define $\mathcal{Q}$ to be dominant, if (i) $\mathcal{Q}_{0}$ is either a
$(1,1)$-mini couple or a regular double chain formed by two _dominant_ regular
chains, and (ii) each regular couple $\mathcal{Q}_{j}$ is dominant.

Figure 18. An example of two equivalent dominant couples with
$(m^{+},m^{-})=(1,2)$ and $((m^{+})^{\prime},(m^{-})^{\prime})=(2,1)$. Here we
assume that (i) couples represented by the same color are equivalent (with the
corresponding $Z$ sets, which are omitted), and (ii) the symbol $[Z]$ means
the value of $j$ corresponding to this branching node belongs to the suitable
$Z^{\pm}$ or $(Z^{\pm})^{\prime}$ set. The code of each mini tree is indicated
beside the node $\mathfrak{n}_{a}^{\pm}$ and
$(\mathfrak{n}_{a}^{\pm})^{\prime}$, and $\mathcal{Q}_{lp}$ and
$\mathcal{Q}_{lp}^{\prime}$ have type 1.

####  4.4.1. An equivalence relation

Given a dominant couple $\mathcal{Q}$, recall that $\mathcal{N}^{*}$ is the
set of branching nodes, and $\mathcal{N}^{ch}\subset\mathcal{N}^{*}$ is
defined in Definition 4.11. Let $Z$ be a _special_ subset of
$\mathcal{N}^{ch}$, which will be defined inductively in Definition 4.18
below; we call $\mathscr{Q}:=(\mathcal{Q},Z)$ an _enhanced dominant couple_ ,
and when $Z=\varnothing$, we will also denote
$\mathscr{Q}=(\mathcal{Q},\varnothing)$ just by $\mathcal{Q}$ for convenience.

######  Definition 4.18.

We inductively define special subsets $Z\subset\mathcal{N}^{ch}$, and an
equivalence relation $\sim$ between enhanced dominant couples
$\mathscr{Q}:=(\mathcal{Q},Z)$, as follows. First $\varnothing$ is a special
subset and the enhanced trivial couple $(\times,\varnothing)$ is only
equivalent to itself, moreover two enhanced dominant couples where the
$\mathcal{Q}$ have different types are never equivalent.

Next, if $\mathscr{Q}=(\mathcal{Q},Z)$ and
$\mathscr{Q}^{\prime}=(\mathcal{Q}^{\prime},Z^{\prime})$, where $\mathcal{Q}$
and $\mathcal{Q}^{\prime}$ have type $1$ (recall the definition of type in
Proposition 4.10), then we have
$\mathcal{N}^{ch}=\mathcal{N}_{1}^{ch}\cup\mathcal{N}_{2}^{ch}\cup\mathcal{N}_{3}^{ch}\cup\\{\mathfrak{r}\\}$
where $\mathfrak{r}$ is the root with $+$ sign, see Definition 4.11. Then $Z$
is special if and only if $Z=Z_{1}\cup Z_{2}\cup Z_{3}$ (i.e. $\mathfrak{r}$
is _not_ in $Z$) where $Z_{j}\subset\mathcal{N}_{j}^{ch}$ is special, and
similarly for $\mathcal{Q}^{\prime}$. Let
$\mathscr{Q}_{j}=(\mathcal{Q}_{j},Z_{j})$, we define
$\mathscr{Q}\sim\mathscr{Q}^{\prime}$ if and only if
$\mathscr{Q}_{j}\sim\mathscr{Q}_{j}^{\prime}$ for $1\leq j\leq 3$.

Now let $\mathscr{Q}$ and $\mathscr{Q}^{\prime}$ be as before, but suppose
$\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ have type $2$. Let $\mathcal{Q}_{0}$
be associated with $\mathcal{Q}$ as in Proposition 4.10, and similarly for
$\mathcal{Q}^{\prime}$ (same for the other objects appearing below). Suppose
the two regular chains of $\mathcal{Q}_{0}$ have scale $2m^{+}$ and $2m^{-}$
respectively, and let the branching nodes in $\mathcal{Q}_{0}$ be
$\mathfrak{n}_{a}^{\pm}(1\leq a\leq 2m^{\pm})$, where
$\mathfrak{n}_{2j-1}^{\pm}$ is paired with $\mathfrak{n}_{2j}^{\pm}$ for
$1\leq j\leq m^{\pm}$, see Figure 18. We will use the notations in Definition
4.11, and note that $\mathcal{Q}$ is dominant and $\mathcal{Q}_{lp}$ is
trivial or has type 1. Recall that

$\mathcal{N}^{ch}=\bigg{(}\bigcup_{j,\epsilon,\iota}\mathcal{N}_{j,\epsilon,\iota}^{ch}\bigg{)}\cup\mathcal{N}_{lp}^{ch}\cup\big{\\{}\mathfrak{n}_{2j-1}^{+}:1\leq
j\leq m^{+}\big{\\}}\cup\big{\\{}\mathfrak{n}_{2j-1}^{-}:1\leq j\leq
m^{-}\big{\\}}$ (4.3)

as in (4.2); then $Z$ is special if and only if

$Z=\bigg{(}\bigcup_{j,\epsilon,\iota}Z_{j,\epsilon,\iota}\bigg{)}\cup
Z_{lp}\cup\big{\\{}\mathfrak{n}_{2j-1}^{+}:j\in
Z^{+}\big{\\}}\cup\big{\\{}\mathfrak{n}_{2j-1}^{-}:j\in Z^{-}\big{\\}}$ (4.4)

for some special subsets
$Z_{j,\epsilon,\iota}\subset\mathcal{N}_{j,\epsilon,\iota}^{ch}$ and
$Z_{lp}\subset\mathcal{N}_{lp}^{ch}$, and some subsets
$Z^{\pm}\subset\\{1,\cdots,m^{\pm}\\}$. Similar representations are defined
for $\mathscr{Q}^{\prime}$. For $\epsilon\in\\{\pm\\}$ and each $1\leq j\leq
m^{\epsilon}$, consider the tuple
$(\mathtt{I}_{j,\epsilon},\mathtt{c}_{j,\epsilon},\mathscr{X}_{j,\epsilon,1},\mathscr{X}_{j,\epsilon,2})$.
Here $\mathtt{I}_{j,\epsilon}=1$ if $j\in Z^{\epsilon}$ and
$\mathtt{I}_{j,\epsilon}=0$ otherwise, $\mathtt{c}_{j,\epsilon}\in\\{1,2,3\\}$
is the _first digit of_ the code of the mini tree associated with the pair
$\\{2j-1,2j\\}\in\mathcal{P}^{\epsilon}$ (see Definition 4.11; this code is
also the code assigned for the pair $\\{2j-1,2j\\}\in\mathcal{P}^{\epsilon}$
as in Proposition 4.7). Moreover $\mathscr{X}_{j,\epsilon,\iota}$ is the
equivalence class of the enhanced dominant couple
$\mathscr{Q}_{j,\epsilon,\iota}=(\mathcal{Q}_{j,\epsilon,\iota},Z_{j,\epsilon,\iota})$
for $\iota\in\\{1,2\\}$, and let $\mathscr{Y}$ be the equivalence class of the
enhanced dominant couple $\mathscr{Q}_{lp}=(\mathcal{Q}_{lp},Z_{lp})$.

We now define $\mathscr{Q}\sim\mathscr{Q}^{\prime}$, if and only if (i)
$m^{+}+m^{-}=(m^{+})^{\prime}+(m^{-})^{\prime}$, and (ii) the tuples coming
from $\mathcal{Q}_{0}$ (there are total $m^{+}+m^{-}$ of them) form _a
permutation of_ the corresponding tuples coming from
$\mathcal{Q}_{0}^{\prime}$ (there are total
$(m^{+})^{\prime}+(m^{-})^{\prime}$ of them), and (iii)
$\mathscr{Y}=\mathscr{Y}^{\prime}$. Finally, note that if
$\mathscr{Q}=(\mathcal{Q},Z)$ and
$\mathscr{Q}^{\prime}=(\mathcal{Q}^{\prime},Z^{\prime})$ are equivalent then
$n(\mathcal{Q})=n(\mathcal{Q}^{\prime})$ and $|Z|=|Z^{\prime}|$. When
$\mathscr{Q}\sim\mathscr{Q}^{\prime}$ with $Z=Z^{\prime}=\varnothing$, we also
say that $\mathcal{Q}\sim\mathcal{Q}^{\prime}$.

###  4.5. Encoded trees

Let $\mathcal{T}$ be a tree, we will assign to each of its _branching_ nodes
$\mathfrak{n}\in\mathcal{N}$ a _code_
$\mathtt{c}=\mathtt{c}_{\mathfrak{n}}\in\\{0,1,2,3\\}$ to form an _encoded
tree_.

Given a encoded tree $\mathcal{T}$, define the _canonical path_ to be the
unique path $\gamma$ starting from the root $\mathfrak{r}$ and ending at
either a leaf or a branching node with code $0$, such that any non-terminal
node $\mathfrak{n}\in\gamma$ is a branching node with code
$\mathtt{c}_{\mathfrak{n}}\in\\{1,2,3\\}$, and the next node
$\mathfrak{n}^{\prime}$ in $\gamma$ is the $\mathtt{c}_{\mathfrak{n}}$-th
child of $\mathfrak{n}$ counting from left to right.

######  Definition 4.19.

An _encoded chain_ is an encoded tree whose canonical path $\gamma$ ends at a
leaf, and for any non-terminal node $\mathfrak{n}\in\gamma$, the two children
of $\mathfrak{n}$ other than $\mathfrak{n}^{\prime}$ are both leaves, where
$\mathfrak{n}^{\prime}$ is the next node in $\Gamma$. We call the endpoint of
$\gamma$, which is a leaf, the _tail leaf_ of $\mathcal{T}$.

######  Proposition 4.20.

Given an encoded tree $\mathcal{T}\neq\bullet$, we say $\mathcal{T}$ has _type
1_ if its root $\mathfrak{r}$ has code $\mathtt{c}_{\mathfrak{r}}=0$,
otherwise we say it has _type 2_. Now, for any type 2 encoded tree
$\mathcal{T}$, there is a unique encoded chain $\mathcal{T}_{0}\neq\bullet$,
such that $\mathcal{T}$ is obtained from $\mathcal{T}_{0}$ by replacing each
leaf with an encoded tree, and that the tail leaf is replaced by either
$\bullet$ (i.e. remains a leaf) or an encoded tree of type 1.

###### Proof.

This is straightforward from the definition. In fact $\mathcal{T}$ has type 1
if and only if $\mathtt{c}_{\mathfrak{r}}=0$ for the root $\mathfrak{r}$;
suppose $\mathtt{c}_{\mathfrak{r}}\in\\{1,2,3\\}$, then $\mathcal{T}_{0}$ must
have the same canonical path $\gamma$ as $\mathcal{T}$. Thus $\mathcal{T}_{0}$
must be the encoded tree formed by selecting each node in $\gamma$ and
collapsing the subtree rooted at this node to a leaf. Such $\mathcal{T}_{0}$
is clearly unique, and is nontrivial when
$\mathtt{c}_{\mathfrak{r}}\in\\{1,2,3\\}$. ∎

######  Definition 4.21.

We define the equivalence relation between encoded trees as follows. First the
trivial tree $\bullet$ is only equivalent to itself, and encoded trees of
different type are not equivalent. Now suppose $\mathcal{T}$ and
$\mathcal{T}^{\prime}$ are two encoded trees of type 1, then define
$\mathcal{T}\sim\mathcal{T}^{\prime}$ if and only if
$\mathcal{T}_{j}\sim\mathcal{T}_{j}^{\prime}$ for $1\leq j\leq 3$, where
$\mathcal{T}_{j}$ and $\mathcal{T}_{j}^{\prime}$ are the subtrees of
$\mathcal{T}$ and $\mathcal{T}^{\prime}$ respectively, from left to right.

Now suppose $\mathcal{T}$ and $\mathcal{T}^{\prime}$ are two encoded trees of
type 2, then by Proposition 4.20 there exists a unique encoded chain
$\mathcal{T}_{0}$ such that $\mathcal{T}$ is formed by replacing each leaf of
$\mathcal{T}_{0}$ with an encoded tree, and the same holds for
$\mathcal{T}^{\prime}$. Let the branching nodes of $\mathcal{T}_{0}$ from top
to bottom be $\mathfrak{n}_{j}\,(1\leq j\leq m)$. For each $1\leq j\leq m$,
let $\mathcal{T}_{j,1}$ and $\mathcal{T}_{j,2}$ be the two encoded trees that
replace the two children of $\mathfrak{n}_{j}$ other than $\mathfrak{n}_{j+1}$
(or the tail leaf), counted from left to right; moreover let
$\mathcal{T}_{ta}$ be the encoded tree replacing the tail leaf, which is
either trivial or has type 1. Consider the triples
$(\mathtt{c}_{j},\mathscr{Z}_{j,1},\mathscr{Z}_{j,2})$ for each $j$, where
$\mathtt{c}_{j}$ is the code of $\mathfrak{n}_{j}$, and
$\mathscr{Z}_{j,\iota}$ is the equivalence class of $\mathcal{T}_{j,\iota}$
for $\iota\in\\{1,2\\}$. Then the encoded trees $\mathcal{T}$ and
$\mathcal{T}^{\prime}$ are equivalent, if and only if (i) $m=m^{\prime}$ where
$m^{\prime}$ is defined similarly for $\mathcal{T}^{\prime}$, (ii) the triples
$(\mathtt{c}_{j},\mathscr{Z}_{j,1},\mathscr{Z}_{j,2})$ form a _permutation_ of
the corresponding triples coming from $\mathcal{T}_{0}^{\prime}$, and (iii)
$\mathcal{T}_{ta}$ is equivalent to $\mathcal{T}_{ta}^{\prime}$.

####  4.5.1. Dominant couples and encoded trees

Given any dominant couple $\mathcal{Q}$, we can inductively define a unique
encoded tree $\mathcal{T}$ associated to $\mathcal{Q}$, as follows.

######  Definition 4.22.

Let $\mathcal{Q}$ be a dominant couple, we define the encoded tree
$\mathcal{T}$ associated with $\mathcal{Q}$ as follows. First if
$\mathcal{Q}=\times$ then define $\mathcal{T}=\bullet$. Suppose $\mathcal{Q}$
has type 1, then let $\mathcal{Q}_{j}\,(1\leq j\leq 3)$ be defined as in
Definition 4.11, then define $\mathcal{T}$ to be the encoded tree such that
the root has code $0$, and the three subtrees are $\mathcal{T}_{j}\,(1\leq
j\leq 3)$ which are associated with $\mathcal{Q}_{j}$, from left to right.

Now suppose $\mathcal{Q}$ is a dominant couple of type 2. Let the relevant
notations like $\mathcal{Q}_{j,\epsilon,\iota}$ and $\mathcal{Q}_{lp}$ be as
in Definition 4.11. Let $m=m^{+}+m^{-}$, consider the triples
$(\mathtt{c}_{j,\epsilon},\mathcal{Q}_{j,\epsilon,1},\mathcal{Q}_{j,\epsilon,2})$
for $\epsilon\in\\{\pm\\}$ and $1\leq j\leq m^{\epsilon}$, where
$\mathtt{c}_{j,\epsilon}$ is the first digit of the code of the mini-tree
associated with the pair $\\{2j-1,2j\\}\in\mathcal{P}^{\epsilon}$ as in
Definition 4.18; we rearrange them putting the $\epsilon=+$ triples before the
$\epsilon=-$ ones, and in increasing order of $j$ for fixed sign. Let the
rearranged tuples be $(\mathtt{c}_{i},\mathcal{Q}_{i,1},\mathcal{Q}_{i,2})$
for $1\leq i\leq m$, then $\mathcal{T}$ is defined as follows. First let
$\mathcal{T}_{0}$ be the encoded chain which has $m$ branching nodes
$\mathfrak{n}_{i}\,(1\leq i\leq m)$ from top to bottom with code
$\mathtt{c}_{i}$, then for each $i$, replace the two children leaves of
$\mathfrak{n}_{i}$ other than $\mathfrak{n}_{i+1}$ (or the tail leaf) with
$\mathcal{T}_{i,1}$ and $\mathcal{T}_{i,2}$ which are the encoded trees
associated to $\mathcal{Q}_{i,1}$ and $\mathcal{Q}_{i,2}$ by induction
hypothesis. Finally the tail leaf is replaced by $\mathcal{T}_{ta}$ which is
the encoded tree associated with $\mathcal{Q}_{lp}$.

######  Proposition 4.23.

The mapping from dominant couples $\mathcal{Q}$ to encoded trees
$\mathcal{T}$, as defined in Definition 4.22, is surjective. Moreover two
dominant couples $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ are equivalent in
the sense of Definition 4.18, if and only if the associated encoded trees
$\mathcal{T}$ and $\mathcal{T}^{\prime}$ are equivalent in the sense of
Definition 4.21. In particular this mapping induces a bijection between the
equivalence classes of dominant couples and equivalence classes of encoded
trees.

###### Proof.

The mapping is surjective because for any $\mathcal{T}$ one can always
construct $\mathcal{Q}$ by reverting the construction in Definition 4.22,
following the same induction process using Proposition 4.20. Now recall that
equivalence between dominant couples $\mathcal{Q}$ is defined as a special
case in Definition 4.18 with $Z=\varnothing$, thus in Definition 4.18 for type
2 (type 1 is similar), the first component $\mathtt{I}_{j,\pm}$ of the tuple
$(\mathtt{I}_{j,\pm},\mathtt{c}_{j,\pm},\mathscr{X}_{j,\pm,1},\mathscr{X}_{j,\pm,2})$
is always $0$. Therefore, part (ii) of the equivalence relation between
$\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ can be described as the triples
$(\mathtt{c}_{j,\pm},\mathscr{X}_{j,\pm,1},\mathscr{X}_{j,\pm,2})$ coming from
$\mathcal{Q}$ being a permutation of the triples coming from
$\mathcal{Q}^{\prime}$ (as well as other similar conditions). By induction
hypothesis, this is equivalent to the triples
$(\mathtt{c}_{i},\mathscr{Z}_{i,1},\mathscr{Z}_{i,2})$ coming from
$\mathcal{Q}$ being a permutation of the triples coming from
$\mathcal{Q}^{\prime}$, where $\mathscr{Z}_{i,\iota}$ is the equivalence class
of the encoded tree associated to $\mathcal{Q}_{i,\iota}$, and the triples
$(\mathtt{c}_{i},\mathcal{Q}_{i,1},\mathcal{Q}_{i,2})$ are rearranged from the
triples
$(\mathtt{c}_{j,\epsilon},\mathcal{Q}_{j,\epsilon,1},\mathcal{Q}_{j,\epsilon,2})$
as in Definition 4.22. Then, using Definition 4.21, we see that this is
equivalent to part (ii) of the equivalence relation between $\mathcal{T}$ and
$\mathcal{T}^{\prime}$. Similarly the other parts also match, therefore
$\mathcal{Q}$ being equivalent to $\mathcal{Q}^{\prime}$ is equivalent to
$\mathcal{T}$ being equivalent to $\mathcal{T}^{\prime}$. ∎

####  4.5.2. A summary

We will be using the equivalence relations between enhanced dominant couples
$\mathscr{Q}=(\mathcal{Q},Z)$ (say $\sim_{1}$), between dominant couples
$\mathcal{Q}$ (say $\sim_{2}$), and between encoded trees (say $\sim_{3}$).
Clearly $\sim_{2}$ is a special case of $\sim_{1}$, and Proposition 4.23
establishes a bijection between equivalence classes under $\sim_{2}$ and
equivalence classes under $\sim_{3}$. By abusing notation, below we will use
the notation $\mathscr{X}$ (and similarly $\mathscr{Y}$ etc.) to denote an
equivalence class in each of these cases; the precise meaning will be clear
from the context. For later use, we list a few easily verified facts about
these equivalence classes below.

