[\mathfrak{z},\mathfrak{z},\mathfrak{z}]_{>M}\big{)},\,$

it is sufficient to set

$\mathfrak{g}^{\\#}_{M}=\lim_{t\to+\infty}e^{-it\Delta}\mathbb{I}_{0}\big{(}\Phi_{\mathfrak{z}_{\leq
M}}[u^{\\#}_{M}]+[\mathfrak{z},\mathfrak{z},\mathfrak{z}]_{>M}\big{)}\,,$

to show that $\mathfrak{g}^{\\#}_{M}\in H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$,
and to show that that (Scattering of global solutions: ) holds. Using (5.1)
and (5.3) we obtain that

$\Big{\|}\Phi_{z_{\leq
M}}[u^{\\#}_{M}]+[z,z,z]_{>M}\Big{\|}_{X^{*,\mathfrak{s}+\varepsilon}([0,+\infty))}\\\
\lesssim\Big{(}\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}([0,+\infty))}+\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}<\infty.$

The existence of the limit in $H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ defining
$\mathfrak{g}^{\\#}_{M}$ and (Scattering of global solutions: ) follow, again,
from (3.4) and (3.4).

Continuous dependence on the multilinear data. We have already argued that for
any $f\in H^{S+\varepsilon}_{x}(\mathbb{R}^{d})$ with $d(0,f)\leq R$ there
exists a unique solution $u\in
C^{0}\big{(}[0,T);H^{S+\varepsilon}_{x}(\mathbb{R}^{d})\big{)}$ of the form
(1.5), which satisfies $\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}([0,T))}<\infty$ and
$u^{\\#}_{M}=\mathcal{J}_{z,M}(u^{\\#}_{M})\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$. Proposition 3.1 and bound (3.1) shows that
$\mathbb{I}_{0}$ is a bounded linear map from
$X^{*,\mathfrak{s}+\varepsilon}([0,T))$ to $X^{\mathfrak{s}}([0,T))$. In Lemma
5.3 we have shown that the map

$\vec{z}_{M}\mapsto[z,z,z]_{>M}$

is Lipschitz contious with respect to the distance we introduced . Finally,
the map

$(\vec{z}_{M},v)\mapsto\Phi_{z_{\leq M}}[v]$

is Lipshitz continuous thanks to estimates (5.1) and (5.1). ∎

##  6\. Probabilistic estimates

In this section, we focus on establishing probabilistic estimates for the
multilinear correction terms $\mathfrak{z}_{k}$, defined in (1). We thereby
prove Theorem 1.6.

To prove these estimates we have to keep track of how the Wiener randomization
of $f$, encated to obtain $\mathfrak{f}$, reflects on the terms
$\mathfrak{z}_{k}$. We do this by introducing efficient bookkeeping notation
by indexing all terms that explicitly depend on the initial datum
$\mathfrak{f}$ using ternary trees, defined below. We stress that trees serve
merely as a notation, and we do not use any graph theory or subtle properties
of trees.

More precisely, we define the set $\mathbb{T}$ of ternary trees as
$\mathbb{T}:=\bigcup_{n\geq 1}\mathbb{T}_{n}$, where for each
$n\in\mathbb{N}\setminus{0}$ the set of trees $\mathbb{T}_{n}$ is given by
induction as follows:

  * •

We say $\tau\in\mathbb{T}_{1}$ if $\tau=[\bullet]$.

  * •

We say $\tau\in\mathbb{T}_{n}$ for $n\geq 1$ if
$\tau=[\tau_{1},\tau_{2},\tau_{3}]$ for some $\tau_{j}\in\mathbb{T}_{n_{j}}$
and $n=n_{1}+n_{2}+n_{3}$ with $1\leq n_{j}<n$.

The index $n$ can be viewed as the number of leaves of of a tree.

Pictorially, our inductive construction of $\tau=[\tau_{1},\tau_{2},\tau_{3}]$
can be represented as follows

$\bullet$$\tau_{1}$$\tau_{2}$$\tau_{3}$

where $\tau_{j}$, $j=1,2,3$ are ternary trees. If $\tau\in\mathbb{T}_{n}$, we
set $|\tau|=n$ In particular, if $\tau=[\tau_{1},\tau_{2},\tau_{3}]$ then
$|\tau|=|\tau_{1}|+|\tau_{2}|+|\tau_{3}|$. Under this convention, $|\tau|$
corresponds to the number of nodes with no descendants, that we refer to as
“leaves”. We omit the proof, as this fact does not explicitly enter our
discussion.

In the literature, one usually allows a node of a general ternary trees to
have zero, one, two, or three descendants. However, in the present manuscript,
we only consider trees in which each node has either three descendants
(children) or no descendants at all (a leaf). By induction, one can show there
are no ternary trees with an even number of leaves, that is,
$\mathbb{T}_{n}=\emptyset$ when $n\in 2\mathbb{N}$. For example, we have

$\Big{|}\,\big{[}[\bullet],[\bullet],[\bullet]\big{]}\,\Big{|}=3\,,\qquad\Big{|}\,\Big{[}[\bullet],\big{[}[\bullet],[\bullet],[\bullet]\big{]},[\bullet]\Big{]}\,\Big{|}=5.$

Graphically, these trees can be represented as

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

To each tree $\tau\in\mathbb{T}_{n}$ with $n\in 2\mathbb{N}+1$ we inductively
assign an $n$-(real) linear tree operator $R_{\tau}$ mapping $n$-tuples of
functions $(f_{1},\ldots,f_{n})\in L^{2}(\mathbb{R}^{d})$ to functions on
$\mathbb{R}\times\mathbb{R}^{d}$.