(1) We know that equivalent (enhanced) dominant couples and encoded trees must
have the same scale and $|Z|$, and the same type. The bijection in Proposition
4.23 also preserves the type; moreover, if $\mathcal{Q}$ has scale $2n$, then
the associated $\mathcal{T}$ has scale $n$. If $\mathscr{X}$ denotes the
equivalence class for both objects, we will define the _half-scale_ of
$\mathscr{X}$ to be $n$.

(2) If the net sign $\zeta^{*}(\mathcal{Q})$ of a dominant couple
$\mathcal{Q}$ is defined by (2.22), and we define the net sign
$\zeta^{*}(\mathcal{T})$ of an encoded tree $\mathcal{T}$ by

$\zeta^{*}(\mathcal{T})=\prod_{\mathfrak{n}\in\mathcal{N}}(-1)^{\mathtt{c}_{\mathfrak{n}}}$
(4.5)

where $\mathtt{c}_{\mathfrak{n}}$ is the code of $\mathfrak{n}$, then these
signs are preserved under equivalence (including $\sim_{1}$), and also under
the bijection in Proposition 4.23 (so
$\zeta^{*}(\mathcal{Q})=\zeta^{*}(\mathcal{T})$ if $\mathcal{T}$ is associated
to $\mathcal{Q}$).

(3) Let $\mathscr{X}$ be an equivalence class of enhanced dominant couples.
Then, if $\mathscr{X}$ has type 1, it can be uniquely determined (bijectively)
by an _ordered_ triple $(\mathscr{X}_{1},\mathscr{X}_{2},\mathscr{X}_{3})$ of
equivalence classes of enhanced dominant couples. If $\mathscr{X}$ has type 2,
it can be uniquely determined (bijectively) by the following objects:

  * •

A positive integer $m\geq 1$;

  * •

An _unordered_ collection of (ordered) tuples
$(\mathtt{I}_{j},\mathtt{c}_{j},\mathscr{X}_{j,1},\mathscr{X}_{j,2})$ for
$1\leq j\leq m$, where each $\mathtt{I}_{j}\in\\{0,1\\}$, each
$\mathtt{c}_{j}\in\\{1,2,3\\}$ and each $\mathscr{X}_{j,1}$ and
$\mathscr{X}_{j,2}$ is an equivalence class of enhanced dominant couples;

  * •

An equivalence class $\mathscr{Y}$ of enhanced dominant couples that is
trivial or has _type 1_.

(4) Let $\mathscr{X}$ be an equivalence class of dominant couples (with
$Z=\varnothing$) or encoded trees. Then the same description in (3) is valid,
except that the tuple
$(\mathtt{I}_{j},\mathtt{c}_{j},\mathscr{X}_{j,1},\mathscr{X}_{j,2})$ should
be replaced by the triple
$(\mathtt{c}_{j},\mathscr{X}_{j,1},\mathscr{X}_{j,2})$.

##  5\. Regular couples I: the $\mathcal{A}$ and $\mathcal{B}$ coefficients

We start with the analysis of $\mathcal{K}_{\mathcal{Q}}$ associated to the
regular couples $\mathcal{Q}$, which will occupy up to Section 7. The first
step is to obtain suitable estimates for the coefficients
$\mathcal{B}_{\mathcal{Q}}$ occurring in (2.24), which is based on
$\mathcal{A}_{\mathcal{T}}$ occurring in (2.12).

###  5.1. Properties of the coefficients $\mathcal{B}_{\mathcal{Q}}$

Recall the coefficients
$\mathcal{A}_{\mathcal{T}}=\mathcal{A}_{\mathcal{T}}(t,\alpha[\mathcal{N}])$
and
$\mathcal{B}_{\mathcal{Q}}=\mathcal{B}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{*}])$
defined in (2.13) and (2.23). By induction, we can also write

$\mathcal{A}_{\mathcal{T}}(t,\alpha[\mathcal{N}])=\int_{\mathcal{D}}\prod_{\mathfrak{n}\in\mathcal{N}}e^{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}},$ (5.1)

where the domain

$\mathcal{D}=\big{\\{}t[\mathcal{N}]:0<t_{\mathfrak{n}^{\prime}}<t_{\mathfrak{n}}<t,\mathrm{\
whenever\ }\mathfrak{n}^{\prime}\mathrm{\ is\ a\ child\ node\ of\
}\mathfrak{n}\big{\\}},$ (5.2)

and similarly

$\mathcal{B}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{*}])=\int_{\mathcal{E}}\prod_{\mathfrak{n}\in\mathcal{N}^{*}}e^{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}},$ (5.3)

where the domain

$\mathcal{E}=\big{\\{}t[\mathcal{N}^{*}]:0<t_{\mathfrak{n}^{\prime}}<t_{\mathfrak{n}},\mathrm{\
whenever\ }\mathfrak{n}^{\prime}\mathrm{\ is\ a\ child\ node\ of\
}\mathfrak{n};\\\ t_{\mathfrak{n}}<t\mathrm{\ whenever\
}\mathfrak{n}\in\mathcal{N}^{+}\mathrm{\ and\ }t_{\mathfrak{n}}<s\mathrm{\
whenever\ }\mathfrak{n}\in\mathcal{N}^{-}\big{\\}}.$ (5.4)

Now suppose $\mathcal{Q}$ is a _regular_ couple. By Proposition 4.3, for any
decoration $\mathscr{E}$ of $\mathcal{Q}$, we must have
$\zeta_{\mathfrak{n}^{\prime}}\Omega_{\mathfrak{n}^{\prime}}=-\zeta_{\mathfrak{n}}\Omega_{\mathfrak{n}}$
for any pair $\\{\mathfrak{n},\mathfrak{n}^{\prime}\\}$ of branching nodes.
Let $\mathcal{N}^{ch}$ be defined as in Definition 4.11, then we may define
$\widetilde{\mathcal{B}}_{\mathcal{Q}}=\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])$
by

$\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])=\mathcal{B}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{*}]),$
(5.5)

assuming that $\alpha[\mathcal{N}^{*}\backslash\mathcal{N}^{ch}]$ is defined
such that
$\zeta_{\mathfrak{n}^{\prime}}\alpha_{\mathfrak{n}^{\prime}}=-\zeta_{\mathfrak{n}}\alpha_{\mathfrak{n}}$
for each pair $\\{\mathfrak{n},\mathfrak{n}^{\prime}\\}$.

####  5.1.1. Structure of $\widetilde{\mathcal{B}}_{\mathcal{Q}}$

For a regular couple $\mathcal{Q}\neq\times$, let $\mathcal{Q}_{0}$ be
uniquely determined by Propositions 4.8 and 4.10, which is either a
$(1,1)$-mini couple, or a nontrivial regular double chain, such that
$\mathcal{Q}$ is obtained from $\mathcal{Q}_{0}$ by replacing each leaf-pair
with a regular couple. We will use the notations of Definition 4.11, including
for example $\mathcal{P}^{\pm}$, $m^{\pm}$ and
$\mathcal{Q}_{j,\epsilon,\iota}$, $\mathcal{Q}_{lp}$ and
$\mathcal{N}_{j,\epsilon,\iota}^{ch}$, $\mathcal{N}_{lp}^{ch}$ etc.

_Case 1_. If $\mathcal{Q}$ has type 1, then by (5.3) and (5.5), we deduce that

$\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])=\int_{0}^{t}\int_{0}^{s}e^{\pi
i\alpha_{\mathfrak{r}}(t_{1}-s_{1})}\prod_{j=1}^{3}\widetilde{\mathcal{B}}_{\mathcal{Q}_{j}}(t_{1},s_{1},\alpha[\mathcal{N}_{j}^{ch}])\,\mathrm{d}t_{1}\mathrm{d}s_{1}.$
(5.6)

We remark that in (5.6), the variables $(t_{1},s_{1})$ appearing in
$\widetilde{\mathcal{B}}_{\mathcal{Q}_{j}}$ may be replaced by $(s_{1},t_{1})$
for some $j$, depending on the signs of the leaves of $\mathcal{Q}_{0}$.

_Case 2_. Suppose $\mathcal{Q}$ has type 2. For each $1\leq j\leq m^{\pm}$,
let $\\{a,b\\}$ be the $j$-th pair in $\mathcal{P}^{\pm}$ where $a<b$, then
$\mathfrak{n}_{a}^{+}\in\mathcal{N}^{ch}$; we shall rename
$\alpha_{\mathfrak{n}_{a}^{+}}:=\alpha_{j}^{+}$, and define
$\beta_{a}^{+}:=\zeta_{\mathfrak{n}_{a}^{+}}\alpha_{\mathfrak{n}_{a}^{+}}=\epsilon_{j}^{+}\alpha_{j}^{+}$
where $\epsilon_{j}^{+}=\zeta_{\mathfrak{n}_{a}^{+}}\in\\{\pm\\}$ and
$\beta_{b}^{+}:=\zeta_{\mathfrak{n}_{b}^{+}}\alpha_{\mathfrak{n}_{b}^{+}}=-\epsilon_{j}^{+}\alpha_{j}^{+}$.
The same is done for the other regular chain $\mathcal{T}^{-}$. Then, by these
definitions, and (5.3) and (5.5), we deduce that

$\displaystyle\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])$
$\displaystyle=\int_{t>t_{1}>\cdots>t_{2m^{+}}>0}e^{\pi
i(\beta_{1}^{+}t_{1}+\cdots+\beta_{2m^{+}}^{+}t_{2m^{+}})}\prod_{j=1}^{m^{+}}\prod_{\iota=1}^{2}\widetilde{\mathcal{B}}_{\mathcal{Q}_{j,+,\iota}}\big{(}t_{a},t_{b},\alpha[\mathcal{N}_{j,+,\iota}^{ch}]\big{)}$
(5.7) $\displaystyle\times\int_{s>s_{1}>\cdots>s_{2m^{-}}>0}e^{\pi
i(\beta_{1}^{-}s_{1}+\cdots+\beta_{2m^{-}}^{-}s_{2m^{-}})}\prod_{j=1}^{m^{-}}\prod_{\iota=1}^{2}\widetilde{\mathcal{B}}_{\mathcal{Q}_{j,-,\iota}}\big{(}s_{a},s_{b},\alpha[\mathcal{N}_{j,-,\iota}^{ch}]\big{)}$
$\displaystyle\times\widetilde{\mathcal{B}}_{\mathcal{Q}_{lp}}\big{(}t_{2m^{+}},s_{2m^{-}},\alpha[\mathcal{Q}_{lp}^{ch}]\big{)}\prod_{j=1}^{m^{+}}\mathrm{d}t_{j}\prod_{j=1}^{m^{-}}\mathrm{d}s_{j}.$

As in _Case 1_ , we remark that in some factors $(t_{a},t_{b})$ may be
replaced by $(t_{b},t_{a})$, and similarly for $(s_{a},s_{b})$ and
$(t_{2m^{+}},s_{2m^{-}})$, depending on the signs of the relevant leaves. Note
also that $\mathcal{Q}_{lp}$ is trivial (in which case
$\widetilde{\mathcal{B}}_{\mathcal{Q}_{lp}}\equiv 1$) or has type 1; this is
not needed here, but will be useful later.

####  5.1.2. Estimates for $\widetilde{\mathcal{B}}_{\mathcal{Q}}$

The goal of this section is to prove the following

######  Proposition 5.1.

Let $\mathcal{Q}$ be a regular couple of scale $2n$. Then, the function

$\widetilde{\mathcal{B}}_{\mathcal{Q}}\big{(}t,s,\alpha[\mathcal{N}^{ch}]\big{)}$

is the sum of at most $2^{n}$ terms. For each term there exists a subset
$Z\subset\mathcal{N}^{ch}$, such that this term has form

$\prod_{\mathfrak{n}\in
Z}\frac{\chi_{\infty}(\alpha_{\mathfrak{n}})}{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}}\cdot\int_{\mathbb{R}^{2}}\mathcal{C}\big{(}\lambda_{1},\lambda_{2},\alpha[\mathcal{N}^{ch}\backslash
Z]\big{)}e^{\pi
i(\lambda_{1}t+\lambda_{2}s)}\,\mathrm{d}\lambda_{1}\mathrm{d}\lambda_{2}$
(5.8)

for $t,s\in[0,1]$, where $\chi_{\infty}$ is as in Section 2.3.1. In (5.8) the
function $\mathcal{C}$ satisfies the estimate

$\int\langle\lambda_{1}\rangle^{1/4}\langle\lambda_{2}\rangle^{1/4}\big{|}\partial_{\alpha}^{\rho}\mathcal{C}\big{(}\lambda_{1},\lambda_{2},\alpha[\mathcal{N}^{ch}\backslash
Z]\big{)}\big{|}\,\mathrm{d}\alpha[\mathcal{N}^{ch}\backslash
Z]\mathrm{d}\lambda_{1}\mathrm{d}\lambda_{2}\leq C^{n}(2|\rho|)!$ (5.9)

for any multi-index $\rho$, and we also have the weighted estimate

$\int\langle\lambda_{1}\rangle^{1/8}\langle\lambda_{2}\rangle^{1/8}\cdot\max_{\mathfrak{n}\in\mathcal{N}^{ch}\backslash
Z}\langle\alpha_{\mathfrak{n}}\rangle^{1/8}\big{|}\mathcal{C}\big{(}\lambda_{1},\lambda_{2},\alpha[\mathcal{N}^{ch}\backslash
Z]\big{)}\big{|}\,\mathrm{d}\alpha[\mathcal{N}^{ch}\backslash
Z]\mathrm{d}\lambda_{1}\mathrm{d}\lambda_{2}\leq C^{n}.$ (5.10)

We will denote the $(\lambda_{1},\lambda_{2})$ integral in (5.8) by
$\widetilde{\mathcal{B}}_{\mathcal{Q},Z}=\widetilde{\mathcal{B}}_{\mathcal{Q},Z}(t,s,\alpha[\mathcal{N}^{ch}\backslash
Z])$, so we have

$\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])=\sum_{Z\subset\mathcal{N}^{ch}}\prod_{\mathfrak{n}\in
Z}\frac{\chi_{\infty}(\alpha_{\mathfrak{n}})}{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}}\cdot\widetilde{\mathcal{B}}_{\mathcal{Q},Z}(t,s,\alpha[\mathcal{N}^{ch}\backslash
Z]).$ (5.11)

The proof of Proposition 5.1 is done by induction, using the recursive
description in (5.6) and (5.7). Clearly the hardest case is _Case 2_ , where
$\mathcal{Q}_{0}$ is a regular double chain. Therefore, before proving
Proposition 5.1 in Section 5.3 below, we first need to analyze the expressions
associated with regular chains. This will be done in Section 5.2.

###  5.2. Regular chain estimates

Let $\mathcal{P}$ be a legal partition of $\\{1,\cdots,2m\\}$. As in Section
5.1.1, we list the pairs $\\{a,b\\}\in\mathcal{P}\,(a<b)$ in the increasing
order of $a$. If the $j$-th pair is $\\{a,b\\}$, we define
$\beta_{a}=\epsilon_{j}\alpha_{j}$ and $\beta_{b}=-\epsilon_{j}\alpha_{j}$,
where $1\leq j\leq m$ and $\epsilon_{j}\in\\{\pm\\}$. For this section, we
also introduce the parameters $\lambda_{a}\,(1\leq a\leq 2m)$ and
$\lambda_{0}$, and define $\mu_{j}=\lambda_{a}+\lambda_{b}$ if the $j$-th pair
is $\\{a,b\\}$. Define now

$K(t,\alpha_{1},\cdots,\alpha_{m},\lambda_{0},(\lambda_{a})_{1\leq a\leq
2m}):=\int_{t>t_{1}>\cdots>t_{2m}>0}e^{\pi
i[(\beta_{1}+\lambda_{1})t_{1}+\cdots+(\beta_{2m}+\lambda_{2m})t_{2m}]+\pi
i\lambda_{0}t_{2m}}\,\mathrm{d}t_{1}\cdots\mathrm{d}t_{2m}.$ (5.12)

If we define the operator

$I_{\beta}f(t)=\int_{0}^{t}e^{\pi i\beta s}f(s)\,\mathrm{d}s,$

then we have

$K(t,\alpha_{1},\cdots,\alpha_{m},\lambda_{0},(\lambda_{a})_{1\leq a\leq
2m})=I_{\beta_{1}+\lambda_{1}}\cdots I_{\beta_{2m}+\lambda_{2m}}(e^{\pi
i\lambda_{0}s})(t).$ (5.13)

By definition, if we replace $\alpha_{j}$ by
$\widetilde{\alpha_{j}}=\alpha_{j}+\epsilon_{j}\lambda_{a}$, where
$\\{a,b\\}\,(a<b)$ is the $j$-the pair in $\mathcal{P}$, and replace
$\lambda_{a}$ by $\widetilde{\lambda_{a}}=0$ and replace $\lambda_{b}$ by
$\widetilde{\lambda_{b}}=\lambda_{a}+\lambda_{b}=\mu_{j}$, it is easily seen
that

$K(t,\alpha_{1},\cdots,\alpha_{m},\lambda_{0},\lambda_{1},\cdots,\lambda_{2m})=K(t,\widetilde{\alpha_{1}},\cdots,\widetilde{\alpha_{m}},\lambda_{0},\widetilde{\lambda_{1}},\cdots,\widetilde{\lambda_{2m}}).$
(5.14)

Therefore, in this section we will assume $\lambda_{a}=0$ and
$\lambda_{b}=\mu_{j}$ for the $j$-th pair $\\{a,b\\}\,(a<b)$.

For the purpose of Section 5.2.1 below, we also define the operators

$J_{\alpha;\gamma_{1},\gamma_{2}}f(t)=\int_{0}^{t}e^{\pi i\alpha(t-s)}e^{\pi
i(\gamma_{1}t+\gamma_{2}s)}f(s)\,\mathrm{d}s$

and

$R_{\alpha,\beta;\gamma_{1},\gamma_{2},\gamma_{3}}f(t)=\int_{0}^{t}\frac{\chi_{\infty}(\alpha+\gamma_{3})}{\alpha+\gamma_{3}}e^{\pi
i\beta(t-s)}e^{\pi i(\gamma_{1}t+\gamma_{2}s)}f(s)\,\mathrm{d}s.$

Given variables $(\alpha_{a},\cdots,\alpha_{b})$, we define a _cluster_ to be
any linear combination $y_{a}\alpha_{a}+\cdots+y_{b}\alpha_{b}$ where
$y_{j}\in\\{-1,0,1\\}$. Moreover, below we always view the operators as
mapping functions on $[0,1]$ to functions on $[0,1]$.