  * •

We set $R_{[\bullet]}[f](t,x)\coloneqq e^{it\Delta}f(x).$

  * •

Inductively, we set

$\displaystyle
R_{[\tau_{1},\tau_{2},\tau_{3}]}[\mathbf{f}_{1}\oplus\mathbf{f}_{2}\oplus\mathbf{f}_{3}](t,\cdot)$
$\displaystyle\coloneqq\mp
i\int_{0}^{t}e^{i(t-s)\Delta}\Big{(}R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot)\Big{)}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}s\,,$

where the choice of the sign is exactly the opposite to the sign on the right
hand side of (1). Here
$\mathbf{f}_{j}=(f_{j,1},\ldots,f_{j,|\tau_{j}|})\in\big{(}L^{2}(\mathbb{R}^{d})\big{)}^{|\tau_{j}|}$
are $|\tau_{j}|$-tuples of $L^{2}(\mathbb{R}^{d})$ functions and

$\mathbf{f}_{1}\oplus\mathbf{f}_{2}\oplus\mathbf{f}_{3}=\big{(}f_{1,1},\ldots,f_{1,|\tau_{1}|},f_{2,1},\ldots,f_{2,|\tau_{2}|},f_{3,1},\ldots,f_{3,|\tau_{3}|}^{3}\big{)}\,.$

To simplify notation, given $\tau\in\mathbb{T}$ and one function $f\in
L^{2}(\mathbb{R}^{d})$, we write

$R_{\tau}[f]:=R_{\tau}\big{[}f,\ldots,f\big{]}\,,$

where the right-hand side contains $|\tau|$ copies of $f$.

The main result of this section shows that the functions
$R_{\tau}[\mathfrak{f}]$ lie, almost surely, in the space $Y^{\mu(|\tau|,S)}$
with an appropriate regularity $\mu(|\tau|,S)$ depending on $S$, the
regularity of $f\in H^{S}(\mathbb{R}^{d})$, and on $|\tau|$, the order of
multilinearity of the operator $R_{\tau}$.

######  Proposition 6.1.

For any $S\in\mathbb{R}$, $n\geq 1$, let $\mu(k,S)$ be as defined in (1.3).

Fix $\tau\in\mathbb{T}$, $S>0$, and let $0<\varepsilon\lesssim_{|\tau|}1$ be
small. Then for every $0<\varepsilon_{0}\lesssim_{\varepsilon,|\tau|,S}1$
there exist constants $C=C(\varepsilon_{0},\varepsilon,|\tau|,S)>0$
independent of $f\in L^{2}(\mathbb{R}^{d})$, such that for any $\lambda>0$ it
holds that

$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}>\lambda\big{\\}}\Big{)}\leq
C\exp\Bigg{(}-\frac{\lambda^{\frac{2}{|\tau|}}}{C\|f\|_{H^{S+\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,,$

where $\mathfrak{f}$ is the unit scale randomization of $f$ given by (1.1). In
particular,

$\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}<\infty\qquad\text{almost
surely}$

for any $f\in H^{S+\varepsilon}_{x}(\mathbb{R}^{d})$.

Before proceeding with the proof we use the above result to deduce the
probability distribution for $\|\mathfrak{z}_{k}\|_{Y^{S}(\mathbb{R})}$,
$\|\mathfrak{z}_{\leq M}\|_{Y^{S}(\mathbb{R})}$, defined in (1.3). We show
that Theorem 1.6 holds and thereby we have for any $\lambda>1$ that

$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega\colon\big{\|}\mathfrak{z}_{\leq
M}\big{\|}_{Y^{S}(\mathbb{R})}>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-\frac{\lambda^{\frac{2}{M}}}{C\|f\|_{H_{x}^{S+\varepsilon}(\mathbb{R}^{d})}^{2}}\bigg{)}\,.$

###### Proof of Theorem 1.6.

The set $\bigcup_{k\leq M}\mathbb{T}_{k}$ is finite, and therefore by (1)
there is a constant $\widetilde{C}=\widetilde{C}(M)$ such that

$\displaystyle\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}\mathfrak{z}_{k}\big{\|}_{Y^{S}(\mathbb{R})}>\lambda\big{\\}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\bigcup_{\begin{subarray}{c}\tau\in\mathbb{T}_{k}\end{subarray}}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(k,S)}(\mathbb{R})}>\widetilde{C}^{-1}\lambda\big{\\}}\Big{)}$
$\displaystyle\leq\sum_{\begin{subarray}{c}\tau\in\mathbb{T}_{k}\end{subarray}}\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(k,S)}(\mathbb{R})}>\widetilde{C}^{-1}\lambda\big{\\}}\Big{)}\,.$
$\displaystyle\leq\sum_{\begin{subarray}{c}\tau\in\mathbb{T}_{k}\end{subarray}}\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(k,S)}(\mathbb{R})}>\widetilde{C}^{-1}\lambda\big{\\}}\Big{)}\,,$

From (6.1) of Proposition 6.1 we obtain that

$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}\mathfrak{z}_{k}\big{\|}_{Y^{S}(\mathbb{R})}>\lambda\big{\\}}\Big{)}\lesssim\widetilde{C}CC\exp\Bigg{(}-\frac{\lambda^{\frac{2}{k}}}{\widetilde{C}^{2}C\|f\|_{H^{S+\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)},$

as required.

To obtain (6) it is sufficient to note that $S\leq\mu(k,S)$ and apply the
triangle inequality. ∎

Next, we proceed to the proof of Proposition 6.1. We rely on the deterministic
estimate of Lemma 6.2, which establishes regularity for the operators
$R_{\tau}$ when evaluated on functions with bounded frequency support. Then,
using Lemma 6.5 below, we show that the probabilistic bounds of Proposition
6.1 follow from estimates on large enough moments of the random variable
$\big{\|}R_{\tau}[\mathfrak{f}]\|_{Y^{\mu(|\tau|,S)}(\mathbb{R})}$. Such
moment bounds are established with help of multi-parameter Wiener chaos
estimates (Lemma 6.3).

First, we state and prove, or provide references for the required lemmata.
Finally, we prove Proposition 6.1.

######  Lemma 6.2.

Let $\mu(n,S)$ be as in (1.3) and fix $\tau\in\mathbb{T}$ and $S,R_{0}>0$.
Choose any $0<\varepsilon\lesssim_{|\tau|}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,|\tau|}1$. For any tuple
$\mathbf{f}=(f_{1},\ldots
f_{|\tau|})\in\big{(}L^{2}(\mathbb{R}^{d})\big{)}^{|\tau|}$ of functions with
$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{j}))\leq 2R_{0}$,
$j\in\\{1,\ldots,|\tau|\\}$, it holds that

$\Big{\|}R_{\tau}\big{[}\mathbf{f}\big{]}\Big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\lesssim\prod_{j=1}^{|\tau|}\|f_{j}\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}.$

The implicit constant may depend on $|\tau|$, $\varepsilon$,
$\varepsilon_{0}$, and $R_{0}$.

The following lemma establishes the relation between the regularity parameter
$\mu(|\tau|,S)$ and the trilinear bound (4.2).

Our probabilistic estimates depend crucially on the Wiener chaos estimates. If
$n=1$ and $g_{j}$ having standard normal distribution, the claim of Lemma 6.3
reduces to the classical Khintchine inequality for Gaussian sums. Here, we
provide a version of the general statement.

######  Lemma 6.3.

[TT10, Proposition 2.4] Fix integers $M,n\geq 1$ and let $(g_{j})_{1\leq j\leq
M}\in\mathcal{N}_{\mathbb{C}}(0,1)$ be a sequence of complex $L^{2}$
normalized independent Gaussian random variables. Given
$c\colon\mathbb{N}^{n}\to\mathbb{C}$ define the random variable

$\Xi_{n,M}\coloneqq\sum_{\mathrlap{k_{1},\ldots,k_{n}\in\\{1,\cdots,M\\}}}\hskip
10.00002ptc(k_{1},\cdots,k_{n})g_{k_{1}}\cdots g_{k_{n}}.$

Then, for all $\gamma\geq 2$ it holds that

$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\Xi_{n,M}\big{|}^{\gamma}\big{)}^{\frac{1}{\gamma}}\lesssim_{n}(\gamma-1)^{\frac{n}{2}}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\Xi_{n,M}\big{|}^{2}\big{)}^{\frac{1}{2}}$

with the implicit constant independent of $M$ and $\gamma$.

######  Remark 6.4.

The assertion of Lemma 6.3 is slightly different compared to [TT10,
Proposition 2.4], where the summation is taken over the simplex

$A(n,M)=\big{\\{}(k_{1},\ldots,k_{n})\colon k_{1}\leq k_{2}\leq\ldots\leq
k_{n}\leq M\big{\\}}\,.$

However, the proof of Lemma 6.3 follows the proof of [TT10, Proposition 2.4]
line by line, with $A(n,M)$ replaced by $\\{1,\cdots,M\\}^{n}$. Otherwise,
Lemma 6.3 can be deduced from [TT10, Proposition 2.4] simply by successive
applications of the triangle inequality.

Finally, we formulate a slight modification of [Tzv09, Lemma 4.5]. We omit the
proof: it coincides with the one in the reference and follows from the
Chebyshev inequality and an appropriate optimization.

######  Lemma 6.5.

Let $F$ be a random variable and suppose that there exist $K>0$,
$\gamma_{0}\geq 1$, and $k\geq 1$ such that for any $\gamma\geq\gamma_{0}$ we
have

$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|F|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\leq\gamma^{\frac{k}{2}}K.$

Then, there exist $c>0$ and $C>0$ depending on $\gamma_{0}$ and $k$, but
independent of $K$ and $\gamma$, such that for every $\lambda>0$,

$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:|F|>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-c\frac{\lambda^{\frac{2}{k}}}{K^{\frac{2}{k}}}\bigg{)}\,.$

In particular, we have

$\mathbb{P}(\\{\omega\in\Omega:|F|<\infty\\})=1.$

From the above we can deduce an estimate on the probability distribution of
the Sobolev norms of the randomized initial data.