####  5.2.1. Class $J$ and $R$ operators

######  Definition 5.2.

Let $E$ be a finite set of positive integers, and $A\subset E$. We define an
operator $\mathcal{J}=\mathcal{J}_{\alpha[A],\mu[E]}$, which depends on the
variables $\alpha[A]$ and $\mu[E]$, to have class $J$ (and norm
$\|\mathcal{J}\|=1$), if we have

$\mathcal{J}_{\alpha[A],\mu[E]}=\int
m(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})J_{\ell;\gamma_{1},\gamma_{2}}\,\mathrm{d}\gamma_{1}\,\mathrm{d}\gamma_{2},$
(5.15)

where $\ell$ is a cluster of $\alpha[A]$, and
$m=m(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})$ is a function such that

${\int\bigg{(}1+\sum_{j\in
A}|\alpha_{j}|+|\gamma_{1}|\bigg{)}^{1/4}\,|\partial_{\alpha}^{\rho}m(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})|\,\mathrm{d}\alpha[A]\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{2}\leq(2|\rho|)!},$
(5.16)

for all $\mu[E]$ and multi-index $\rho$. Note that the weight on the left hand
side of (5.16) does not involve $|\gamma_{2}|$.

We also define an operator $\mathcal{R}=\mathcal{R}_{\alpha[A],\mu[E]}$, which
again depends on the variables $\alpha[A]$ and $\mu[E]$, to have class $R$
(and norm $\|\mathcal{R}\|=1$), if $1\in A$ (called the _special index_), and
we have

$\mathcal{R}_{\alpha[A],\mu[E]}=\int
m(\alpha[A\backslash\\{1\\}],\mu[E],\gamma_{1},\gamma_{2},\gamma_{3})R_{\ell_{1}+\epsilon\alpha_{1},\ell_{2}+\epsilon\alpha_{1};\gamma_{1},\gamma_{2},\gamma_{3}}\,\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}$
(5.17)

where $\epsilon\in\\{\pm\\}$, $\ell_{1}$ and $\ell_{2}$ are two clusters of
$\alpha[A\backslash\\{1\\}]$, and
$m=m(\alpha[A\backslash\\{1\\}],\mu[E],\gamma_{1},\gamma_{2},\gamma_{3})$ is a
function such that

${\int\bigg{(}1+\sum_{j\in
A\backslash\\{1\\}}|\alpha_{j}|+|\gamma_{1}|+|\gamma_{3}|\bigg{)}^{1/4}|\partial_{\alpha}^{\rho}m(\alpha[A\backslash\\{1\\}],\mu[E],\gamma_{1},\gamma_{2},\gamma_{3})|\,\mathrm{d}\alpha[A\backslash\\{1\\}]\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\leq(2|\rho|)!},$
(5.18)

for all $\mu[E]$ and multi-index $\rho$. Note that
$m=m(\alpha[A\backslash\\{1\\}],\mu[E],\gamma_{1},\gamma_{2},\gamma_{3})$ does
not depend on $\alpha_{1}$, and the weight on the left hand side of (5.18)
also does not involve $|\gamma_{2}|$.

More generally, we also define an operator to have class $J$ (or $R$) if it
can be written as a linear combination of operators satisfying (5.15)–(5.16)
(or (5.17)–(5.18)) for different choices of $\ell$ (or $(\ell_{1},\ell_{2})$);
define the norm $\|\mathcal{J}\|$ and $\|\mathcal{R}\|$ to be the infimum of
the sum of absolute value of the coefficients. Below we will study
compositions of class $J$ and $R$ operators, and compositions of them with
other explicit operators; when doing so we always understand that the
variables $\alpha_{j}$ and $\mu_{j}$ involved in different operators are
different.

####  5.2.2. Compositions of class $J$ and $R$ operators

######  Lemma 5.3.

The composition of two class $J$ operators is of class $J$, and the norms
satisfy that $\|\mathcal{J}^{(1)}\mathcal{J}^{(2)}\|\leq
C\|\mathcal{J}^{(1)}\|\cdot\|\mathcal{J}^{(2)}\|$ (the same will be true for
subsequent lemmas).

###### Proof.

Let $\mathcal{J}_{\alpha[A],\mu[E]}^{(1)}$ and
$\mathcal{J}_{\alpha[B],\mu[F]}^{(2)}$ be of class $J$; we may assume that
each satisfies (5.15)–(5.16). Let $\mathcal{J}^{(3)}$ be their composition,
which is an operator depending on the variables $(\alpha[A\cup B],\mu[E\cup
F])$, of form

$\mathcal{J}_{\alpha[A\cup B],\mu[E\cup F]}^{(3)}=\int
m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})m^{(2)}(\alpha[B],\mu[F],\gamma_{3},\gamma_{4})J_{\ell_{1};\gamma_{1},\gamma_{2}}J_{\ell_{2};\gamma_{3},\gamma_{4}}\,\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\mathrm{d}\gamma_{4},$

where $\ell_{1}$ is a cluster of $\alpha[A]$, and $\ell_{2}$ is a cluster of
$\alpha[B]$.

Now look at the operator
$J:=J_{\ell_{1};\gamma_{1},\gamma_{2}}J_{\ell_{2};\gamma_{3},\gamma_{4}}$, we
have

$\displaystyle Jf(t)$ $\displaystyle=\int_{0}^{t}e^{\pi i\ell_{1}(t-z)+\pi
i\gamma_{1}t+\pi i\gamma_{2}z}\,\mathrm{d}z\int_{0}^{z}e^{\pi
i\ell_{2}(z-s)+\pi i\gamma_{3}z+\pi i\gamma_{4}s}f(s)\,\mathrm{d}s$
$\displaystyle=\int_{0}^{t}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i\ell_{1}(t-s-u)+\pi i\ell_{2}u+\pi i(\gamma_{1}t+\gamma_{4}s)+\pi
i(\gamma_{2}+\gamma_{3})(s+u)}\,\mathrm{d}u$ $\displaystyle=\int_{0}^{t}e^{\pi
i\ell_{1}(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})u}\,\mathrm{d}u.$

We decompose $J=J^{\prime}+J^{\prime\prime}$ where in $J^{\prime}$ we multiply
the kernel by $\chi_{0}(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})$ and in
$J^{\prime\prime}$ we multiply by
$\chi_{\infty}(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})$.

To deal with $J^{\prime}$, notice that $\chi_{0}(\gamma)\int_{0}^{v}e^{\pi
i\gamma u}\,\mathrm{d}u$ equals a compactly supported Gevrey $2$ function in
$\gamma$ and $v$ for $v\in[0,1]$, so we may rewrite

$\displaystyle J^{\prime}f(t)$ $\displaystyle=\int_{0}^{t}e^{\pi
i\ell_{1}(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s}f(s)\,\mathrm{d}s\int_{\mathbb{R}}M(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3},\sigma)e^{\pi
i\sigma(t-s)}\,\mathrm{d}\sigma$
$\displaystyle=\int_{\mathbb{R}}M(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3},\sigma)\,\mathrm{d}\sigma\int_{0}^{t}e^{\pi
i\ell_{1}(t-s)+\pi i(\gamma_{1}+\sigma)t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4}-\sigma)s}f(s)\,\mathrm{d}s$

where $M$ is a fixed decaying Gevrey $2$ function in two real variables.
Therefore, the contribution of $J^{\prime}$ to $\mathcal{J}^{(3)}$ equals

$\int
m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})m^{(2)}(\alpha[B],\mu[F],\gamma_{3},\gamma_{4})\\\
\times
M(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3},\sigma)J_{\ell_{1};\gamma_{1}+\sigma,\gamma_{2}+\gamma_{3}+\gamma_{4}-\sigma}\,\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\mathrm{d}\gamma_{4}\mathrm{d}\sigma.$

Note that $\ell_{1}$ is also a cluster of $\alpha[A\cup B]$, we can choose
(note that the $\gamma_{j}$ associated with the composition $m^{(3)}$ are
called $\gamma_{j}^{\prime}$; this will be assumed for subsequent lemmas as
well)

$m^{(3)}(\alpha[A\cup B],\mu[E\cup
F],\gamma_{1}^{\prime},\gamma_{2}^{\prime})=\int_{\gamma_{1}+\sigma={\gamma_{1}^{\prime}}}\int_{\gamma_{2}+\gamma_{3}+\gamma_{4}=\sigma+{\gamma_{2}^{\prime}}}m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})\\\
\times
m^{(2)}(\alpha[B],\mu[F],\gamma_{3},\gamma_{4})M(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3},\sigma)\,\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\mathrm{d}\gamma_{1}.$

which takes care of the contribution of $J^{\prime}$. Then, if we do not take
derivatives and do not count the weight in (5.16), the norm for $m^{(3)}$ is
easily bounded using the corresponding norms for $m^{(1)}$ and $m^{(2)}$. If
we do not take derivatives but include the weight

$\bigg{(}1+\sum_{j\in A\cup
B}|\alpha_{j}|+|{\gamma_{1}^{\prime}}|\bigg{)}^{1/4}$

in (5.16), we may decompose it into three parts $(1+\sum_{j\in
A}|\alpha_{j}|)^{1/4}$, $(\sum_{j\in B}|\alpha_{j}|)^{1/4}$ and
$|{\gamma_{1}^{\prime}}|^{1/4}$. The first two parts can be estimated using
the corresponding norms for $m^{(1)}$ or $m^{(2)}$, while for
$|{\gamma_{1}^{\prime}}|^{1/4}$ we may use
$|{\gamma_{1}^{\prime}}|\leq|\gamma_{1}|+|\sigma|$, together with the
corresponding norms for $m^{(1)}$ and the decay in $\sigma$.

Next consider the higher order derivative estimates. The argument will be the
same for the subsequent lemmas, so we will not repeat this later. Note that
$m^{(3)}$ is a trilinear expression of the functions $m^{(1)}$, $m^{(2)}$ and
$M$; in subsequent lemmas we may have higher degrees of multilinearity, but
they will never exceed $9$. Now by Leibniz rule we have

$\partial_{\alpha}^{\rho}m^{(3)}(\alpha[A\cup B],\mu[E\cup
F],{\gamma_{1}^{\prime}},{\gamma_{2}^{\prime}})=\sum_{\rho^{1}+\rho^{2}+\rho^{3}=\rho}\frac{\rho!}{(\rho^{1})!(\rho^{2})!(\rho^{3})!}\int_{\gamma_{1}+\sigma={\gamma_{1}^{\prime}}}\int_{\gamma_{2}+\gamma_{3}+\gamma_{4}=\sigma+{\gamma_{2}^{\prime}}}\\\
\partial_{\alpha}^{\rho^{1}}m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})\cdot\partial_{\alpha}^{\rho^{2}}m^{(2)}(\alpha[B],\mu[F],\gamma_{3},\gamma_{4})\partial_{\alpha}^{\rho^{3}}M(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3},\sigma)\,\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\mathrm{d}\gamma_{1},$

so the norm of $\partial_{\alpha}^{\rho}m^{(3)}$ can be bounded in the same
way as above, but using the norms of $\partial_{\alpha}^{\rho^{1}}m^{(1)}$,
$\partial_{\alpha}^{\rho^{2}}m^{(2)}$ and $\partial_{\alpha}^{\rho^{3}}M$.
Compared to the versions without the derivatives, we now have extra factors
$(2|\rho^{1}|)!$, $(2|\rho^{2}|)!$ and $(2|\rho^{3}|)!$ in view of (5.16) and
the fact that $M$ is Gevrey $2$. Therefore it suffices to show that

$\sum_{\rho^{1}+\rho^{2}+\rho^{3}=\rho}\frac{\rho!}{(\rho^{1})!(\rho^{2})!(\rho^{3})!}(2|\rho^{1}|)!(2|\rho^{2}|)!(2|\rho^{3}|)!\leq
C(2|\rho|)!,$

which follows from Lemma A.4.

Now for $J^{\prime\prime}$, we continue to calculate

$\displaystyle J^{\prime\prime}f(t)$ $\displaystyle=\int_{0}^{t}e^{\pi
i\ell_{1}(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s}f(s)\cdot\frac{\chi_{\infty}(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}{\pi
i(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}(e^{\pi
i(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})(t-s)}-1)\,\mathrm{d}s$
$\displaystyle=\frac{\chi_{\infty}(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}{i(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}\int_{0}^{t}(e^{\pi
i\ell_{2}(t-s)+\pi i(\gamma_{1}+\gamma_{2}+\gamma_{3})t+\pi
i\gamma_{4}s}-e^{\pi i\ell_{1}(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s})f(s)\,\mathrm{d}s.$

Note that both $\ell_{1}$ and $\ell_{2}$ are clusters of
$(\alpha_{1},\cdots,\alpha_{B})$, just like the above, we can choose either

$m^{(3)}(\alpha[A\cup B],\mu[E\cup
F],{\gamma_{1}^{\prime}},{\gamma_{2}^{\prime}})=\int_{\gamma_{1}+\gamma_{2}+\gamma_{3}={\gamma_{1}^{\prime}}}m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})\\\
\times
m^{(2)}(\alpha[B],\mu[F],\gamma_{3},{\gamma_{2}^{\prime}})\frac{\chi_{\infty}(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}{\pi
i(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}\,\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3},\quad\mathrm{or}\quad$
(5.19) $m^{(3)}(\alpha[A\cup B],\mu[E\cup
F],{\gamma_{1}^{\prime}},{\gamma_{2}^{\prime}})=\int_{\gamma_{2}+\gamma_{3}+\gamma_{4}={\gamma_{2}^{\prime}}}m^{(1)}(\alpha[A],\mu[E],{\gamma_{1}^{\prime}},\gamma_{2})\\\
\times
m^{(2)}(\alpha[B],\mu[F],\gamma_{3},\gamma_{4})\frac{\chi_{\infty}(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}{\pi
i(\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3})}\,\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3},$
(5.20)

which settles the contribution of $J^{\prime\prime}$ if we do not take
derivatives and do not count the weight. As for the weight, notice that for
(5.19) we need to use
$|\gamma_{1}^{\prime}|\leq|\gamma_{1}|+|\gamma_{2}|+|\gamma_{3}|$, which
seemingly involves $\gamma_{2}$; however in view of the denominator
$\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3}$ in (5.19), we may replace
$|\gamma_{2}|^{1/4}$ by either
$|\ell_{2}-\ell_{1}+\gamma_{2}+\gamma_{3}|^{1/4}$, which is estimated using
this denominator, or $|\ell_{2}-\ell_{1}+\gamma_{3}|^{1/4}$, which is
estimated using the corresponding norms for $m^{(1)}$ and $m^{(2)}$.
Similarly, one can treat (5.20). The higher order derivatives can be treated
in the same way as $J^{\prime}$ above. ∎

######  Lemma 5.4.

The composition of a class $J$ operator and a class $R$ operator is of class
$J$.

###### Proof.

Let $\mathcal{J}_{\alpha[A],\mu[E]}^{(1)}$ be of class $J$ and
$\mathcal{R}_{\alpha[B],\mu[F]}^{(2)}$ be of class $R$ with special index
$1\in B$. We first consider
$\mathcal{J}^{(3)}=\mathcal{J}^{(1)}\mathcal{R}^{(2)}$. Similar to Lemma 5.3,
we only need to look at
$J:=J_{\ell_{1};\gamma_{1},\gamma_{2}}R_{\ell_{2}+\epsilon\alpha_{1},\ell_{3}+\epsilon\alpha_{1};\gamma_{3},\gamma_{4},\gamma_{5}}$,
where $\ell_{1}$ is a cluster of $\alpha[A]$, $\ell_{2}$ and $\ell_{3}$ are
two clusters of $\alpha[B\backslash\\{1\\}]$. We have

$\displaystyle Jf(t)$ $\displaystyle=\int_{0}^{t}e^{\pi i\ell_{1}(t-z)+\pi
i\gamma_{1}t+\pi
i\gamma_{2}z}\,\mathrm{d}z\int_{0}^{z}\frac{\chi_{\infty}(\ell_{2}+\epsilon\alpha_{1}+\gamma_{5})}{\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}}e^{\pi
i(\ell_{3}+\epsilon\alpha_{1})(z-s)}e^{\pi
i(\gamma_{3}z+\gamma_{4}s)}f(s)\,\mathrm{d}s$
$\displaystyle=\frac{\chi_{\infty}(\ell_{2}+\epsilon\alpha_{1}+\gamma_{5})}{\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}}\int_{0}^{t}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i\ell_{1}(t-s-u)+\pi i(\ell_{3}+\epsilon\alpha_{1})u+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s+\pi
i(\gamma_{2}+\gamma_{3})u}\,\mathrm{d}u$
$\displaystyle=\frac{\chi_{\infty}(\ell_{2}+\epsilon\alpha_{1}+\gamma_{5})}{\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}}\int_{0}^{t}e^{\pi
i\ell_{1}(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3})u}\,\mathrm{d}u.$

Again decompose $J=J^{\prime}+J^{\prime\prime}$ where for $J^{\prime}$ and
$J^{\prime\prime}$ we multiply by
$\chi_{0}(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3})$ and
$\chi_{\infty}(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3})$
respectively, then as in Lemma 5.3, the contribution of the $J^{\prime}$ term
to $\mathcal{J}^{(3)}$ will be

$\int
m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})m^{(2)}(\alpha[B\backslash\\{1\\}],\mu[F],\gamma_{3},\gamma_{4},\gamma_{5})\\\
\times
M(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3},\sigma)\frac{\chi_{\infty}(\ell_{2}+\epsilon\alpha_{1}+\gamma_{5})}{\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}}J_{\ell_{1};\gamma_{1}+\sigma,\gamma_{2}+\gamma_{3}+\gamma_{4}-\sigma}\,\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\mathrm{d}\gamma_{4}\mathrm{d}\gamma_{5}\mathrm{d}\sigma,$

which can be rewritten in the form of $\mathcal{J}_{\alpha[A\cup B],\mu[E\cup
F]}^{(3)}$ with

$m^{(3)}(\alpha[A\cup B],\mu[E\cup
F],{\gamma_{1}^{\prime}},{\gamma_{2}^{\prime}})=\int_{\gamma_{1}+\sigma={\gamma_{1}^{\prime}}}\int_{\gamma_{2}+\gamma_{3}+\gamma_{4}=\sigma+{\gamma_{2}^{\prime}}}m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})\\\
\times
m^{(2)}(\alpha[B\backslash\\{1\\}],\mu[F],\gamma_{3},\gamma_{4},\gamma_{5})M(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3},\sigma)\frac{\chi_{\infty}(\ell_{2}+\epsilon\alpha_{1}+\gamma_{5})}{\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}}\,\mathrm{d}\gamma_{2}\mathrm{d}\gamma_{3}\mathrm{d}\gamma_{1}\mathrm{d}\gamma_{5}.$

This $m^{(3)}$ can be controlled if we do not take derivatives and do not
count the weight, using the fact that

$\int_{\mathbb{R}}|M(\zeta_{1}+\epsilon\alpha_{1},\sigma)|\cdot\bigg{|}\frac{\chi_{\infty}(\zeta_{2}+\epsilon\alpha_{1})}{\zeta_{2}+\epsilon\alpha_{1}}\bigg{|}\,\mathrm{d}\alpha_{1}\leq
C\langle\sigma\rangle^{-10}$ (5.21)

uniformly in $(\sigma,\zeta_{1},\zeta_{2})$. As for the weight, we can again
decompose it into different parts; compared to Lemma 5.3, the new part that
needs consideration is $|\alpha_{1}|^{1/4}$. But we may replace it by either
$|\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}|^{1/4}$, which does not affect
(5.21), or $|\ell_{2}+\gamma_{5}|^{1/4}$, which can be estimated using the
weighted norm for $m^{(1)}$ or $m^{(2)}$. The higher order derivatives are
also treated in the same way as in Lemma 5.3.