######  Corollary 6.6.

For some $C,c>0$ it holds that

$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\|\mathfrak{f}\|_{H^{S}_{x}(\mathbb{R}^{d})}>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-c\frac{\lambda^{2}}{\|f\|^{2}_{H^{S}_{x}(\mathbb{R}^{d})}}\bigg{)}\,.$

###### Proof.

For any $\gamma\geq 2$, the Minkowski inequality implies

$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\|\mathfrak{f}\|_{L^{2}_{x}(\mathbb{R}^{d})}^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\leq\big{\|}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\big{\|}_{L^{2}_{x}(\mathbb{R}^{d})}$

According to the definition of the unit-scale Wiener randomization (1.1), the
pointwise value $\mathfrak{f}(x)$ has the form required by Lemma 6.3, and
therefore

$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\lesssim(\gamma-1)^{\nicefrac{{1}}{{2}}}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{2}\big{)}^{\nicefrac{{1}}{{2}}}\,.$

Finally, using independence of the random variables $g_{k}$ and Plancherel’s
identity, we obtain that

$\big{\|}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{2}\big{)}^{\nicefrac{{1}}{{2}}}\big{\|}_{L^{2}_{x}(\mathbb{R}^{d})}^{2}\begin{aligned}
&=\sum_{k,k^{\prime}\in\mathbb{Z}^{d}}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}(g_{k}\overline{g_{k^{\prime}}})\int_{\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f(x)\overline{\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k^{\prime}}f(x)}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\\\
&\lesssim\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\big{|}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f(x)\big{|}^{2}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}|\chi_{k}(\xi)|^{2}\big{|}\widehat{f}(\xi)\big{|}^{2}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\\\
&\lesssim\|f\|_{L^{2}_{x}(\mathbb{R}^{d})}^{2}\,,\end{aligned}$

where in the last inequality we used that
$\sum_{k\in\mathbb{Z}^{d}}|\chi_{k}(\xi)|^{2}\lesssim 1$, since
$|\chi_{k}(\xi)|\leq 1$ and for any $\xi\in\mathbb{R}^{d}$ there are finitely
many $k\in\mathbb{Z}^{d}$ for which $\xi\in\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\chi_{k}$. Thus,

$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\|\mathfrak{f}\|_{L^{2}_{x}(\mathbb{R}^{d})}^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\lesssim(\gamma-1)^{\nicefrac{{1}}{{2}}}\|f\|_{L^{2}_{x}(\mathbb{R}^{d})}^{2}$

and from Lemma 6.5 follows that

$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\|\mathfrak{f}\|_{L^{2}_{x}(\mathbb{R}^{d})}>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-c\frac{\lambda^{2}}{\|f\|_{L^{2}_{x}(\mathbb{R}^{d})}^{2}}\bigg{)}\,.$

Since the unit-scale Wiener randomization commutes with the operator
$\langle\Delta\rangle^{S}$, we can replace $f$ with
$\langle\Delta\rangle^{S}f$ and the claim follows. ∎

Next, we prove Lemma 6.2, and finally we show Proposition 6.1.

###### Proof of Lemma 6.2 .

We prove the claim by induction on $|\tau|$. First, we claim that for each
$t\in\mathbb{R}$ and $\tau\in\mathbb{T}$ we have

$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{\tau}[\mathbf{f}](t))\big{)}\Big{)}\leq
2R_{0}|\tau|.$

For the base step, if $|\tau|=1$, that is, $\tau=[\bullet]$, then
$R_{[\bullet]}[f]=e^{it\Delta}f$. Thus, for every $t\in\mathbb{R}$, we have
$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{[\bullet]}[f](t))\big{)}\big{)}\leq 2R_{0}$,
since $\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(e^{it\Delta}f)(\xi)=e^{-4\pi^{2}i|\xi|^{2}}\widehat{f}(\xi)$
and $\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\hat{f}\big{)}\big{)}\leq 2R_{0}$.

Next, fix $n>1$ and assume that (6) holds for any $\tau\in\mathbb{T}$ with
$|\tau|<n$. Choose any $\tau\in\mathbb{T}$ with $|\tau|=n$ and let
$\tau_{j}\in\mathbb{T}$, $j\in\\{1,2,3\\}$, be such that
$\tau=[\tau_{1},\tau_{2},\tau_{3}]$. For any functions $h_{j}\in
L^{2}(\mathbb{R}^{d})$, $j\in\\{1,2,3\\}$ it holds that

$\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{1}\overline{h_{2}}h_{3}}\big{)}\subset\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{h}_{1})-\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{h}_{2})+\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{h}_{3})\,,$

and thus

$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{1}\overline{h_{2}}h_{3}}\big{)}\Big{)}\leq\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{1})\big{)}+\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{2})\big{)}+\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{3})\big{)}.$

Since $n=|\tau|=|\tau_{1}|+|\tau_{2}|+|\tau_{3}|$ and $|\tau_{j}|\geq 1$, we
obtain that $|\tau_{j}|<n$, $j\in\\{1,2,3\\}$, and we can use the induction
hypothesis to deduce for any $s\in\mathbb{R}$ that

$\displaystyle\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot))\big{)}\Big{)}$
$\displaystyle\leq\sum_{j=1}^{3}\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{\tau_{j}}[\mathbf{f}_{j}](s,\cdot))\big{)}\Big{)}\leq\sum_{j=1}^{3}2R_{0}|\tau_{j}|=2R_{0}|\tau|.$

The induction step follows since the multiplication by
$e^{4\pi^{2}i(t-s)|\xi|^{2}}$, or integration in time does not change the
support in Fourier space.

Next, let us inductively prove the bound

$\Big{\|}R_{\tau}\big{[}\mathbf{f}\big{]}\Big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\lesssim_{R_{0},|\tau|,S}\prod_{j=1}^{|\tau|}\|f_{j}\|_{H^{S+2|\tau|\varepsilon}(\mathbb{R}^{d})}\,,$

from which (6.2) follows, since $\varepsilon\lesssim_{|\tau|}1$ is small. If
$|\tau|=1$, that is, $\tau=[\bullet]$, then $\mu(|\tau|,S)=S$. Also,
$R_{[\bullet]}[f]=e^{it\Delta}\mathop{\kern 0.0pt\mathrm{Q}}\mathopen{}_{k}f$
and (6) follows from (3.1) with $h=0$.

Fix $n\in 2\mathbb{N}+1$, $n>1$ and assume that (6) holds for all
$\tau\in\mathbb{T}$ with $|\tau|<n$. As above, let
$\tau_{j}\in\mathbb{T}_{n_{j}}$ be such that
$\tau=[\tau_{1},\tau_{2},\tau_{3}]$ with $|\tau_{j}|<n$ for each
$j\in\\{1,2,3\\}$. For the rest of the proof, we allow all our constants to
depend on $|\tau|,S$, and $R_{0}$.

Since $R_{\tau}[\mathbf{f}]$ has bounded support as shown in (6), then from
(3.1) with $v_{0}=0$ and
$\mu(|\tau|,S+\varepsilon)\geq\mu(|\tau|,S)+\varepsilon$ we obtain that

$\big{\|}R_{\tau}[\mathbf{f}]\big{\|}_{Y^{\mu(|\tau|,S)+\varepsilon}(\mathbb{R})}\begin{aligned}
&\lesssim\big{\|}R_{\tau}[\mathbf{f}]\big{\|}_{Y^{\mu(|\tau|,S+\varepsilon)}(\mathbb{R})}\\\
&\lesssim\Big{\|}R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot)\Big{\|}_{X^{*,\mu(|\tau|,S+2\varepsilon)}(\mathbb{R})}\,.\end{aligned}$

Consequently, (4.4) yields

$\displaystyle\Big{\|}R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot)\Big{\|}_{X^{*,\mu(|\tau|,S+2\varepsilon)}(\mathbb{R})}$
$\displaystyle\lesssim\prod_{j=1}^{3}\Big{\|}R_{\tau_{j}}[\mathbf{f}_{j}]\Big{\|}_{Y^{\mu(|\tau_{j}|,S+2\varepsilon)+\varepsilon}(\mathbb{R})}\lesssim\prod_{j=1}^{3}\Big{\|}R_{\tau_{j}}[\mathbf{f}_{j}]\Big{\|}_{Y^{\mu(|\tau_{j}|,S+3\varepsilon)}(\mathbb{R})}.$

Using the inductive hypothesis with $S+3\varepsilon$ in place of $S$ we obtain

$\Big{\|}R_{\tau_{j}}[\mathbf{f}_{j}]\Big{\|}_{Y^{\mu(|\tau_{j}|,S+3\varepsilon)}(\mathbb{R})}\lesssim\prod_{k=1}^{|\tau_{j}|}\|f_{j}\|_{H^{S+(2|\tau_{j}|+3)\varepsilon}}.$

Since $|\tau_{j}|\leq|\tau|-2$, then $2|\tau_{j}|+3\leq 2|\tau|$, and
therefore

$\Big{\|}R_{\tau}[\mathbf{f}]\Big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\lesssim\prod_{k=1}^{|\tau|}\|f_{k}\|_{H^{S+2|\tau|\varepsilon}}\,,$

as desired ∎

###### Proof of Proposition 6.1.

Fix $\tau\in\mathbb{T}$. Since $\mathbb{T}_{n}=\emptyset$ if $n\in
2\mathbb{N}$, we only consider $|\tau|\in 2\mathbb{N}+1$. According to Lemma
6.5, it suffices to prove that

$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim_{|\tau|}\gamma^{\frac{|\tau|}{2}}\|f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{|\tau|}$

for all $\gamma$ large enough. Fix $\gamma>\frac{2}{\varepsilon_{0}}$ and
recall that $\varepsilon_{0}$ is a small, fixed constant appearing in the
definition (3) of the space $Y$ depending on $S$, $|\tau|$, $\varepsilon$.

Since $\gamma>\frac{2}{\varepsilon_{0}}>2$ and $\varepsilon_{0}<2^{-100}$,
from the definition of $Y^{S}$ and Minkowski’s inequality it follows that

$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\begin{aligned}
&=\Big{(}\mathop{\kern 0.0pt\mathbb{E}}\mathopen{}\big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{\|}_{Y_{N}(\mathbb{R})}^{2}\big{)}^{\frac{\gamma}{2}}\Big{)}^{\frac{1}{\gamma}}\\\
&\leq\Big{(}\sum_{N\in 2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{\|}^{\gamma}_{Y_{N}(\mathbb{R})}\big{)}^{\frac{2}{\gamma}}\Big{)}^{\frac{1}{2}}.\end{aligned}$

Minkowski’s inequality also gives that $(\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\|f\|_{L^{p}}^{\gamma})^{\nicefrac{{1}}{{\gamma}}}\leq\big{\|}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|f|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\big{\|}_{L^{p}}$
whenever $\gamma\geq p$, where the $L^{p}$ norm is over $t,x_{1}$, or
$x^{\prime}$. Having assumed that $\gamma$ is larger than any $L^{p}$
integrability exponent of appearing in the definition of the norms
$Y_{N}(\mathbb{R})$ (see (3)), we obtain that

$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}\Big{\|}_{Y_{N}(\mathbb{R})}^{2}\Big{)}^{\frac{1}{2}}.$

Henceforth, we fix $N\in 2^{\mathbb{N}}$ and we focus on bounding each term of
the sum on the right-hand side of (6) individually.