As for $J^{\prime\prime}$, similarly we have

$J^{\prime\prime}f(t)=\frac{\chi_{\infty}(\ell_{2}+\epsilon\alpha_{1}+\gamma_{5})}{\ell_{2}+\epsilon\alpha_{1}+\gamma_{5}}\cdot\frac{\chi_{\infty}(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3})}{\pi
i(\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3})}\\\
\times\int_{0}^{t}(e^{\pi i({\ell_{3}}+\epsilon\alpha_{1})(t-s)+\pi
i(\gamma_{1}+\gamma_{2}+\gamma_{3})t+\pi i\gamma_{4}s}-e^{\pi
i\ell_{1}(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{3}+\gamma_{4})s})f(s)\,\mathrm{d}s,$ (5.22)

so similar arguments as above imply that the corresponding contribution is of
class $J$, where we have used the fact that both
${\ell_{3}}+\epsilon\alpha_{1}$ and $\ell_{1}$ are clusters of $\alpha[A\cup
B]$, and that

$\int_{\mathbb{R}}\bigg{|}\frac{\chi_{\infty}(\zeta_{1}+\epsilon\alpha_{1})}{\zeta_{1}+\epsilon\alpha_{1}}\bigg{|}\cdot\bigg{|}\frac{\chi_{\infty}(\zeta_{2}+\epsilon\alpha_{1})}{\zeta_{2}+\epsilon\alpha_{1}}\bigg{|}\,\mathrm{d}\alpha_{1}\leq
C$

uniformly in $(\zeta_{1},\zeta_{2})$. The part $|\alpha_{1}|^{1/4}$ of the
weight can be treated in the same way as above, while the part
$|\gamma_{2}|^{1/4}$ of the weight (which is part of
$|\gamma_{1}^{\prime}|^{1/4}$ in one of the two terms in (5.22)) can be
replaced by either
$|\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{2}+\gamma_{3}|^{1/4}$ or
$|\ell_{3}-\ell_{1}+\epsilon\alpha_{1}+\gamma_{3}|^{1/4}$ and treated in the
same way either as above or as $|\alpha_{1}|^{1/4}$.

Now we look at $\mathcal{R}^{(2)}\mathcal{J}^{(1)}$. The proof is similar,
where we now have
$J:=R_{{\ell_{2}}+\epsilon\alpha_{1},{\ell_{3}}+\epsilon\alpha_{1};\gamma_{3},\gamma_{4},\gamma_{5}}J_{\ell_{1};\gamma_{1},\gamma_{2}}$.
Similar calculations yield that

$\displaystyle Jf(t)$
$\displaystyle=\int_{0}^{t}\frac{\chi_{\infty}({\ell_{2}}+\epsilon\alpha_{1}+\gamma_{5})}{{\ell_{2}}+\epsilon\alpha_{1}+\gamma_{5}}e^{\pi
i({\ell_{3}}+\epsilon\alpha_{1})(t-z)+\pi i\gamma_{3}t+\pi
i\gamma_{4}z}\,\mathrm{d}z\int_{0}^{z}e^{\pi i\ell_{1}(z-s)}e^{\pi
i(\gamma_{1}z+\gamma_{2}s)}f(s)\,\mathrm{d}s$
$\displaystyle=\frac{\chi_{\infty}({\ell_{2}}+\epsilon\alpha_{1}+\gamma_{5})}{{\ell_{2}}+\epsilon\alpha_{1}+\gamma_{5}}\int_{0}^{t}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i({\ell_{3}}+\epsilon\alpha_{1})(t-s-u)+\pi i\ell_{1}u+\pi i\gamma_{3}t+\pi
i(\gamma_{4}+\gamma_{1}+\gamma_{2})s+\pi
i(\gamma_{4}+\gamma_{1})u}\,\mathrm{d}u$
$\displaystyle=\frac{\chi_{\infty}({\ell_{2}}+\epsilon\alpha_{1}+\gamma_{5})}{{\ell_{2}}+\epsilon\alpha_{1}+\gamma_{5}}\int_{0}^{t}e^{\pi
i({\ell_{3}}+\epsilon\alpha_{1})(t-s)+\pi i\gamma_{3}t+\pi
i(\gamma_{4}+\gamma_{1}+\gamma_{2})s}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i(\ell_{1}-\ell_{3}-\epsilon\alpha_{1}+\gamma_{4}+\gamma_{1})u}\,\mathrm{d}u.$

We proceed in basically the same way as for
$\mathcal{J}^{(1)}\mathcal{R}^{(2)}$, except that (i) the roles of $\ell_{1}$
and ${\ell_{3}}+\epsilon\alpha_{1}$ are switched, but both are still clusters
of $\alpha[A\cup B]$; (ii) we now need to deal with the weight
$|\gamma_{4}|^{1/4}$, but also $\gamma_{4}$ will be a part of the denominator
so this will not affect the proof. ∎

######  Lemma 5.5.

The composition of two class $R$ operators is of class $J$.

###### Proof.

Let $\mathcal{R}_{\alpha[A],\mu[E]}^{(1)}$ and
$\mathcal{R}_{\alpha[B],\mu[F]}^{(2)}$ be of class $R$, with special indices
$1\in A$ and $2\in B$. Again we first consider the operator
$J:=R_{{\ell_{1}}+\epsilon_{1}\alpha_{1},{\ell_{2}}+\epsilon_{1}\alpha_{1};\gamma_{1},\gamma_{2},\gamma_{3}}R_{{\ell_{3}}+\epsilon_{2}\alpha_{2},{\ell_{4}}+\epsilon_{2}\alpha_{2};\gamma_{4},\gamma_{5},\gamma_{6}}$,
where $\ell_{1}$ and $\ell_{2}$ are clusters of $\alpha[A\backslash\\{1\\}]$,
and $\ell_{3}$ and $\ell_{4}$ are clusters of $\alpha[B\backslash\\{2\\}]$. We
have

$\displaystyle Jf(t)$
$\displaystyle=\int_{0}^{t}\frac{\chi_{\infty}({\ell_{1}}+\epsilon_{1}\alpha_{1}+\gamma_{3})}{{\ell_{1}}+\epsilon_{1}\alpha_{1}+\gamma_{3}}e^{\pi
i({\ell_{2}}+\epsilon_{1}\alpha_{1})(t-z)+\pi i\gamma_{1}t+\pi
i\gamma_{2}z}\,\mathrm{d}z$ $\displaystyle\hskip
14.22636pt\times\int_{0}^{z}\frac{\chi_{\infty}({\ell_{3}}+\epsilon_{2}\alpha_{2}+\gamma_{6})}{{\ell_{3}}+\epsilon_{2}\alpha_{2}+\gamma_{6}}e^{\pi
i({\ell_{4}}+\epsilon_{2}\alpha_{2})(z-s)+\pi i\gamma_{4}z+\pi
i\gamma_{5}s}f(s)\,\mathrm{d}s$
$\displaystyle=\frac{\chi_{\infty}({\ell_{1}}+\epsilon_{1}\alpha_{1}+\gamma_{3})}{{\ell_{1}}+\epsilon_{1}\alpha_{1}+\gamma_{3}}\frac{\chi_{\infty}({\ell_{3}}+\epsilon_{2}\alpha_{2}+\gamma_{6})}{{\ell_{3}}+\epsilon_{2}\alpha_{2}+\gamma_{6}}\int_{0}^{t}f(s)\,\mathrm{d}s$
$\displaystyle\hskip 14.22636pt\times\int_{0}^{t-s}e^{\pi
i({\ell_{2}}+\epsilon_{1}\alpha_{1})(t-s-u)+\pi
i({\ell_{4}}+\epsilon_{2}\alpha_{2})u+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{4}+\gamma_{5})s+\pi
i(\gamma_{2}+\gamma_{4})u}\,\mathrm{d}u$
$\displaystyle=\frac{\chi_{\infty}({\ell_{1}}+\epsilon_{1}\alpha_{1}+\gamma_{3})}{{\ell_{1}}+\epsilon_{1}\alpha_{1}+\gamma_{3}}\frac{\chi_{\infty}({\ell_{3}}+\epsilon_{2}\alpha_{2}+\gamma_{6})}{{\ell_{3}}+\epsilon_{2}\alpha_{2}+\gamma_{6}}\int_{0}^{t}e^{\pi
i({\ell_{2}}+\epsilon_{1}\alpha_{1})(t-s)+\pi i\gamma_{1}t+\pi
i(\gamma_{2}+\gamma_{4}+\gamma_{5})s}f(s)\,\mathrm{d}s$ $\displaystyle\hskip
14.22636pt\times\int_{0}^{t-s}e^{\pi
i(\ell_{4}-\ell_{2}+\epsilon_{2}\alpha_{2}-\epsilon_{1}{\alpha_{1}}+\gamma_{2}+\gamma_{4})u}\,\mathrm{d}u.$

Now, we make the decomposition again by multiplying

$\chi_{0}(\ell_{4}-\ell_{2}+\epsilon_{2}\alpha_{2}-\epsilon_{1}{\alpha_{1}}+\gamma_{2}+\gamma_{4})\quad\textrm{or}\quad\chi_{\infty}(\ell_{4}-\ell_{2}+\epsilon_{2}\alpha_{2}-\epsilon_{1}{\alpha_{1}}+\gamma_{2}+\gamma_{4}),$

and denote the resulting terms by $J^{\prime}$ and $J^{\prime\prime}$. Then
repeating the same arguments before we can show that the contribution of both
$J^{\prime}$ and $J^{\prime\prime}$ are class $J$ operators. The key points
here are that (i) both ${\ell_{2}}+\epsilon_{1}\alpha_{1}$ and
${\ell_{4}}+\epsilon_{2}\alpha_{2}$ are clusters of $\alpha[A]$ and
$\alpha[B]$ respectively, and that (ii) the bound

$\int_{\mathbb{R}^{2}}|M(\zeta_{1}+\epsilon_{2}\alpha_{2}-\epsilon_{1}\alpha_{1},\sigma)|\cdot\bigg{|}\frac{\chi_{\infty}(\zeta_{2}+\epsilon_{1}\alpha_{1})}{\zeta_{2}+\epsilon_{1}\alpha_{1}}\bigg{|}\cdot\bigg{|}\frac{\chi_{\infty}(\zeta_{3}+\epsilon_{2}\alpha_{2})}{\zeta_{3}+\epsilon_{2}\alpha_{2}}\bigg{|}\,\mathrm{d}\alpha_{1}\mathrm{d}\alpha_{2}\leq
C\langle\sigma\rangle^{-10}$

holds uniformly in $(\sigma,\zeta_{1},\zeta_{2},\zeta_{3})$, and similarly

$\int_{\mathbb{R}^{2}}\bigg{|}\frac{\chi_{\infty}(\zeta_{1}+\epsilon_{2}\alpha_{2}-\epsilon_{1}\alpha_{1})}{\zeta_{1}+\epsilon_{2}\alpha_{2}-\epsilon_{1}\alpha_{1}}\bigg{|}\cdot\bigg{|}\frac{\chi_{\infty}(\zeta_{2}+\epsilon_{1}\alpha_{1})}{\zeta_{2}+\epsilon_{1}\alpha_{1}}\bigg{|}\cdot\bigg{|}\frac{\chi_{\infty}(\zeta_{3}+\epsilon_{2}\alpha_{2})}{\zeta_{3}+\epsilon_{2}\alpha_{2}}\bigg{|}\,\mathrm{d}\alpha_{1}\mathrm{d}\alpha_{2}\leq
C$

holds uniformly in $(\zeta_{1},\zeta_{2},\zeta_{3})$, which follow from
elementary calculus. The parts of the weight that need consideration are (i)
$|\alpha_{1}|^{1/4}$ and $|\alpha_{2}|^{1/4}$, which can be treated using the
denominators $\ell_{1}+\epsilon_{1}\alpha_{1}+\gamma_{3}$ and
$\ell_{3}+\epsilon_{2}\alpha_{2}+\gamma_{6}$ respectively, and (2)
$|\gamma_{2}|^{1/4}$ (which is part of $|\gamma_{1}^{\prime}|^{1/4}$ in one of
the terms), which can be treated using the denominator
$\ell_{4}-\ell_{2}+\epsilon_{2}\alpha_{2}-\epsilon_{1}{\alpha_{1}}+\gamma_{2}+\gamma_{4}$.
∎

######  Lemma 5.6.

Suppose $\mathcal{J}$ is an operator of class $J$, then the operator
$I_{\epsilon\alpha_{1}}\mathcal{J}I_{\mu_{1}-\epsilon\alpha_{1}}$, where
$\epsilon\in\\{\pm\\}$, can be decomposed into an operator of class $J$ and an
operator of class $R$ (with special index $1$).

###### Proof.

Let $\mathcal{J}=\mathcal{J}_{\alpha[A],\mu[E]}^{(1)}$ be of class $J$, where
we assume $1\not\in E$. Again we first consider the operator
$X:=I_{\epsilon\alpha_{1}}J_{\ell;\gamma_{1},\gamma_{2}}I_{\mu_{1}-\epsilon\alpha_{1}}$,
where $\ell$ is a cluster of $\alpha[A]$. Then

$\displaystyle Xf(t)$ $\displaystyle=\int_{0}^{t}e^{\epsilon\pi
i\alpha_{1}z}\,\mathrm{d}z\int_{0}^{z}e^{\pi i\ell(z-v)+\pi i\gamma_{1}z+\pi
i\gamma_{2}v}\,\mathrm{d}v\int_{0}^{v}e^{-\epsilon\pi i\alpha_{1}s+\pi
i\mu_{1}s}f(s)\,\mathrm{d}s$ $\displaystyle=\int_{0}^{t}e^{\pi
i(\gamma_{1}+\gamma_{2}+\mu_{1})s}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})u}\,\mathrm{d}u\int_{0}^{t-s-u}e^{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})w}\,\mathrm{d}w.$

Our estimates will be uniform in $\mu_{1}$ due to the only position it
appears, and the fact that the left hand side of (5.16) does not involve
$\gamma_{2}$. When $s$ is fixed, by making the $(\chi_{0},\chi_{\infty})$
decomposition twice, we can reduce the inner $(u,w)$ integral to $6$ different
terms, namely:

$\displaystyle\mathrm{I}$
$\displaystyle:=\int_{\mathbb{R}^{2}}M(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1}-\sigma_{1},\sigma_{2})M(\ell+\epsilon\alpha_{1}+\gamma_{1},\sigma_{1})\cdot
e^{\pi
i(\sigma_{1}+\sigma_{2})(t-s)}\,\mathrm{d}\sigma_{1}\mathrm{d}\sigma_{2},$
$\displaystyle\mathrm{II}$
$\displaystyle:=\int_{\mathbb{R}}M(\ell+\epsilon\alpha_{1}+\gamma_{1},\sigma)\cdot\frac{\chi_{\infty}(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1}-\sigma)}{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1}-\sigma)}(e^{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})(t-s)}-e^{\pi i\sigma(t-s)}),$
$\displaystyle\mathrm{III}$
$\displaystyle:=\int_{\mathbb{R}}M(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1},\sigma)\frac{\chi_{\infty}(\ell+\epsilon\alpha_{1}+\gamma_{1})}{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})}\cdot e^{\pi
i\sigma(t-s)}\,\mathrm{d}\sigma,$ $\displaystyle\mathrm{IV}$
$\displaystyle:=\frac{\chi_{\infty}(\ell+\epsilon\alpha_{1}+\gamma_{1})}{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})}\cdot\frac{\chi_{\infty}(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})}{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})}(e^{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})(t-s)}-1),$
$\displaystyle\mathrm{V}$
$\displaystyle:=\int_{\mathbb{R}}M(\gamma_{2}-\ell,\sigma)\frac{\chi_{\infty}(\ell+\epsilon\alpha_{1}+\gamma_{1})}{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})}e^{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1}+\sigma)(t-s)}\,\mathrm{d}\sigma,$
$\displaystyle\mathrm{VI}$
$\displaystyle:=\frac{\chi_{\infty}(\ell+\epsilon\alpha_{1}+\gamma_{1})}{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})}\cdot\frac{\chi_{\infty}(\gamma_{2}-\ell)}{\pi
i(\gamma_{2}-\ell)}(e^{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})(t-s)}-e^{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})(t-s)}).$

Now, the terms $\mathrm{I}\sim\mathrm{IV}$ will give rise to class $J$
operators; for example, consider the term $\mathrm{IV}$ where we choose the
term $e^{\pi i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})(t-s)}$ from the last
parenthesis, then we may express the contribution of this term as a class $J$
operator with coefficient

$m^{(2)}(\alpha[A\cup\\{1\\}],\mu[E\cup\\{1\\}],{\gamma_{1}^{\prime}},{\gamma_{2}^{\prime}})=\bm{\delta}({\gamma_{2}^{\prime}}-\mu_{1})\\\
\times\int_{\gamma_{1}+\gamma_{2}={\gamma_{1}^{\prime}}}\frac{\chi_{\infty}(\ell+\epsilon\alpha_{1}+\gamma_{1})}{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})}\cdot\frac{\chi_{\infty}(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})}{\pi
i(\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1})}m^{(1)}(\alpha[A],\mu[E],\gamma_{1},\gamma_{2})\,\mathrm{d}\gamma_{1}.$

Here $m^{(1)}$ is the coefficient associated with $\mathcal{J}^{(1)}$, and the
Dirac $\bm{\delta}$ function can be removed by using the fact that

$e^{\pi i\mu_{1}s}=\int_{\mathbb{R}}M(\sigma)e^{\pi
i(\mu_{1}+\sigma)s}\,\mathrm{d}\sigma$ (5.23)

for some analytic function $M$ and all $s\in[0,1]$, which allows to replace
$\bm{\delta}({\gamma_{2}^{\prime}}-\mu_{1})$ by
$M({\gamma_{2}^{\prime}}-\mu_{1})$; moreover the integral in $\alpha_{1}$ is
uniformly bounded given the other variables.

Finally, terms $\mathrm{V}\sim\mathrm{VI}$ lead to class $R$ operators. For
example, consider the term $\mathrm{VI}$ where we choose the term $e^{\pi
i(\ell+\epsilon\alpha_{1}+\gamma_{1})(t-s)}$ from the last parenthesis, then
we may express the contribution of this term as a class $R$ operator with
$\ell_{1}$ and $\ell_{2}$ replaced by $\ell$, and the coefficient

$m^{(2)}(\alpha[A],\mu[E],\gamma_{1}^{\prime},\gamma_{2}^{\prime},\gamma_{3}^{\prime})=\frac{1}{\pi
i}\bm{\delta}(\gamma_{1}^{\prime}-\gamma_{3}^{\prime})\cdot\frac{\chi_{\infty}(\gamma_{2}^{\prime}-\mu_{1}-\ell)}{\pi
i(\gamma_{2}^{\prime}-\mu_{1}-\ell)}m^{(1)}(\alpha[A],\mu[E],\gamma_{1}^{\prime},\gamma_{2}^{\prime}-\mu_{1}).$

The Dirac $\bm{\delta}$ function can again be removed using (5.23), and in all
cases (both for $\mathrm{I}\sim\mathrm{IV}$ and $\mathrm{V}\sim\mathrm{VI}$)
the weight can be treated in the same way as before, using the denominator
$\ell+\epsilon\alpha_{1}+\gamma_{1}$ to estimate $|\alpha_{1}|^{1/4}$, and
using the denominator $\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1}$ or
$\gamma_{2}-\ell$ to estimate $|\gamma_{2}|^{1/4}$ (which may appear as part
of $|\gamma_{1}^{\prime}|^{1/4}$ for some terms). ∎

######  Lemma 5.7.

Suppose $\mathcal{R}$ is an operator of class $R$, then the operator
$I_{\epsilon\alpha_{1}}\mathcal{R}I_{\mu_{1}-\epsilon\alpha_{1}}$, where
$\epsilon\in\\{\pm\\}$, can be decomposed into an operator of class $J$ and an
operator of class $R$ (with special index $1$).