Recall that the function $\mathfrak{f}$ is obtained via the randomization
procedure $\mathfrak{f}\coloneqq\sum_{k\in\mathbb{Z}^{d}}g_{k}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f$. We then use that the map
$\mathbf{f}=(f_{1},\ldots,f_{|\tau|})\mapsto R_{\tau}[\mathbf{f}]$ is linear
in the odd entries and anti-linear in even entries $f_{j}$ to obtain that

$\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}](t,x)\begin{aligned}
&=\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\Big{(}\sum_{\mathrlap{\mathbf{k}\in(\mathbb{Z}^{d})^{|\tau|}}}g_{\mathbf{k}}R_{\tau}\big{[}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{1}}f,\ldots,\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{|\tau|}}f\big{]}(t,x)\Big{)}\\\
&=\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\Big{(}\sum_{\mathrlap{\mathbf{k}\in(\mathbb{Z}^{d})^{|\tau|}}}g_{\mathbf{k}}R_{\tau}^{\mathbf{k}}\big{[}f\big{]}(t,x)\Big{)}\,,\end{aligned}$

where $\mathbf{k}=(k_{1},\ldots,k_{|\tau|})$, $g_{\mathbf{k}}\coloneqq
g_{k_{1}}\overline{g_{k_{2}}}\ldots
g_{k_{|\tau|-2}}\overline{g_{k_{|\tau|-1}}}g_{k_{|\tau|}}$, and we use the
notation $R_{\tau}^{\mathbf{k}}[f]\coloneqq R_{\tau}\big{[}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{1}}f,\ldots,\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{|\tau|}}f\big{]}(t,x)$. We claim that the
following crucial estimate

$\Big{\|}\Big{(}\mathop{\kern 0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}\Big{\|}_{Y_{N}(\mathbb{R})}^{2}\lesssim_{|\tau|}\begin{aligned}
&(\gamma-1)^{|\tau|}\\\
&\times\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\end{aligned}$

holds.

Assuming (6) holds, let us prove (6.1) first. Applying bound (6) to (6) we
obtain that

$\displaystyle\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}$
$\displaystyle\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}$
$\displaystyle\times\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\bigg{)}^{\frac{1}{2}}\,.$

Using the Cauchy-Schwarz inequality in $N$ and the definition (3) of then norm
$Y^{\sigma}(\mathbb{R})$, it follows from the bound (6.2) of Lemma 6.2 that

$\sum_{N\in 2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\\\
\begin{aligned} &\leq\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}^{2}_{Y_{N}(\mathbb{R})}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}^{2}_{Y_{N}(\mathbb{R})}\Big{)}^{\frac{1}{2}}\\\
&=\big{\|}R_{\tau}^{\mathbf{k}}[f]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\big{\|}R_{\tau}^{\mathbf{l}}[f]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\\\
&\lesssim\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{l_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\,,\end{aligned}$

and hence,

$\displaystyle\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}$
$\displaystyle\qquad\times\bigg{(}\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\Big{(}\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\Big{)}\Big{(}\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{l_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\Big{)}\bigg{)}^{\frac{1}{2}}\,.$

Since $(g_{k})_{k\in\mathbb{Z}^{d}}$ satisfy $\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\lesssim_{|\tau|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}|g_{0}|^{2|\tau|}\big{)}\lesssim_{|\tau|}1$
and $\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}=0$
if any fixed $m\in\mathbb{Z}^{d}$ appears an odd number of times in sequence
$(k_{1},\cdots,k_{|\tau|},l_{1},\cdots,l_{|\tau|})$ of elements in
$\mathbb{Z}^{d}$. Then (6) becomes

$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\begin{aligned}
&\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}\sum_{\mathrlap{\mathbf{k}\in(\mathbb{Z}^{d})^{|\tau|}}}\hskip
20.00003pt\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\bigg{)}^{\frac{1}{2}}\\\
&\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}\sum_{k\in\mathbb{Z}^{d}}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\bigg{)}^{\frac{|\tau|}{2}}\,.\end{aligned}$

We claim that

$\sum_{k\in\mathbb{Z}^{d}}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\leq\|f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\,.$

Indeed, since $0\leq\psi(\xi)\leq 1$, then
$\sum_{k\in\mathbb{Z}^{d}}|\psi(\xi-k)|^{2}\leq\sum_{k\in\mathbb{Z}^{d}}|\psi(\xi-k)|=1$,
and therefore

$\sum_{k\in\mathbb{Z}^{d}}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\begin{aligned}
&=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}(1+|\xi|)^{2S+2\varepsilon}|\psi(\xi-k)|^{2}|\hat{f}(\xi)|^{2}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}\xi\\\
&\leq\int_{\mathbb{R}^{d}}(1+|\xi|)^{2S+2\varepsilon}|\hat{f}(\xi)|^{2}d\xi\end{aligned}$

and the required bound (6) follows.