###### Proof.

Let $\mathcal{R}=\mathcal{R}_{\alpha[A],\mu[E]}^{(1)}$ be of class $R$ with
special index $2$, where $1\not\in E$. Consider
$X=I_{\epsilon\alpha_{1}}R_{\ell_{1}+\epsilon_{2}\alpha_{2},\ell_{2}+\epsilon_{2}\alpha_{2};\gamma_{1},\gamma_{2},\gamma_{3}}I_{\mu_{1}-\epsilon\alpha_{1}}$,
where $\ell_{1}$ and $\ell_{2}$ are clusters of $\alpha[A\backslash\\{2\\}]$.
Our estimates will be uniform in $\mu_{1}$ as in Lemma 5.6. We proceed in
basically the same way as in Lemma 5.6, and obtain the same expressions
$\mathrm{I}\sim\mathrm{VI}$, except that here $\ell$ is replaced by
$\ell_{2}+\epsilon_{2}\alpha_{2}$, and that we have an extra factor
$\frac{\chi_{\infty}(\ell_{1}+\epsilon_{2}\alpha_{2}+\gamma_{3})}{\ell_{1}+\epsilon_{2}\alpha_{2}+\gamma_{3}}$.
The terms $\mathrm{I}\sim\mathrm{IV}$ still give rise to class $J$ operators,
since the factors in the coefficients that depend on
$(\alpha_{1},\alpha_{2})$, which are bounded by

$\frac{1}{\langle\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1}\rangle}\cdot\frac{1}{\langle\ell_{2}+\epsilon\alpha_{1}+\epsilon_{2}\alpha_{2}+\gamma_{1}\rangle}\cdot\frac{1}{\langle\ell_{1}+\epsilon_{2}\alpha_{2}+\gamma_{3}\rangle},$

are integrable in $(\alpha_{1},\alpha_{2})$ uniformly in the other variables.

As for terms $\mathrm{V}\sim\mathrm{VI}$, they will give rise to class $R$
operators with special index 1. For this we only need the integrability in
$\alpha_{2}$ (uniformly in the other variables) of the factors in the
coefficients that depend on $\alpha_{2}$, which follows from the upper bound

$\frac{1}{\langle\ell_{1}+\epsilon_{2}\alpha_{2}+\gamma_{3}\rangle}\cdot\frac{1}{\langle\gamma_{2}-\epsilon_{2}\alpha_{2}-\ell_{2}\rangle}.$

Again, in all cases the weight can be treated in the same way as before, where
for terms $\mathrm{I}\sim\mathrm{IV}$ we use the denominator
$\ell_{1}+\epsilon_{2}\alpha_{2}+\gamma_{3}$ to estimate $|\alpha_{2}|^{1/4}$,
use the denominator
$\ell_{2}+\epsilon\alpha_{1}+\epsilon_{2}\alpha_{2}+\gamma_{1}$ to estimate
$|\alpha_{1}|^{1/4}$, and use the denominator
$\gamma_{1}+\gamma_{2}+\epsilon\alpha_{1}$ to estimate $|\gamma_{2}|^{1/4}$
which may appear as part of $|\gamma_{1}^{\prime}|^{1/4}$ for some terms. For
terms $\mathrm{V}\sim\mathrm{VI}$ we use the denominator
$\ell_{1}+\epsilon_{2}\alpha_{2}+\gamma_{3}$ to estimate $|\alpha_{2}|^{1/4}$,
and use the denominator $\gamma_{2}-\epsilon_{2}\alpha_{2}-\ell_{2}$ to
estimate $|\gamma_{2}|^{1/4}$. ∎

######  Lemma 5.8.

The operator $I_{\epsilon\alpha_{1}}I_{\mu_{1}-\epsilon\alpha_{1}}$, where
$\epsilon\in\\{\pm\\}$, can be decomposed as

$I_{\epsilon\alpha_{1}}I_{\mu_{1}-\epsilon\alpha_{1}}=\frac{\chi_{\infty}(\alpha_{1})}{\epsilon\pi
i\alpha_{1}}\mathcal{J}_{\alpha[\
],\mu_{1}}+\mathcal{J}_{\alpha_{1},\mu_{1}}+\mathcal{R}_{\alpha_{1},\mu_{1}},$

where $\mathcal{J}_{(\cdots)}$ and $\mathcal{R}_{(\cdots)}$ are of class $J$
and $R$ (with special index $1$) respectively, and $\alpha[\ ]$ indicates the
corresponding set $A=\varnothing$ in (5.15).

###### Proof.

We directly calculate $X=I_{\epsilon\alpha_{1}}I_{\mu_{1}-\epsilon\alpha_{1}}$
such that

$Xf(t)=\int_{0}^{t}e^{\epsilon\pi i\alpha_{1}z}\,\mathrm{d}z\int_{0}^{z}e^{\pi
i(\mu_{1}-\epsilon\alpha_{1})s}f(s)\,\mathrm{d}s=\int_{0}^{t}e^{\pi
i\mu_{1}s}f(s)\,\mathrm{d}s\int_{0}^{t-s}e^{\epsilon\pi
i\alpha_{1}u}\,\mathrm{d}u,$

note that the estimates are again uniform in $\mu_{1}$. Now $X$ can be
decomposed into three terms,

$\displaystyle\mathrm{I}$
$\displaystyle:=\int_{\mathbb{R}}M(\epsilon\alpha_{1},\sigma)J_{0;\sigma,\mu_{1}-\sigma}\,\mathrm{d}\sigma,$
$\displaystyle\mathrm{II}$
$\displaystyle:=\frac{\chi_{\infty}(\epsilon\alpha_{1})}{\epsilon\pi
i\alpha_{1}}J_{0;0,\mu_{1}},$ $\displaystyle\mathrm{III}$
$\displaystyle:=R_{\epsilon\alpha_{1},\epsilon\alpha_{1};0,\mu_{1},0}.$

Clearly $\mathrm{I}=\mathcal{J}_{\alpha_{1},\mu_{1}}$ has class $J$, as we can
always convolve by a decaying analytic function as in (5.23); likewise
$\mathrm{II}=\frac{\chi_{\infty}(\alpha_{1})}{\epsilon\pi
i\alpha_{1}}\cdot\mathcal{J}_{\alpha[\ ],\mu_{1}}$ with the operator also
having class $\mathcal{J}$. Now looking at $\mathrm{III}$, we can introduce
the integration in $(\gamma_{1},\gamma_{2})$ by convolution; to see it is of
class $R$, which involves $\gamma_{3}$ integration, we simply rewrite

$\frac{\chi_{\infty}(\epsilon\alpha_{1})}{\epsilon\pi
i\alpha_{1}}=\int_{\mathbb{R}}M(\sigma)\frac{\chi_{\infty}(\epsilon\alpha_{1})}{\epsilon\pi
i\alpha_{1}}\,\mathrm{d}\sigma$

with some fixed compactly supported Gevrey $2$ function $M$; then we replace
$\frac{\chi_{\infty}(\epsilon\alpha_{1})}{\epsilon\pi i\alpha_{1}}$ by
$\frac{\chi_{\infty}(\sigma+\epsilon\alpha_{1})}{\pi
i(\sigma+\epsilon\alpha_{1})}$ to produce the $\gamma_{3}$ integral (which is
just the $\sigma$ integral here), noticing that

$\int_{\mathbb{R}}\bigg{|}\frac{\chi_{\infty}(\epsilon\alpha_{1})}{\epsilon\pi
i\alpha_{1}}-\frac{\chi_{\infty}(\sigma+\epsilon\alpha_{1})}{\pi
i(\sigma+\epsilon\alpha_{1})}\bigg{|}\,\mathrm{d}\alpha_{1}\leq C$

for $|\sigma|\leq 1$, so the error term introduced in this way is of form
$\mathcal{J}_{\alpha_{1},\mu_{1}}$ which has class $J$. ∎

####  5.2.3. Regular chain expressions

######  Lemma 5.9.

Let $\mathcal{P}$ be a legal partition of $\\{1,\cdots,2m\\}$, where $m\geq
1$, and consider the operator

$I:=X_{0}I_{\beta_{1}+\lambda_{1}}X_{1}I_{\beta_{2}+\lambda_{2}}\cdots
X_{2m-1}I_{\beta_{2m}+\lambda_{2m}}X_{2m},$

again assume $\lambda_{a}=0$ and $\lambda_{b}=\mu_{j}$ for the $j$-th pair
$\\{a,b\\}\,(a<b)$. Suppose that each $X_{a}\,(0\leq a\leq 2m)$ is either of
class $J$ or $R$, or $X_{a}=\mathrm{Id}$, such that $X_{a}\neq\mathrm{Id}$ if
$\\{a,a+1\\}\in\mathcal{P}$. Then $I$ can be decomposed into an operator of
class $J$ and an operator of class $R$ (with special index 1). Moreover the
norms of these operators are at most

$C^{m}\prod_{a=0}^{2m}\|X_{a}\|,$

with $\|X_{a}\|=1$ if $X_{a}=\mathrm{Id}$.

###### Proof.

First note that we may always assume $X_{0}=X_{2m}=\mathrm{Id}$ in view of
Lemmas 5.3–5.5. Now we induct on $m$ (for convenience of induction we may
replace the power $C^{m}$ by $C^{2m-1}$). When $m=1$ we may assume
$I=I_{\epsilon\alpha_{1}}\widetilde{I}I_{\mu_{1}-\epsilon\alpha_{1}}$ where
$\widetilde{I}$ has class $J$ or $R$, so the result follows from Lemmas
5.6–5.7. Suppose the result is true for $m^{\prime}<m$, and consider any legal
partition $\mathcal{P}$ of $\\{1,\cdots,2m\\}$. We know that $\mathcal{P}$ is
formed either by concatenating two smaller legal partitions
$\mathcal{P}^{\prime}$ and $\mathcal{P}^{\prime\prime}$, or by enclosing a
legal partition $\mathcal{P}^{\prime}$ into the pair $\\{1,2m\\}$. In the
first case we have $I=I^{\prime}I^{\prime\prime}$ where $I^{\prime}$ and
$I^{\prime\prime}$ are the operators corresponding to $\mathcal{P}^{\prime}$
and $\mathcal{P}^{\prime\prime}$ respectively (with obvious choices of the
$X_{a}$’s), so the result follows from Lemmas 5.3–5.5. In the second case we
have $I=I_{\epsilon\alpha_{1}}I^{\prime}I_{\mu_{1}-\epsilon\alpha_{1}}$ where
$I^{\prime}$ is the operator corresponding to $\mathcal{P}^{\prime}$ (with
obvious choices of the $X_{a}$’s), so the result follows from Lemmas 5.6–5.7.
∎

######  Lemma 5.10.

Consider now the operator $I=I_{\beta_{1}+\lambda_{1}}\cdots
I_{\beta_{2m}+\lambda_{2m}}$ with $\lambda_{a}$ satisfying the same conditions
as in Lemma 5.9. Then, $I$ is a sum of at most $2^{m}$ terms: for each term
there exists a set $Z\subset\\{1,\cdots,m\\}$, such that this term has form

$\prod_{j\in Z}\frac{\chi_{\infty}(\alpha_{j})}{\epsilon_{j}\pi
i\alpha_{j}}\cdot\widetilde{I},$

where $\widetilde{I}$ is another operator that depends only on the variables
$(\mu_{1},\cdots,\mu_{m})$ and $\alpha[W]$ with $W=\\{1,\cdots,m\\}\backslash
Z$, and has either class $J$ or class $R$, with norm $\|\widetilde{I}\|\leq
C^{m}$.

###### Proof.

We first consider all adjacent pairs $\\{a,a+1\\}\in\mathcal{P}$. For such
pairs we have a factor in $I$ that is
$I_{\epsilon_{j}\alpha_{j}}I_{\mu_{j}-\epsilon_{j}\alpha_{j}}$ for some $j$,
so by Lemma 5.8 we can decompose it into
$\frac{\chi_{\infty}(\alpha_{j})}{\epsilon_{j}\pi
i\alpha_{j}}\mathcal{J}_{\alpha[\
],\mu_{j}}+\mathcal{J}_{\alpha_{j},\mu_{j}}+\mathcal{R}_{\alpha_{j},\mu_{j}}$.
Now if we select the term $\frac{\chi_{\infty}(\alpha_{j})}{\epsilon_{j}\pi
i\alpha_{j}}\mathcal{J}_{\alpha[\ ],\mu_{j}}$ we already get one factor
$\frac{\chi_{\infty}(\alpha_{j})}{\epsilon_{j}\pi i\alpha_{j}}$, while the
remaining part of the operator will no long depend on $\alpha_{j}$; if we
select $\mathcal{J}_{\alpha_{j},\mu_{j}}$ or
$\mathcal{R}_{\alpha_{j},\mu_{j}}$ we will leave it as is, and note that in
any case our operator has class $J$ or class $R$.

Now, after removing all adjacent pairs $\\{a,a+1\\}$ we can reduce
$\mathcal{P}$ to a smaller legal partition $\mathcal{P}^{\prime}$. Then, apart
from the possible
$\frac{\chi_{\infty}(\alpha_{j})}{\epsilon_{\pi}i\alpha_{j}}$ factors, the
remaining part of the operator, denoted by $\widetilde{I}$, will be of form
$I$ described in Lemma 5.9. Note that if $\mathcal{P}^{\prime}=\varnothing$,
then $\widetilde{I}$ is already a composition of at most $m$ class $J$ or $R$
operators, so the result follows directly from Lemmas 5.3–5.5. If
$\mathcal{P}^{\prime}\neq\varnothing$, applying Lemma 5.9 then yields that
$\widetilde{I}$ has class $J$ or $R$, and that the norm

$\|\widetilde{I}\|\leq C^{m^{\prime}}\prod_{a=0}^{2m^{\prime}}\|X_{a}\|.$

Here we assume that (after relabeling form smallest to largest)
$\mathcal{P}^{\prime}$ is a legal partition of $\\{1,\cdots,2m^{\prime}\\}$,
and each $X_{a}$ is in fact a composition of class $J$ and $R$ operators
appearing in Lemma 5.8, and the number $n_{a}$ of such operators is the number
of adjacent pairs in $\mathcal{P}$ between the elements $a$ and $a+1$ (again
after relabeling) of $\mathcal{P}^{\prime}$ (note that $n_{a}\geq 1$ if
$\\{a,a+1\\}\in\mathcal{P}^{\prime}$). Therefore $\|X_{a}\|\leq C^{n_{a}}$ by
iterating Lemmas 5.3–5.5, and since
$n_{0}+\cdots+n_{2m^{\prime}}=m-m^{\prime}$, we conclude that
$\|\widetilde{I}\|\leq C^{m}$, which completes the proof. ∎

###  5.3. Proof of Proposition 5.1

In this section we prove Proposition 5.1. The proof is done by induction on
the scale of $\mathcal{Q}$, and Lemma 5.10 plays a key role in the inductive
step.

###### Proof of Proposition 5.1.

We induct on $n$. The base case $n=0$ is trivial as
$\widetilde{\mathcal{B}}_{\times}(t,s)\equiv 1$. Suppose the result is true
for regular couples of smaller scales, and consider a regular couple
$\mathcal{Q}$ of scale $2n$. By the discussion in Section 5.1.1, we know that
$\widetilde{\mathcal{B}}_{\mathcal{Q}}$ can be expressed as in either (5.6) or
(5.7), such that the regular couples appearing on the right hand sides all
have scale strictly less than $2n$.

_Case 1_. Suppose we have (5.6), then by induction hypothesis we have

$\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])=\prod_{j=1}^{3}\prod_{\mathfrak{n}\in
Z_{j}}\frac{\chi_{\infty}(\alpha_{\mathfrak{n}})}{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}}\int_{\mathbb{R}^{6}}\prod_{j=1}^{3}\mathcal{C}_{j}(\lambda_{2j-1},\lambda_{2j},\alpha[\mathcal{N}_{j}^{ch}\backslash
Z_{j}])\prod_{j=1}^{6}\mathrm{d}\lambda_{j}\\\
\times\int_{0}^{t}\int_{0}^{s}e^{\pi
i\alpha_{\mathfrak{r}}(t_{1}-s_{1})}e^{\pi
i(\lambda^{*}t_{1}+\lambda^{**}s_{1})}\,\mathrm{d}t_{1}\mathrm{d}s_{1}.$
(5.24)

Here in (5.24), each $Z_{j}$ is a subset of $\mathcal{N}_{j}^{ch}$, each
$\mathcal{C}_{j}=\mathcal{C}_{j}(\lambda_{2j-1},\lambda_{2j},\alpha[\mathcal{N}_{j}^{ch}\backslash
Z_{j}])$ is a function satisfying (5.9), and
$(\lambda^{*},\lambda^{**})=(\lambda_{1}+\lambda_{3}+\lambda_{5},\lambda_{2}+\lambda_{4}+\lambda_{6})$.
Note that $\mathcal{C}_{j}$ is either the function $\mathcal{C}$ associated
with $\mathcal{Q}_{j}$ and $Z_{j}$ as in (5.8), or is the same function with
the variables $(\lambda_{2j-1},\lambda_{2j})$ replaced by
$(\lambda_{2j},\lambda_{2j-1})$. The same is true in _Case 2_ below.

By integrability in $(\lambda_{1},\cdots,\lambda_{6})$, we may fix the choices
of these parameters, and also exploit the weight
$\langle\lambda_{1}\rangle^{1/4}\cdots\langle\lambda_{6}\rangle^{1/4}$ from
(5.9) if needed. We then explicitly calculate the expression

$\int_{0}^{t}\int_{0}^{s}e^{\pi i\alpha_{\mathfrak{r}}(t_{1}-s_{1})}e^{\pi
i(\lambda^{*}t_{1}+\lambda^{**}s_{1})}\,\mathrm{d}t_{1}\mathrm{d}s_{1}=\frac{e^{\pi
i(\alpha_{\mathfrak{r}}+\lambda^{*})t}-1}{\pi
i(\alpha_{\mathfrak{r}}+\lambda^{*})}\cdot\frac{e^{\pi
i(-\alpha_{\mathfrak{r}}+\lambda^{**})s}-1}{\pi
i(-\alpha_{\mathfrak{r}}+\lambda^{**})}.$ (5.25)

By inserting the cutoffs $\chi_{0}(\alpha_{\mathfrak{r}}+\lambda^{*})$ or
$\chi_{\infty}(\alpha_{\mathfrak{r}}+\lambda^{*})$, and
$\chi_{0}(-\alpha_{\mathfrak{r}}+\lambda^{**})$ or
$\chi_{\infty}(-\alpha_{\mathfrak{r}}+\lambda^{**})$ as in Section 5.2.1, we
can easily show that the above expression, as a function of
$(t,s,\alpha_{\mathfrak{r}})$, can be written in the form

$\int_{\mathbb{R}^{2}}\mathcal{C}^{\prime}(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\alpha_{\mathfrak{r}})e^{\pi
i(\lambda_{1}^{\prime}t+\lambda_{2}^{\prime}s)}\,\mathrm{d}\lambda_{1}^{\prime}\mathrm{d}\lambda_{2}^{\prime},$
(5.26)

where $\mathcal{C}^{\prime}$ is such that

${\int\langle\lambda_{1}^{\prime}\rangle^{1/4}\langle\lambda_{2}^{\prime}\rangle^{1/4}|\partial_{\alpha_{\mathfrak{r}}}^{\rho}\mathcal{C}^{\prime}(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\alpha_{\mathfrak{r}})|\,\mathrm{d}\lambda_{1}^{\prime}\mathrm{d}\lambda_{2}^{\prime}\mathrm{d}\alpha_{\mathfrak{r}}\leq
C(2|\rho|)!}$ (5.27)

uniformly in the choices of $\lambda_{j}$. For example, if we insert the
cutoffs $\chi_{\infty}(\alpha_{\mathfrak{r}}+\lambda^{*})$ and
$\chi_{\infty}(-\alpha_{\mathfrak{r}}+\lambda^{**})$, and choose the terms
$e^{\pi i(\alpha_{\mathfrak{r}}+\lambda^{*})t}$ and $e^{\pi
i(-\alpha_{\mathfrak{r}}+\lambda^{**})s}$ from the numerators in (5.25), then
by using (5.23) we can write

$\mathcal{C}^{\prime}(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\alpha_{\mathfrak{r}})=M(\lambda_{1}^{\prime}-\alpha_{\mathfrak{r}}-\lambda^{*})M(\lambda_{2}^{\prime}+\alpha_{\mathfrak{r}}-\lambda^{**})\cdot\frac{\chi_{\infty}(\alpha_{\mathfrak{r}}+\lambda^{*})}{\pi
i(\alpha_{\mathfrak{r}}+\lambda^{*})}\cdot\frac{\chi_{\infty}(-\alpha_{\mathfrak{r}}+\lambda^{**})}{\pi
i(-\alpha_{\mathfrak{r}}+\lambda^{**})},$

for which (5.27) is easily verified, noticing also that $\chi_{\infty}$ is
Gevrey $2$. The other terms can be treated similarly.

Now, for the regular couple $\mathcal{Q}$ and the associated
$\widetilde{\mathcal{B}}_{\mathcal{Q}}$, we may choose $Z=Z_{1}\cup Z_{2}\cup
Z_{3}$. Using (5.24), and rewriting (5.25) as the form (5.26), we can then
easily prove that it has the form (5.8) and satisfies (5.9); in fact, assume
$\mathcal{Q}_{j}$ has scale $n_{j}$, then the induction hypothesis together
with (5.27) bounds the left hand side of (5.9) without derivatives by

$C^{n_{1}}\cdot C^{n_{2}}\cdot C^{n_{3}}\cdot
C=C^{n_{1}+n_{2}+n_{3}+1}=C^{n}.$

As for the higher order derivatives estimate, notice that the whole expression
(5.24) is a linear combination of terms that are _factorized_ as a product of
functions of $\alpha[\mathcal{N}_{j}^{ch}]$ for $1\leq j\leq 3$ and a function
of $\alpha_{\mathfrak{r}}$. For any multi-index $\rho$, suppose we want to
control the $\partial_{\alpha}^{\rho}$ derivative of the relevant quantity,
and let the multi-index of derivatives falling on each of the above four sets
of variables be $\rho_{j}\,(1\leq j\leq 4)$, then by induction hypothesis and
(5.27), the left hand side of (5.9) is at most

$C^{n}(2|\rho_{1}|)!\cdots(2|\rho_{4}|)!\leq C^{n}(2|\rho|)!,$

which is what we need.

_Case 2_. Suppose we have (5.7), again by induction hypothesis we have

$\displaystyle\widetilde{\mathcal{B}}_{\mathcal{Q}}(t,s,\alpha[\mathcal{N}^{ch}])$
$\displaystyle=\prod_{\epsilon\in\\{\pm\\}}\prod_{j=1}^{m^{\epsilon}}\prod_{\iota=1}^{2}\prod_{\mathfrak{n}\in
Z_{j,\epsilon,\iota}}\frac{\chi_{\infty}(\alpha_{\mathfrak{n}})}{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}}\cdot\prod_{\mathfrak{n}\in
Z_{lp}}\frac{\chi_{\infty}(\alpha_{\mathfrak{n}})}{\zeta_{\mathfrak{n}}\pi
i\alpha_{\mathfrak{n}}}$ (5.28)
$\displaystyle\times\int\prod_{\epsilon\in\\{\pm\\}}\prod_{j=1}^{m^{\epsilon}}\prod_{\iota=1}^{2}\mathcal{C}_{j,\epsilon,\iota}\big{(}\lambda_{a,\epsilon,\iota},\lambda_{b,\epsilon,\iota},\alpha[\mathcal{N}_{j,\epsilon,\iota}^{ch}\backslash
Z_{j,\epsilon,\iota}]\big{)}\,\mathrm{d}\lambda_{a,\epsilon,\iota}\lambda_{b,\epsilon,\iota}$
$\displaystyle\times\int\mathcal{C}_{lp}(\lambda_{lp,+},\lambda_{lp,-},\alpha[\mathcal{N}_{lp}^{ch}\backslash
Z_{lp}])\,\mathrm{d}\lambda_{lp,+}\mathrm{d}\lambda_{lp,-}$
$\displaystyle\times\int_{t>t_{1}>\cdots>t_{2m^{+}}>0}e^{\pi
i[(\beta_{1}^{+}+\lambda_{1}^{+})t_{1}+\cdots+(\beta_{2m^{+}}^{+}+\lambda_{2m^{+}}^{+})t_{2m^{+}}]+\pi
i\lambda_{0}^{+}t_{2m^{+}}}\,\mathrm{d}t_{1}\cdots\mathrm{d}t_{2m^{+}}$
$\displaystyle\times\int_{s>s_{1}>\cdots>s_{2m^{-}}>0}e^{\pi
i[(\beta_{1}^{-}+\lambda_{1}^{-})s_{1}+\cdots+(\beta_{2m^{-}}^{-}+\lambda_{2m^{-}}^{-})s_{2m^{-}}]+\pi
i\lambda_{0}^{-}s_{2m^{-}}}\,\mathrm{d}s_{1}\cdots\mathrm{d}s_{2m^{-}}$

Here in (5.28) each $Z_{j,\epsilon,\iota}$ is a subset of
$\mathcal{N}_{j,\epsilon,\iota}^{ch}$ and $Z_{lp}$ is a subset of
$\mathcal{N}_{lp}^{ch}$, each $\mathcal{C}_{j,\epsilon,\iota}$ and
$\mathcal{C}_{lp}$ is a function satisfying (5.9), and
$\lambda_{a}^{\pm}=\lambda_{a,\pm,1}+\lambda_{a,\pm,2}$ for $1\leq a\leq
2m^{\pm}$, $\lambda_{0}^{\pm}=\lambda_{lp,\pm}$.

By integrability in $(\lambda_{a,\epsilon,\iota})$ and $(\lambda_{lp,\pm})$,
we may fix the choices of these parameters. Note that we can also exploit the
weight

$\prod_{j,\epsilon,\iota}\langle\lambda_{a,\epsilon,\iota}\rangle^{1/4}\langle\lambda_{b,\epsilon,\iota}\rangle^{1/4}\cdot\langle\lambda_{lp,+}\rangle^{1/4}\langle\lambda_{lp,-}\rangle^{1/4}$
(5.29)

from (5.9), whenever needed. Once these parameters are fixed, the relevant
term in (5.28) is then reduced to the product of a function of
$(t,\alpha_{1}^{+},\cdots,\alpha_{m^{+}}^{+})$, and a function of $s$ and
$(s,\alpha_{1}^{-},\cdots,\alpha_{m^{-}}^{-})$. Let us look at the function
depending on $t$, since the other can be treated in the same way.

As in (5.12), this function can be written as

$K(t,\alpha_{1}^{+},\cdots,\alpha_{m^{+}}^{+},\lambda_{0}^{+},\lambda_{1}^{+},\cdots,\lambda_{2m^{+}}^{+}).$
(5.30)

For $1\leq j\leq m^{+}$, as in (5.14), define
$\widetilde{\alpha_{j}}=\alpha_{j}^{+}+\epsilon_{j}^{+}\lambda_{a}^{+}$ and
$\mu_{j}^{+}=\lambda_{a}^{+}+\lambda_{b}^{+}$ where $(a,b)$ is the $j$-th
pair, then using (5.13) and Lemma 5.10, we can write

$K(t,\alpha_{1}^{+},\cdots,\alpha_{m^{+}}^{+},\lambda_{0}^{+},\lambda_{1}^{+},\cdots,\lambda_{2m^{+}}^{+})=\sum_{Z^{+}\subset\\{1,\cdots,m^{+}\\}}\prod_{j\in
Z^{+}}\frac{\chi_{\infty}(\widetilde{\alpha_{j}})}{\epsilon_{j}^{+}\pi
i\widetilde{\alpha_{j}}}\cdot\widetilde{I}(e^{\pi i\lambda_{0}^{+}s})(t),$
(5.31)

where $Z^{+}$ is a subset of $\\{1,\cdots,m^{+}\\}$, $\widetilde{I}$ is an
operator depending on the variables $(\mu_{1}^{+},\cdots,\mu_{m^{+}}^{+})$ and
$\widetilde{\alpha}[W^{+}]:=(\widetilde{\alpha_{j}})_{j\in W^{+}}$ where
$W^{+}:=\\{1,\cdots,m^{+}\\}\backslash Z^{+}$, that is the sum of a class $J$
operator and a class $R$ operator in the sense of Definition 5.2. There are
then two sub cases.

_Case 2.1_. Suppose $\widetilde{I}$ is of class $J$, then $\widetilde{I}$ has
form (5.15). The point is that, when the variables
$(\lambda_{0}^{+},\gamma_{1},\gamma_{2},\mu_{1}^{+},\cdots,\mu_{m^{+}}^{+})$
and $\widetilde{\alpha}[W^{+}]$ are fixed, and $\ell$ is a cluster of
$\widetilde{\alpha}[W^{+}]$, then we can write

$J_{\ell;\gamma_{1},\gamma_{2}}(e^{\pi
i\lambda_{0}^{+}s})(t)=\int_{\mathbb{R}}G(\lambda)e^{\pi
it\lambda}\,\mathrm{d}\lambda$ (5.32)

for $t\in[0,1]$, where (viewing $\widetilde{\alpha}[W^{+}]$ as parameters)

${\int_{\mathbb{R}}\langle\lambda\rangle^{1/4}|\partial_{\alpha}^{\rho}G(\lambda)|\,\mathrm{d}\lambda\leq
C(2|\rho|)!\bigg{(}1+\sum_{j\in
W^{+}}|\widetilde{\alpha_{j}}|+|\gamma_{1}|\bigg{)}^{1/4}}.$ (5.33)

This is in fact obvious by calculating

$J_{\ell;\gamma_{1},\gamma_{2}}(e^{\pi i\lambda_{0}^{+}s})(t)=e^{\pi
i(\ell+\gamma_{1})t}\int_{0}^{t}e^{\pi
i(\gamma_{2}+\lambda_{0}^{+}-\ell)s}\,\mathrm{d}s$

and inserting the cutoffs $\chi_{0}(\gamma_{2}+\lambda_{0}^{+}-\ell)$ or
$\chi_{\infty}(\gamma_{2}+\lambda_{0}^{+}-\ell)$ as in _Case 1_ above,
noticing that if part of the weight is
$|\gamma_{2}+\lambda_{0}^{+}-\ell|^{1/4}$, it can be treated by exploiting the
denominator which contains the same expression
$\gamma_{2}+\lambda_{0}^{+}-\ell$. Now using (5.15) to expand $\widetilde{I}$
as a linear combination of $J_{\ell;\gamma_{1},\gamma_{2}}$, and combining
(5.33) with (5.16) (using Leibniz rule and Lemma A.4 if necessary), we obtain
that

$\widetilde{I}(e^{\pi
i\lambda_{0}^{+}s})(t)=\int_{\mathbb{R}}H(\lambda,\lambda_{0}^{+},\mu_{1}^{+},\cdots,\mu_{m^{+}}^{+},\widetilde{\alpha}[W^{+}])e^{\pi
it\lambda}\,\mathrm{d}\lambda$ (5.34)

for $t\in[0,1]$, where

${\int\langle\lambda\rangle^{1/4}|\partial_{\alpha}^{\rho}H(\lambda,\lambda_{0}^{+},\mu_{1}^{+},\cdots,\mu_{m^{+}}^{+},\widetilde{\alpha}[W^{+}])|\,\mathrm{d}\widetilde{\alpha}[W^{+}]\mathrm{d}\lambda\leq
C^{m^{+}}(2|\rho|)!}.$ (5.35)

_Case 2.2_. Suppose $\widetilde{I}$ is of class $R$, then $\widetilde{I}$ has
form (5.17), say with special index $1$. The arguments are similar to _Case
2.1_ , except that now we are considering the function

$R_{\ell_{1}+\epsilon\widetilde{\alpha_{1}},\ell_{2}+\epsilon\widetilde{\alpha_{1}};\gamma_{1},\gamma_{2},\gamma_{3}}(e^{\pi
i\lambda_{0}^{+}s})(t)=\frac{\chi(\ell_{1}+\epsilon\widetilde{\alpha_{1}}+\gamma_{3})}{\ell_{1}+\epsilon\widetilde{\alpha_{1}}+\gamma_{3}}e^{\pi
i(\ell_{2}+\epsilon\widetilde{\alpha_{1}}+\gamma_{1})t}\int_{0}^{t}e^{\pi
i(\gamma_{2}+\lambda_{0}^{+}-\ell_{2}-\epsilon\widetilde{\alpha_{1}})s}\,\mathrm{d}s.$

By inserting the cutoffs
$\chi_{0}(\gamma_{2}+\lambda_{0}^{+}-\ell_{2}-\epsilon\widetilde{\alpha_{1}})$
or
$\chi_{\infty}(\gamma_{2}+\lambda_{0}^{+}-\ell_{2}-\epsilon\widetilde{\alpha_{1}})$
as above, and using the fact that the resulting coefficient depending on
$\widetilde{\alpha_{1}}$ is bounded by

$\frac{1}{\langle\ell_{1}+\epsilon\widetilde{\alpha_{1}}+\gamma_{3}\rangle}\cdot\frac{1}{\langle\gamma_{2}+\lambda_{0}^{+}-\ell_{2}-\epsilon\widetilde{\alpha_{1}}\rangle}$
(5.36)

which is integrable in $\widetilde{\alpha_{1}}$ uniformly in other variables,
we can conclude that, similar to (5.32) and (5.33), we have

$R_{\ell_{1}+\epsilon\widetilde{\alpha_{1}},\ell_{2}+\epsilon\widetilde{\alpha_{1}};\gamma_{1},\gamma_{2},\gamma_{3}}(e^{\pi
i\lambda_{0}^{+}s})(t)=\int_{\mathbb{R}}G_{1}(\lambda,\widetilde{\alpha_{1}})e^{\pi
it\lambda}\,\mathrm{d}\lambda$ (5.37)

for $t\in[0,1]$, where (again viewing
$\widetilde{\alpha}[W^{+}\backslash\\{1\\}]$ as parameters)

${\int_{\mathbb{R}^{2}}\langle\lambda\rangle^{1/4}|\partial_{\alpha}^{\rho}G_{1}(\lambda,\widetilde{\alpha_{1}})|\,\mathrm{d}\lambda\mathrm{d}\widetilde{\alpha_{1}}\leq
C(2|\rho|)!\bigg{(}1+\sum_{1\neq j\in
W^{+}}|\widetilde{\alpha_{j}}|+|\gamma_{1}|+|\gamma_{3}|\bigg{)}^{1/4}}.$
(5.38)

Note that in deducing (5.38) we have used the fact that the integrability of
(5.36) remains true uniformly in the other variables, even if one of the
denominators is raised to the $3/4$-th power. Here if part of the weight is
$|\gamma_{2}+\lambda_{0}^{+}-\ell_{2}-\epsilon\widetilde{\alpha_{1}}|^{1/4}$,
it can be estimated using the denominator which contains the same expression
$\gamma_{2}+\lambda_{0}^{+}-\ell_{2}-\epsilon\widetilde{\alpha_{1}}$; if part
of the weight is $|\widetilde{\alpha_{1}}|^{1/4}$, it can be estimated using
the denominator $\ell_{1}+\epsilon\widetilde{\alpha_{1}}+\gamma_{3}$. Now
using (5.17) to expand $\widetilde{I}$ as a linear combination of
$R_{\ell_{1}+\epsilon\widetilde{\alpha_{1}},\ell_{2}+\epsilon\widetilde{\alpha_{1}};\gamma_{1},\gamma_{2},\gamma_{3}}$,
and combining (5.38) with (5.18) as in _Case 2.1_ above, we obtain that (5.34)
and (5.35) remain true in this case.

Now, in either case, we have obtained the formula (5.34) and the estimate
(5.35), which are enough to treat the $\widetilde{I}(e^{\pi
i\lambda_{0}^{+}s})$ part of $K$ in (5.31), noticing that
$\widetilde{\alpha_{j}}$ is a translation of $\alpha_{j}^{+}$ given the
$(\lambda_{a,\epsilon,\iota})$ and $(\lambda_{lp,\pm})$ variables. However,
for $j\in Z^{+}$, we still have
$\widetilde{\alpha_{j}}=\alpha_{j}^{+}+\epsilon_{j}^{+}\lambda_{a}^{+}$
instead of $\alpha_{j}^{+}$ in (5.31). But this is easily resolved using the
simple fact that

$\int_{\mathbb{R}}\bigg{|}\frac{\chi_{\infty}(\alpha_{j}^{+}+\epsilon_{j}^{+}\lambda_{a}^{+})}{\epsilon_{j}^{+}\pi
i(\alpha_{j}^{+}+\epsilon_{j}^{+}\lambda_{a}^{+})}-\frac{\chi_{\infty}(\alpha_{j}^{+})}{\epsilon_{j}^{+}\pi
i\alpha_{j}^{+}}\bigg{|}\,\mathrm{d}\alpha_{j}^{+}\leq
C\log(2+|\lambda_{a}^{+}|)\leq C\langle\lambda_{a}^{+}\rangle^{1/12}.$ (5.39)

Therefore, if we encounter the difference term
$\frac{\chi_{\infty}(\alpha_{j}^{+}+\epsilon_{j}^{+}\lambda_{a}^{+})}{\epsilon_{j}^{+}\pi
i(\alpha_{j}^{+}+\epsilon_{j}^{+}\lambda_{a}^{+})}-\frac{\chi_{\infty}(\alpha_{j}^{+})}{\epsilon_{j}^{+}\pi
i\alpha_{j}^{+}}$ for some $j$, we simply move this $j$ from $Z^{+}$ to
$W^{+}$ and exploit the integrability of this difference in $\alpha_{j}^{+}$.
The higher order derivatives can be treated in the same way as in _Case 1_ ,
and the total loss caused by the right hand sides of (5.39) is bounded by

$C^{m^{+}}\prod_{j\in
Z^{+}}\prod_{\iota}\langle\lambda_{a,+,\iota}\rangle^{1/12}\langle\lambda_{b,+,\iota}\rangle^{1/12}$
(5.40)

(where $Z^{+}$ is the set before moving the elements $j$). Clearly 5.40 can be
controlled by the part in (5.29) with sign $+$, up to a $C^{m^{+}}$ factor.