We conclude the proof by showing that (6) holds. By (6) and Lemma 6.3 applied
point-wise in $(t,x)$, we have

$\displaystyle\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}](t,x)\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}$
$\displaystyle\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}](t,x)\big{|}^{2}\Big{)}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\Big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\Big{)}\;R_{\tau}^{\mathbf{k}}\big{[}f\big{]}(t,x)\overline{R_{\tau}^{\mathbf{l}}\big{[}f\big{]}(t,x)}\bigg{)}^{\nicefrac{{1}}{{2}}}\,.$

Next, for any $v\colon\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{C}$ we define
the norm $Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})$ by halving all
integrability exponents in the expression (3) for the norm
$Y_{N}(\mathbb{R})$:

$\|v\|_{Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})}\coloneqq\|v\|_{L_{t}^{\frac{1}{\varepsilon_{0}}}L_{x}^{\frac{1}{1-\varepsilon_{0}}}(\mathbb{R}\times\mathbb{R}^{d})}+\|v\|_{L_{t}^{\frac{1}{1-\varepsilon_{0}}}L_{x}^{\frac{d}{d-2}\frac{1}{1-\varepsilon_{0}}}(\mathbb{R}\times\mathbb{R}^{d})}\\\
\begin{aligned}
&+\sum_{l=1}^{d}\Big{(}N^{-\frac{1}{2}}\|v\|_{L_{e_{l}}^{(\frac{1}{1-\varepsilon_{0}},\frac{1}{\varepsilon_{0}},\frac{1}{\varepsilon_{0}})}(\mathbb{R})}+N^{-\frac{1}{2}}\|v\|_{L_{e_{l}}^{(\frac{1}{1-\varepsilon_{0}},\frac{1}{\varepsilon_{0}},\frac{\mathfrak{c}_{0}}{2(1-\varepsilon_{0})})}(\mathbb{R})}\Big{)}\\\
&+\sum_{l=1}^{d}\Big{(}N^{\frac{1}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}v\|_{L_{e_{l}}^{(\frac{1}{\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}})}(\mathbb{R})}+N^{\frac{1}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}v\|_{L_{e_{l}}^{(\frac{1}{\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}})}(\mathbb{R})}\Big{)}.\end{aligned}$

Note that
$\big{\|}|v|^{\nicefrac{{1}}{{2}}}\big{\|}_{Y_{N}(\mathbb{R})}^{2}\approx_{\varepsilon_{0}}\|v\|_{Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})}$,
and consequently after taking the $Y_{N}(\mathbb{R})$ norm on both sides of
(6) and using the triangle inequality for
$Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})$ we have that

$\displaystyle\Big{\|}\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}\Big{\|}_{Y_{N}(\mathbb{R})}^{2}$
$\displaystyle\hskip
50.00008pt\lesssim_{|\tau|,\varepsilon_{0}}(\gamma-1)^{|\tau|}\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})}.$

Finally, by Hölder’s inequality one has
$\big{\|}v_{1}v_{2}\big{\|}_{Y_{N}^{1/2}(\mathbb{R})}\lesssim\big{\|}v_{1}\big{\|}_{Y_{N}(\mathbb{R})}\big{\|}v_{2}\big{\|}_{Y_{N}(\mathbb{R})}$
and (6) follows. ∎

##  7\. Conclusion - Proof of main theorems

In this section, we use Theorem 1.5 and Theorem 1.6 proved respectively in
Section 5 and Section 6 to deduce the remaining theorems stated in Section 1.

###### Proof of Theorem 1.3 .

By Theorem 1.6 the assumptions of Theorem 1.5 on $z_{k}$ are satisfied almost
surely. Then, the first assertion follows from (Time-continuity and scattering
of multilinear data: ) and Theorem 1.6 and the second one follows from the
“Time-continuity and scattering of the multilinear data” claim of Theorem 1.5.
∎

###### Proof of Theorems 1.1, 1.2, and 1.4.

Let us relabel $S$ as $S^{\prime}$ in the theorems above. Since
$\lim_{M\to\infty}\mu(M,S_{\mathrm{min}})\geq\mathfrak{s}_{c}$, we fix $M$
such that $\mu(M+1,S^{\prime})>\mathfrak{s}_{c}$ and an arbitrary
$\mathfrak{s}$ with $\mathfrak{s}_{c}<\mathfrak{s}<\mu(M+1,S^{\prime})$, for
Theorem 1.4.

Next, we fix $S>S_{\mathrm{min}}$ such that $S<S^{\prime}<\min(2S,S+1)$ and
$\mu(M+1,S^{\prime})>\mathfrak{s}$. Finally we choose
$0<\varepsilon\lesssim_{M,S,S^{\prime}}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,M,S,S^{\prime}}1$ such that such that
$\varepsilon<\frac{\min(S,1,S^{\prime}-S)}{3M}$ and the conditions of Theorem
1.5 and of Theorem 1.6 are fulfilled.