Now we can return to the formula (5.28). Clearly the arguments for the
functions depending on $t$ can be repeated for the function depending on $s$,
obtaining a set $Z^{-}$. By combining these arguments, as well as the
arguments moving some of the $j\in Z^{\pm}$ to $W^{\pm}$, we can write
$\widetilde{\mathcal{B}}_{\mathcal{Q}}$ in the form of (5.8), where

$Z=\bigg{(}\bigcup_{j,\epsilon,\iota}Z_{j,\epsilon,\iota}\bigg{)}\cup
Z_{lp}\cup\big{\\{}\mathfrak{n}_{a}^{+}:a<b,\,j\in
Z^{+}\big{\\}}\cup\big{\\{}\mathfrak{n}_{a}^{-}:a<b,\,j\in Z^{-}\big{\\}};$

note also that $\epsilon_{j}^{+}=\zeta_{\mathfrak{n}_{a}^{+}}$ for $j\in
Z^{+}$ (hence $\mathfrak{n}_{a}^{+}\in Z$) as in Section 5.1.1, and that the
sets $Z^{\pm}$ may have been modified after moving the elements $j$ as
described above. The bound (5.9) without derivatives then follows from the
estimates obtained above including (5.35) and (5.39); in fact, if the regular
couples $\mathcal{Q}_{j,\epsilon,\iota}$ has scale $n_{j,\epsilon,\iota}$
etc., then the induction hypothesis together with the above estimates bounds
the left hand side of (5.9) without derivatives by

$\prod_{j,\epsilon,\iota}C^{n_{j,\epsilon,\iota}}\cdot C^{n_{lp}}\cdot
C^{m^{+}}\cdot C^{m^{-}}=C^{n},$

noticing that

$\sum_{j,\epsilon,\iota}n_{j,\epsilon,\iota}+n_{lp}+m^{+}+m^{-}=n.$

The higher order derivative estimates in (5.9) can be proved in the same way
as in _Case 1_ using the factorized structure. This completes the proof of
(5.8) and (5.9).

Finally we prove (5.10) using a modification of the above inductive arguments.
The proof scheme is the same, except that we induct (5.10) in addition to
(5.9). In the inductive step, if
$\mathfrak{n}_{*}:=\arg\\!\max\langle\alpha_{\mathfrak{n}}\rangle$ belongs to
one of $\mathcal{N}_{j,\epsilon,\iota}^{ch}\backslash Z_{j,\epsilon,\iota}$ or
$\mathcal{N}_{lp}^{ch}\backslash Z_{lp}$, then we repeat the above arguments
using (5.10) for $\mathcal{C}_{j,\epsilon,\iota}$ or $\mathcal{C}_{lp}$, and
noticing that changing the exponent $1/4$ to $1/8$ does not affect any part of
the above proof.

Now suppose $\mathfrak{n}_{*}=\mathfrak{n}_{a}^{\pm}$ with $a<b$ and $j\in
W^{\pm}$ (in _Case 2_ , or $\mathfrak{n}_{*}=\mathfrak{r}$ in _Case 1_ which
is similar), then for the functions $\mathcal{C}_{j,\epsilon,\iota}$ and
$\mathcal{C}_{lp}$ we do not need to gain the $\alpha_{\mathfrak{n}}$ weight,
so we can use the bound (5.9) for them. We shall replace the weight
$\langle\lambda\rangle^{1/4}$ in (5.33) and (5.38) by
$\langle\lambda\rangle^{1/8}\langle\alpha_{\mathfrak{n}_{*}}\rangle^{1/8}$.
For example, if $\mathfrak{n}_{*}=\mathfrak{n}_{a}^{+}$ in _Case 2.1_ then
instead of (5.35) we now have

$\int\langle\lambda\rangle^{1/8}\langle\alpha_{\mathfrak{n}_{*}}\rangle^{1/8}|H(\lambda,\lambda_{0}^{+},\mu_{1}^{+},\cdots,\mu_{m^{+}}^{+},\widetilde{\alpha}[W^{+}])|\,\mathrm{d}\widetilde{\alpha}[W^{+}]\mathrm{d}\lambda\leq
C^{m^{+}}\langle\lambda_{a}^{+}\rangle^{1/8},$

using also that
$\widetilde{\alpha_{j}}=\alpha_{\mathfrak{n}_{*}}\pm\lambda_{a}^{+}$; the
other cases are similar. Since
$\lambda_{a}^{+}=\lambda_{a,+,1}+\lambda_{a,+,2}$, we know that the factor
$\langle\lambda_{a}^{+}\rangle^{1/8}$ can be added to (5.40) which is then
still controlled by the part in (5.29) with sign $+$. This means that we can
insert the power $\langle\alpha_{\mathfrak{n}_{*}}\rangle^{1/8}$ at the price
of weakening $\langle\lambda\rangle^{1/4}$ to $\langle\lambda\rangle^{1/8}$.
Finally, if $\mathfrak{n}_{*}$ occurs in the process of moving $j$ from
$Z^{+}$ to $W^{+}$, then the same result is true because (5.39) is still true,
with the right hand side replaced by $C\langle\lambda_{a}^{+}\rangle^{1/7}$,
if the integrand on the left hand side is multiplied by
$\langle\alpha_{j}^{+}\rangle^{1/8}$. This proves (5.10). ∎

##  6\. Regular couples II: approximation by integrals

With the properties of $\mathcal{B}_{\mathcal{Q}}$ obtained in Section 5, we
now calculate the asymptotics of $\mathcal{K}_{\mathcal{Q}}$ as in (2.24) for
regular $\mathcal{Q}$, using number theoretic methods.

###  6.1. A general approximation result

We prove here a general approximation result, which we apply to
$\mathcal{K}_{\mathcal{Q}}$ in Section 6.2.

######  Proposition 6.1.

Fix $\beta\in(\mathbb{R}^{+})^{d}\backslash\mathfrak{Z}$. Consider the
following expression

$I:=\sum_{(x_{1},\cdots,x_{n})}\sum_{(y_{1},\cdots,y_{n})}W(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})\cdot\Psi(L^{2}\delta\langle
x_{1},y_{1}\rangle_{\beta},\cdots,L^{2}\delta\langle
x_{n},y_{n}\rangle_{\beta}).$ (6.1)

where $(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})\in(\mathbb{Z}_{L}^{d})^{2n}$ in
(6.1). Assume there is a (strict) partial ordering $\prec$ on
$\\{1,\cdots,n\\}$, and that the followings hold for the functions $W$ and
$\Psi$:

(1) The function $W$ satisfies the bound (here $\widehat{W}$ denotes the
Fourier transform in $(\mathbb{R}^{d})^{2n}$)

$\|\widehat{W}\|_{L^{1}}+\|\widehat{\partial W}\|_{L^{1}}\leq(C^{+})^{n}.$
(6.2)

(2) This $W$ is supported in the set

$E:=\big{\\{}(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}):|\widetilde{x_{j}}-a_{j}|,\,|\widetilde{y_{j}}-b_{j}|\leq\lambda_{j},\,\forall
1\leq j\leq n\big{\\}},$ (6.3)

where $1\leq\lambda_{j}\leq(\log L)^{4}$ are constants, $a_{j}$ and $b_{j}$
are constant vectors. Each $\widetilde{x_{j}}$ is a linear function that
equals either $x_{j}$, or $x_{j}\pm x_{j^{\prime}}$ or $x_{j}\pm
y_{j^{\prime}}$ for some $j^{\prime}\prec j$, similarly each
$\widetilde{y_{j}}$ equals either $y_{j}$, or $y_{j}\pm x_{j^{\prime\prime}}$
or $y_{j}\pm y_{j^{\prime\prime}}$ for some $j^{\prime\prime}\prec j$.

(3) For some set $J\subset\\{1,\cdots,n\\}$, the function $\Psi$ has the
expression

$\Psi(\Omega_{1},\cdots,\Omega_{n})=\prod_{j\in
J}\frac{\chi_{\infty}(\Omega_{j})}{\Omega_{j}}\cdot\Psi_{1}(\Omega[J^{c}]),$
(6.4)

where $\chi_{\infty}$ is as in Section 2.3.1, and for any $|\rho|\leq 10n$ we
have

$\|\partial^{\rho}\Psi_{1}\|_{L^{1}}\leq
C^{n}(4|\rho|)!,\,\,\big{\|}\max_{j\in
J^{c}}\langle\Omega_{j}\rangle^{1/8}\cdot\Psi_{1}\big{\|}_{L^{1}}\leq C^{n}.$
(6.5)

Assume $n\leq(\log L)^{3}$. Then we have

$\bigg{|}I-L^{2dn}\int_{(\mathbb{R}^{d})^{2n}}W(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})\cdot\Psi(L^{2}\delta\langle
x_{1},y_{1}\rangle_{\beta},\cdots,L^{2}\delta\langle
x_{n},y_{n}\rangle_{\beta})\,\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}\mathrm{d}y_{1}\cdots\mathrm{d}y_{n}\bigg{|}\\\
\leq(\lambda_{1}\cdots\lambda_{n})^{C}(C^{+}L^{2d-2}\delta^{-1})^{n}L^{-2\nu}.$
(6.6)

Moreover, defining

$I_{\mathrm{app}}=(L^{2d-2}\delta^{-1})^{n}\int\Psi_{1}\mathrm{d}\Omega[J^{c}]\cdot\int_{(\mathbb{R}^{d})^{2n}}W(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})\\\
\times\prod_{j\in J}\frac{1}{\langle
x_{j},y_{j}\rangle_{\beta}}\prod_{j\not\in J}\bm{\delta}(\langle
x_{j},y_{j}\rangle_{\beta})\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}\mathrm{d}y_{1}\cdots\mathrm{d}y_{n},$
(6.7)

where the singularities $1/\langle x_{j},y_{j}\rangle_{\beta}$ are treated
using the Cauchy principal value, we have

$|I_{\mathrm{app}}|\leq(\lambda_{1}\cdots\lambda_{n})^{C}(C^{+}L^{2d-2}\delta^{-1})^{n},\quad|I-I_{\mathrm{app}}|\leq(\lambda_{1}\cdots\lambda_{n})^{C}(C^{+}L^{2d-2}\delta^{-1})^{n}L^{-2\nu}.$
(6.8)

Before proving Proposition 6.1, we first need to establish a few auxiliary
results. In these results we will use the notation $e(z)=e^{2\pi iz}$, and fix
$\lambda$ such that $1\leq\lambda\leq(\log L)^{4}$; moreover we will set
$\upsilon=1/40$, so that $\nu\ll\upsilon$ by our choice.

######  Lemma 6.2.

Suppose $\Phi:\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{C}$
is a function satisfying the bounds

$\sup_{s,x,y}|\partial_{x}^{\alpha}\partial_{y}^{\beta}\Phi(s,x,y)|\leq D$
(6.9)

for all multi-indices $|\alpha|,|\beta|\leq 10d$. Then we have:

(1) The following bound

$\left|\int_{\mathbb{R}}\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(s,x,y)\cdot
e(\xi\cdot x+\eta\cdot y+s\langle
x,y\rangle_{\beta})\,\mathrm{d}x\mathrm{d}y\mathrm{d}s\right|\lesssim
D\lambda^{2d}$ (6.10)

holds uniformly in $(\xi,\eta,a,b)\in\mathbb{R}^{4d}$.

(2) Suppose, in addition, that $\Phi$ satisfies one of the following two
requirements:

  1. (a)

$\Phi(s,x,y)=\chi_{0}(\frac{s}{K})\widehat{\psi}(\frac{s}{\delta
L^{2}})\Phi^{\prime}(x,y)$, where $\psi:=\frac{\chi_{\infty}(t)}{t}$ and
$\Phi^{\prime}$ satisfies (6.9) without $s$, or

  2. (b)

$\Phi(s,x,y)=\chi_{0}(\frac{s}{K})\Phi^{\prime}(\frac{s}{\delta L^{2}},x,y)$,
where $\Phi^{\prime}$ satisfies (6.9), and
$\mathcal{F}_{s}\Phi^{\prime}(\cdot,x,y)$ is supported on an interval of
length $O(1)$ in $\mathbb{R}$ which does not depend on
$(x,y)\in\mathbb{R}^{2d}$.

Here $K:=\lambda^{-1}L^{1-\upsilon}\gg 1$. Then, there holds

$\begin{split}\bigg{|}\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\int_{\mathbb{R}}\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(s,x,y)\cdot
e[(Lg+\xi)\cdot x+(Lh+\eta)\cdot y+s\langle x,y\rangle_{\beta}]\bigg{|}\\\
\lesssim
D\lambda^{2d}\min\left[(1+|\xi|+|\eta|)L^{-\frac{4}{10}\upsilon},1\right]\end{split}$
(6.11)

uniformly in $(a,b)\in\mathbb{R}^{2d}$. In particular, we have

$\bigg{|}\sum_{(g,h)\in\mathbb{Z}^{2d}}\int_{\mathbb{R}}\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(s,x,y)\cdot
e[(Lg+\xi)\cdot x+(Lh+\eta)\cdot y+s\langle
x,y\rangle_{\beta}]\bigg{|}\lesssim D\lambda^{2d}$ (6.12)

uniformly in $(\xi,\eta,a,b)\in\mathbb{R}^{4d}$.

###### Proof.

(1) By translating $x$ and $y$, it is enough to consider the case $a=b=0$. The
result then follows by an application of the stationary phase lemma (or direct
integration of the Gaussian phase) to bound the integral in $(x,y)$ by
$\lambda^{2d}\langle s\rangle^{-d}$ which is integrable.

(2) Let us denote the left hand side of (6.11) without absolute value by
$M(\xi,\eta)$ and also $\Omega=\langle x,y\rangle_{\beta}$. Here, we split the
discussion into two cases depending on the size of $a$ and $b$:

_Case 1: if $\max(|a|,|b|)\leq\lambda L^{\upsilon/2}$._ Here, we argue via a
stationary phase analysis for the phase function $\varphi(x,y)=(Lg+\xi)\cdot
x+(Lh+\eta)\cdot y+s\Omega$. Noting that

$\nabla_{x}\varphi=Lg+\xi+s(\beta^{1}y^{1},\ldots,\beta^{d}y^{d}),\quad\nabla_{y}\varphi=Lh+\eta+s(\beta^{1}x^{1},\ldots,\beta^{d}x^{d}),$

and using our assumption on the $s$ support of $\Phi$, we can bound the norms
of $s(\beta^{1}y^{1},\ldots,\beta^{d}y^{d})$ and
$s(\beta^{1}x^{1},\ldots,\beta^{d}x^{d})$ by $L/10$ if $L$ is large enough. If
$|Lg+\xi|+|Lh+\eta|\geq L/5$ (which happens for all but one value of $(g,h)$),
we integrate by parts at most $2d$ times in $x$ or $y$ in the
$\mathrm{d}x\mathrm{d}y$ integral, and gain a denominator that is bounded
below by $(L+|Lg+\xi|+|Lh+\eta|)^{2d}$. For the only remaining value of
$(g,h)$, we use a stationary phase estimate similar to part (1). In the end we
get

$\displaystyle|M(\xi,\eta)|\lesssim$ $\displaystyle
D\lambda^{2d}\sum_{(g,h)\neq
0}\int_{|s|\leq\lambda^{-1}L^{1-\upsilon}}\left[(L+|Lg+\xi|+|Lh+\eta|)^{-2d}+\langle
s\rangle^{-d}\mathbf{1}_{|Lg+\xi|+|Lh+\eta|<L/5}\right]$
$\displaystyle\lesssim$ $\displaystyle
D\lambda^{2d}\bigg{[}L^{-\upsilon}+\sum_{(g,h)\neq
0}\mathbf{1}_{|Lg+\xi|+|Lh+\eta|<L/5}\bigg{]}$ $\displaystyle\lesssim$
$\displaystyle
D\lambda^{2d}\left(L^{-\upsilon}+\mathbf{1}_{|\xi|+|\eta|\gtrsim
L}\right)\lesssim
D\lambda^{2d}\min\left(1,(1+|\xi|+|\eta|)L^{-\upsilon}\right),$

as needed.

_Case 2: if $\max(|a|,|b|)\geq\lambda L^{\upsilon/2}$._ Here the analysis is
slightly more delicate, and we will obtain the estimate

$|M(\xi,\eta)|\lesssim
D\lambda^{2d}\min\left[1,(1+|\xi|+|\eta|)L^{-\frac{4}{10}\upsilon}\right]$
(6.13)

uniformly in $(a,b)$, which finishes the proof of the lemma.

We start by splitting the integral appearing in $M(\xi,\eta)$ into two parts:
one with $|\nabla_{x}\varphi|+|\nabla_{y}\varphi|\leq L^{1-2\upsilon}$ giving
a contribution $M_{1}(\xi,\eta)$ and the complementary region giving a
contribution $M_{1}^{\prime}(\xi,\eta)$. More precisely, define

$\displaystyle M_{1}(\xi,\eta)=$
$\displaystyle\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\int_{\mathbb{R}}\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(s,x,y)\chi_{0}\left(\frac{\nabla_{x}\varphi}{L^{1-2\upsilon}}\right)\chi_{0}\left(\frac{\nabla_{y}\varphi}{L^{1-2\upsilon}}\right)$
(6.14) $\displaystyle\qquad\qquad\qquad\times e[(Lg+\xi)\cdot x+(Lh+\eta)\cdot
y+s\Omega]\,\mathrm{d}x\mathrm{d}y\mathrm{d}s,$ $\displaystyle
M_{1}^{\prime}(\xi,\eta)=$ $\displaystyle M(\xi,\eta)-M_{1}(\xi,\eta).$

The contribution of $M_{1}^{\prime}(\xi,\eta)$ can be bounded easily by
integrating by parts $10d$ times in either $x$ or $y$ (depending on whether
$|\nabla_{x}\varphi|$ or $|\nabla_{y}\varphi|$ is $\gtrsim L^{1-2\upsilon}$)
and estimated by $(\lambda^{-1}L^{1-\upsilon})\lambda^{2d}L^{-9d}$ which is
more than acceptable. As such, we reduce to obtaining the bound (6.13) for
$M_{1}(\xi,\eta)$.