According to Theorem 1.6 with $S$ replaced by $S+\varepsilon$, and because
$\mu(k,S+\varepsilon)\geq\mu(k,S)+\varepsilon$, we obtain for all $k\leq M$
that

$\mathbb{P}\Big{(}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}>\lambda\Big{)}\leq
C\exp\Bigg{(}-\frac{\lambda^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,.$

This allows us to restrict our attention from now on to the probability set
$\Omega_{0}$ with $\mathbb{P}(\Omega_{0})=1$ for which

$\max_{k\leq
M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}<+\infty$

Define the random time

$T=\begin{dcases}\frac{1}{C}\big{(}1+\max_{k\leq
M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\big{)}^{-\nicefrac{{3}}{{c}}}&\text{if
}R>\delta_{0}\,,\\\ +\infty&\text{if }\begin{aligned}
&\|\mathfrak{f}\|_{H^{S+\varepsilon}_{x}}\leq\delta_{0}\\\
&\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\quad
k\in\\{1,\ldots,M\\}\end{aligned}\end{dcases}$

with $\delta_{0}$, $c$, and $C$ as in Theorem 1.5. The local existence of a
random solution $u=\mathfrak{z}_{\leq M}+u^{\\#}_{M}$ on $[0,T)$ with

$\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}([0,T))}+\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,T);H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})\big{)}}<\infty$

is guaranteed by the local and global existence claims Theorem 1.5. Such a
solution is unique among those satisfying $u^{\\#}_{M}\in
C^{0}\big{(}[0,T);H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ as claimed by the
uniqueness claim of Theorem 1.5. Furthermore, since
$\mathfrak{s}<\mu(M+1,S^{\prime})$ is arbitrary, the uniqueness claim of
Theorem 1.5 shows that

$\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,T);H^{\widetilde{\mathfrak{s}}}_{x}(\mathbb{R}^{d})\big{)}}<\infty$

for any $\widetilde{\mathfrak{s}}<\mu(M+1,S^{\prime})$, as required by Theorem
1.4.

The measurability of $u$ follows from the fact that $u^{\\#}_{M}$ depends
continuously on the multilinear data $(\mathfrak{f},\vec{\mathfrak{z}}_{M})$.

To establish almost-certain continuity of the full solution $u$ we recall that
Theorem 1.3 applied with $S+\varepsilon$ in place of $\varepsilon$ shows that
almost surely

$\|\mathfrak{z}_{k}\|_{C^{0}\big{(}[0,\infty);H^{\mu(k,S)}_{x}(\mathbb{R}^{d})\big{)}}<\infty\,,$

while for $k=1$ it holds that
$\|\mathfrak{z}_{1}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}<\infty$
since, using Corollary 6.6, we get

$\|\mathfrak{z}_{1}(t)\|_{H^{S^{\prime}}_{x}(\mathbb{R}^{d})}=\|e^{it\Delta}\mathfrak{f}\|_{H^{S^{\prime}}_{x}(\mathbb{R}^{d})}=\|\mathfrak{f}\|_{H^{S^{\prime}}_{x}(\mathbb{R}^{d})}<\infty\,.$

Since we chose $S$ such $\mu(k,S)>S^{\prime}$ for $k\geq 2$, the triangle
inequality implies

$\|u\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}\leq\begin{aligned}
\|\mathfrak{z}_{1}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}&+\sum_{k=2}^{M}\|\mathfrak{z}_{k}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}\\\
&+\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}<\infty\,,\end{aligned}$

as required.

Next, since $T\geq\frac{1}{C}\big{(}1+\max_{k\leq
M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\big{)}^{-\nicefrac{{3}}{{c}}}$,
accroding to the defintion (7). The probability estimates (7) imply for any
$\lambda\geq\delta_{0}$ and $k\leq M$ that

$\mathbb{P}\Big{(}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}>\lambda\Big{)}\leq
C\exp\Bigg{(}-\frac{\lambda^{\frac{2}{M}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,.$

Then (1.1) follows after standard algebraic manipulations.

To prove Theorem 1.2, choose $\delta_{0}$ as in Theorem 1.5 and note that by
choosing appropriate $\varepsilon,\varepsilon_{0},\mathfrak{s}_{c}$, and $M$
in Theorem 1.5, then $\delta_{0}$ depends only on $S$ and $\mathfrak{s}_{c}$.
Set

$\Omega_{\mathrm{glob}}=\\{\omega:\|\mathfrak{f}\|_{H^{S+\varepsilon}_{x}}\leq\delta_{0},\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\quad
k\in\\{1,\ldots,M\\}\\}$

and note that by (7), $T=\infty$ on $\Omega_{\mathrm{glob}}$. Then (7) implies

(7.3)
$\mathbb{P}\Big{(}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\Big{)}\geq
1-C\exp\Bigg{(}-\frac{\delta_{0}^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,,$

and therefore

(7.4)
$\mathbb{P}\Big{(}\max_{k=1,\ldots,M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\Big{)}\geq
1-CM\exp\Bigg{(}-\frac{\delta_{0}^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,.$

Moreover, by (6.6) we have

(7.5)
$\mathbb{P}\Big{(}\|\mathfrak{f}\|_{H^{S+\varepsilon}_{x}}\leq\delta_{0}\Big{)}\geq
1-C\exp\Bigg{(}-\frac{\delta_{0}^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}$

and (1.2) follows.

Scattering for the random mulitilinear terms $\mathfrak{z}_{k}$ follows from
Theorem 1.3 while scattering in $H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ of
$u^{\\#}_{M}$ follows from (Scattering of global solutions: ) since
$T=+\infty$ on $\Omega_{\mathrm{glob}}$. ∎

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