Next, we would like to localize in $\Omega$. For this let
$J:={\max(|a|,|b|)}\geq\lambda L^{\upsilon/2}$. Here, the analysis is
different depending on whether we make the assumption (a) or (b) on
$\Phi(s,x,y)$. Under assumption (a), we note that $\widehat{\psi}(s)$ is odd
and fastly decaying at infinity, and has a jump discontinuity at 0; all its
derivatives are also uniformly bounded and decaying, except at 0 where they
are not defined. We would like to integrate by parts in $s$ once, in the
region when $|\Omega|\gtrsim JL^{-1+\frac{21}{10}\upsilon}$. Such integration
by parts produces a new $s$-integrand which has the same form and is bounded
as

$\lesssim|\Omega|^{-1}\left(\delta^{-1}L^{-2}+\frac{1}{K}+\frac{J}{L^{1-2\upsilon}}\right)=|\Omega|^{-1}\left(\frac{\lambda}{L^{1-\upsilon}}+\frac{JL^{\upsilon}}{L^{1-\upsilon}}\right)\lesssim\frac{JL^{2\upsilon}}{|\Omega|L}\lesssim
L^{-\upsilon/10},$

and a boundary term (due to the discontinuity of $\widehat{\psi}(s)$ at 0)
that gives a contribution of

$\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\chi_{0}\left(\frac{Lg+\xi}{L^{1-2\upsilon}}\right)\chi_{0}\left(\frac{Lh+\eta}{L^{1-2\upsilon}}\right)\int_{\mathbb{R}^{2d}}\Omega^{-1}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\\\
\times\Phi^{\prime}(x,y)\cdot e[(Lg+\xi)\cdot x+(Lh+\eta)\cdot
y]\,\mathrm{d}x\mathrm{d}y,$

which can be estimated using Lemma 6.3 below by

$\lesssim D\lambda^{2d}\sum_{(g,h)\neq
0}\mathbf{1}_{|Lg+\xi|+|Lh+\eta|<L/10}\lesssim
D\lambda^{2d}\min\left(1,(1+|\xi|+|\eta|)L^{-\upsilon}\right),$

as needed. Repeating this integration by parts $C\upsilon^{-1}$ times, we are
reduced to obtaining the bound (6.13) for

$\displaystyle M_{2}(\xi,\eta)=$
$\displaystyle\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\int_{\mathbb{R}}\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(s,x,y)\chi_{0}\left(\frac{\nabla_{x}\varphi}{L^{1-2\upsilon}}\right)\chi_{0}\left(\frac{\nabla_{y}\varphi}{L^{1-2\upsilon}}\right)\chi_{0}\left(\frac{\Omega}{JL^{-1+\frac{21\upsilon}{10}}}\right)$
$\displaystyle\qquad\qquad\qquad\times e[(Lg+\xi)\cdot x+(Lh+\eta)\cdot
y+s\cdot\Omega]\,\mathrm{d}x\mathrm{d}y\mathrm{d}s.$

Now, under assumption (b) on $\Phi(s,x,y)$, we have, for some
$p\in\mathbb{R}$, $\Phi^{\prime}(s,x,y)=e^{-2\pi
isp}\Phi^{\prime\prime}(s,x,y)$ where
$\mathcal{F}_{s}\Phi^{\prime\prime}(\cdot,x,y)$ is supported in (say) $[-1,1]$
for any $(x,y)\in\mathbb{R}^{2d}$. As before, we split $M_{1}(\xi,\eta)$ in
two parts depending on the size of $|\Omega-\frac{p}{\delta L^{2}}|$. Let
$\widetilde{M}_{2}(\xi,\eta)$ denote the contribution of the region where
$|\Omega-\frac{p}{\delta L^{2}}|\leq JL^{-1+\frac{21}{10}\upsilon}$ and
$\widetilde{M}_{2}^{\prime}(\xi,\eta)$ the contribution of the complementary
region, namely

$\displaystyle\widetilde{M}_{2}(\xi,\eta)=$
$\displaystyle\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\int_{\mathbb{R}}\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\chi_{0}\big{(}\frac{s}{K}\big{)}\Phi^{\prime\prime}\big{(}\frac{s}{\delta
L^{2}},x,y\big{)}\chi_{0}\bigg{(}\frac{\nabla_{x}\varphi}{L^{1-2\upsilon}}\bigg{)}\chi_{0}\bigg{(}\frac{\nabla_{y}\varphi}{L^{1-2\upsilon}}\bigg{)}$
$\displaystyle\qquad\qquad\qquad\chi_{0}\bigg{(}\frac{\Omega-\frac{p}{\delta
L^{2}}}{JL^{-1+\frac{21\upsilon}{10}}}\bigg{)}e\big{[}(Lg+\xi)\cdot
x+(Lh+\eta)\cdot y+s\big{(}\Omega-\frac{p}{\delta
L^{2}}\big{)}\big{]}\,\mathrm{d}x\mathrm{d}y\mathrm{d}s,$
$\displaystyle\widetilde{M}_{2}^{\prime}(\xi,\eta)=$ $\displaystyle
M_{1}(\xi,\eta)-\widetilde{M_{2}}(\xi,\eta).$

The contribution of $\widetilde{M}_{2}^{\prime}$ can be bounded by integrating
sufficiently many times in $s$: Each integration by parts produces a factor of
$(\Omega-\frac{p}{\delta L^{2}})^{-1}$ at the expense of having an $s$
derivative fall on either $\chi_{0}(s/K)$ or $\Phi^{\prime\prime}(s/(\delta
L^{2}),x,y)$ (giving a factor bounded by $D(\delta L^{2})^{-1}$ given the
Fourier support assumption on $\Phi^{\prime\prime}$) or the $\nabla\varphi$
factors in the other spatial cutoffs (which gives a factor bounded by
$\frac{JL^{\upsilon}}{L^{1-\upsilon}}$). In effect, the net gain of this
integration by part step is

$\lesssim\big{|}\Omega-\frac{p}{\delta L^{2}}\big{|}^{-1}\left(\frac{1}{\delta
L^{2}}+\frac{1}{K}+\frac{J}{L^{1-2\upsilon}}\right)\lesssim\frac{JL^{2\upsilon}}{|\Omega-\frac{p}{\delta
L^{2}}|L}\lesssim L^{-\upsilon/10},$

using the lower bound on $|\Omega-\frac{p}{\delta L^{2}}|$ in
$\widetilde{M}_{2}^{\prime}$. As such, we can integrate by parts
$C\upsilon^{-1}$ times in $s$ to obtain that the contribution of
$\widetilde{M}_{2}^{\prime}$ is acceptable as well.

As such, in both cases (a) and (b), we are left with the contribution of
$M_{2}(\xi,\eta)$ and $\widetilde{M}_{2}(\xi,\eta)$ respectively, which we
will estimate in the same way (thanks to the bounds
$|\Phi^{\prime}|,|\Phi^{\prime\prime}|\lesssim D$) by

$\displaystyle
D\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\chi_{0}\bigg{(}\frac{\Omega-\frac{p}{\delta
L^{2}}}{JL^{-1+\frac{21\upsilon}{10}}}\bigg{)}\int_{\mathbb{R}}\chi_{0}\big{(}\frac{s}{K}\big{)}\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\chi_{0}\left(\frac{\nabla_{x}\varphi}{L^{1-2\upsilon}}\right)\chi_{0}\bigg{(}\frac{\nabla_{y}\varphi}{L^{1-2\upsilon}}\bigg{)}\mathrm{d}s\mathrm{d}x\mathrm{d}y.$

Now notice that the volume of the set of $(x,y)$ satisfying
$|x-a|\lesssim\lambda,|y-b|\lesssim\lambda$ and $|\Omega-\frac{p}{\delta
L^{2}}|\lesssim JL^{-1+\frac{21\upsilon}{10}}\big{\\}}$ is bounded by
$\lambda^{2d-1}L^{-1+\frac{21\upsilon}{10}}$ since $|\nabla_{x,y}\Omega|\sim
J$ (using the coarea formula or change of variables). Furthermore, for fixed
$(\xi,\eta,x,y)$, when $(g,h)$ varies, we claim that the measure of the set
$S(x,y)$ on which the $s$ integrand is supported is bounded by
$L^{1-2\upsilon}\cdot\frac{(1+JL^{-\upsilon})}{J}$. To see this, suppose
without loss of generality that $|y^{1}|\sim J$ for example, then $s\in
S(x,y)$ implies that

$\bigg{|}\frac{s\beta^{1}y^{1}}{L}+g^{1}+\frac{\xi^{1}}{L}\bigg{|}\lesssim
L^{-2\upsilon}\Rightarrow\bigg{\\{}\frac{s\beta^{1}y^{1}}{L}+\frac{\xi^{1}}{L}\bigg{\\}}\leq
L^{-2\upsilon},$

where $\\{\cdot\\}$ denotes the distance to the nearest integer. Since also
$|\frac{s\beta^{1}y^{1}}{L}|\lesssim JKL^{-1}=JL^{-\upsilon}$, we conclude
that $\frac{s\beta^{1}y^{1}}{L}$ belongs to an interval of length $\lesssim
JL^{-\upsilon}$, intersected by the $L^{-2\upsilon}$ neighborhood of the
lattice $\frac{\xi_{1}}{L}+\mathbb{Z}$. Thus $\frac{sy^{1}}{L}$ belongs to a
set of measure at most $(1+JL^{-\upsilon})L^{-2\upsilon}$ and hence

$|S(x,y)|\lesssim\frac{L}{J}(1+JL^{-\upsilon})L^{-2\upsilon},$

as claimed. With this estimate in hand, we can bound both $M_{2}$ and
$\widetilde{M}_{2}$ by

$\displaystyle\lesssim$ $\displaystyle
D\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\chi_{0}\bigg{(}\frac{\Omega-\frac{p}{\delta
L^{2}}}{JL^{-1+\frac{21\upsilon}{10}}}\bigg{)}\int_{S(x,y)}\sum_{0\neq(g,h)\in\mathbb{Z}^{2d}}\chi_{0}\bigg{(}\frac{\nabla_{x}\varphi}{L^{1-2\upsilon}}\bigg{)}\chi_{0}\bigg{(}\frac{\nabla_{y}\varphi}{L^{1-2\upsilon}}\bigg{)}\mathrm{d}s\mathrm{d}x\mathrm{d}y$
$\displaystyle\lesssim$ $\displaystyle
D\int_{\mathbb{R}^{2d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\chi_{0}\bigg{(}\frac{\Omega-\frac{p}{\delta
L^{2}}}{JL^{-1+\frac{21\upsilon}{10}}}\bigg{)}|S(x,y)|\,\mathrm{d}x\mathrm{d}y$
$\displaystyle\lesssim$ $\displaystyle
D\lambda^{2d-1}L^{-1+\frac{21\upsilon}{10}}\times\frac{L}{J}(1+JL^{-1})L^{-2\upsilon}=D\lambda^{2d-1}L^{\upsilon/10}(J^{-1}+L^{-1})\lesssim
D\lambda^{2d-1}L^{-\frac{4\upsilon}{10}}.$

This finishes the proof of (6.13), and hence that of (6.11). ∎

######  Lemma 6.3.

Suppose that $\Theta=\Theta(x,y)$ satisfies (6.9) without $s$.

(1) The following bound holds

$\left|\int_{\mathbb{R}^{2d}}\Omega^{-1}\chi_{\infty}(\mu\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Theta(x,y)e(\xi\cdot
x+\eta\cdot y)\,\mathrm{d}x\mathrm{d}y\right|\lesssim D\lambda^{2d},$

uniformly in $\mu>1$ and $(a,b,\xi,\eta)\in\mathbb{R}^{4d}$. In particular,

$\lim_{\mu\to\infty}\int_{\mathbb{R}^{2d}}\Omega^{-1}\chi_{\infty}(\mu\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Theta(x,y)e(\xi\cdot
x+\eta\cdot y)\,\mathrm{d}x\mathrm{d}y$

exists and is $\lesssim D\lambda^{2d}$ uniformly in
$(a,b,\xi,\eta)\in\mathbb{R}^{4d}$.

(2) The following estimate holds for the difference

$\left|\int_{\mathbb{R}^{2d}}\Omega^{-1}\chi_{0}(\mu\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Theta(x,y)e(\xi\cdot
x+\eta\cdot y)\,\mathrm{d}x\mathrm{d}y\right|\lesssim
D\lambda^{2d}\mu^{-1}(1+|\xi|+|\eta|),$

uniformly in $(a,b)$.

###### Proof.

(1) It is enough to consider the region $|\Omega|\lesssim 1$. Also, without
loss of generality we only need to consider the region with
$|x^{1}|\sim\max(|x|,|y|)$. Recall that $\chi_{0}$ is extended to
$\mathbb{R}^{d}$ as in Section 2.3.1; by abusing notation we may also use some
$\chi_{0}$ of different support. Let $x=(x^{1},x^{\prime})$, and
$\beta=(\beta^{1},\beta^{\prime})$ etc., it is enough to consider

$\displaystyle N(\mu,\xi,\eta):=$
$\displaystyle\int_{\mathbb{R}^{2d}}\Omega^{-1}\chi_{\infty}(\mu\Omega)\chi_{0}(\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Theta(x,y)\chi_{0}\big{(}\frac{x^{\prime}}{|x^{1}|}\big{)}\chi_{0}\big{(}\frac{y}{|x^{1}|}\big{)}e(\xi\cdot
x+\eta\cdot y)\mathrm{d}x\mathrm{d}y$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{x^{\prime}}{|x^{1}|}\big{)}e(\xi\cdot
x)\mathrm{d}x\int_{\mathbb{R}^{d}}\Omega^{-1}\chi_{\infty}(\mu\Omega)\chi_{0}(\Omega)\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Theta(x,y)\chi_{0}\big{(}\frac{y}{|x^{1}|}\big{)}e(\eta\cdot
y)\mathrm{d}y.$

We change variables in the $y^{1}$ integral by setting
$u=\Omega=\beta^{1}x^{1}y^{1}+\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}$, and write

$\displaystyle N(\mu,\xi,\eta)=$
$\displaystyle\int_{\mathbb{R}^{d}}|\beta^{1}x^{1}|^{-1}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{x^{\prime}}{|x^{1}|}\big{)}e(\xi\cdot
x)\mathrm{d}x\int_{\mathbb{R}^{d-1}}\chi_{0}\big{(}\frac{y^{\prime}-b^{\prime}}{\lambda}\big{)}e\bigg{[}\eta^{\prime}\cdot
y^{\prime}-\frac{\eta^{1}\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}}{\beta^{1}x^{1}}\bigg{]}\mathrm{d}y^{\prime}$
$\displaystyle\qquad\times\int_{\mathbb{R}}\frac{\chi_{\infty}(\mu
u)\chi_{0}(u)}{u}\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Theta}(x,u,y^{\prime})e\big{(}\frac{\eta^{1}u}{\beta^{1}x^{1}}\big{)}\mathrm{d}u.$
(6.15)

where
$\widetilde{\Theta}(x,u,y^{\prime})=\Theta(x,y)\chi_{0}(\frac{y}{|x^{1}|})$
evaluated at $y^{1}=\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}}{\beta^{1}x^{1}}$.

Now, we write the integral over $u$ as

$\int_{0}^{\infty}\frac{\chi_{\infty}(\mu
u)\chi_{0}(u)}{u}\Delta(u,y^{\prime},x)\,\mathrm{d}u,$

where

$\displaystyle\Delta(u,y^{\prime},x)$
$\displaystyle=\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Theta}(x,u,y^{\prime})e\big{(}\frac{\eta^{1}u}{\beta^{1}x^{1}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\chi_{0}\bigg{(}\frac{-u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Theta}(x,-u,y)e\big{(}\frac{-\eta^{1}u}{\beta^{1}x^{1}}\big{)}$
$\displaystyle=2i\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Theta}(x,u,y^{\prime})\sin\big{(}2\pi\frac{\eta^{1}u}{\beta^{1}x^{1}}\big{)}+D\cdot
O\bigg{(}\frac{|u|}{\beta^{1}|x^{1}|}+\frac{|u|}{\beta^{1}|x^{1}|^{2}}\bigg{)}.$

Noting that the above integral is actually supported on the interval
$[(2\mu)^{-1},2]$, the contribution of the terms
$O(\frac{u}{\beta^{1}|x^{1}|}+\frac{u}{\beta^{1}|x^{1}|^{2}})$ is acceptable.
Hence, we are left with bounding

$\displaystyle\int_{0}^{\infty}\chi_{\infty}(\mu
u)\chi_{0}(u)\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Theta}(x,u,y^{\prime})\sin\big{(}2\pi\frac{\eta^{1}u}{\beta^{1}x^{1}}\big{)}\frac{\mathrm{d}u}{u}$
$\displaystyle=\pm\int_{0}^{\infty}\chi_{\infty}\big{(}\frac{\mu\beta^{1}|x^{1}|u}{|\eta^{1}|}\big{)}\chi_{0}\big{(}\frac{\beta^{1}|x^{1}|u}{|\eta^{1}|}\big{)}\chi_{0}\big{(}\frac{u-Q}{\lambda|\eta^{1}|}\big{)}\widetilde{\Theta}\big{(}x,\frac{\beta^{1}x^{1}u}{\eta^{1}},y^{\prime}\big{)}\sin(2\pi
u)\frac{\mathrm{d}u}{u}$

for some $Q$ depending on $x,y^{\prime},\beta,\lambda$. The integral for
$0\leq u\leq 5$ is bounded by $O(1)$, so we may use a smooth cutoff
$\varphi(u)$ to restrict to $u\geq 5$. Then we integrate by parts in $u$ once.
If the derivative falls on $1/u$ or $\varphi(u)$ the resulting contribution is
bounded by $O(1)$. If the derivative falls on anything but the first factor,
then the new integrand is bounded by
$\frac{\beta^{1}|x^{1}|}{|\eta^{1}|}(1+\frac{1}{|x^{1}|}+\frac{1}{|x^{1}|^{2}})$,
so the resulting contribution is bounded by
$1+\frac{1}{|x^{1}|}+\frac{1}{|x^{1}|^{2}}$ even without using the $1/u$
factor, as $u$ belongs to an interval of length
$\frac{|\eta^{1}|}{\beta^{1}|x^{1}|}$ due to the second factor. If the
derivative falls on the first factor, then in the new integrand we will have
$\frac{\mu\beta^{1}|x^{1}|}{|\eta^{1}|}$ instead of
$\frac{\beta^{1}|x^{1}|}{|\eta^{1}|}$, but $u$ also belongs to a smaller
interval of length $\frac{|\eta^{1}|}{\mu\beta^{1}|x^{1}|}$, so the conclusion
will be the same.

This leads to an acceptable contribution to $N(\mu,\xi,\eta)$ in (6.15), and
gives a bound that is uniform in $\mu,\xi,\eta$. The statement about the
$\lim_{\mu\to\infty}$ follows directly from the above argument and dominated
convergence.

(2) Arguing exactly as above, it is enough to bound

$\displaystyle\int_{0}^{\infty}\frac{\chi_{0}(\mu
u)}{u}\Delta(u,y^{\prime},x)du$
$\displaystyle\lesssim\int_{0}^{\infty}\frac{\chi_{0}(\mu
u)}{u}u\left(\frac{1}{\beta^{1}|x^{1}|}+\frac{1}{\beta^{1}|x^{1}|^{2}}+\frac{|\eta^{1}|}{\beta^{1}|x^{1}|}\right)\,du$
$\displaystyle\lesssim\mu^{-1}\left(\frac{1}{\beta^{1}|x^{1}|}+\frac{1}{\beta^{1}|x^{1}|^{2}}+\frac{|\eta^{1}|}{\beta^{1}|x^{1}|}\right)$

which gives the needed bound when substituted in (6.15). ∎

######  Lemma 6.4.

Suppose that
$\Phi(s,x,y):\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{C}$
is a function satisfying (6.9).

(1) If $\Phi$ is supported on $|s|<L^{2}$, then the following bound holds
uniformly in $(a,b,\xi,\eta)\in\mathbb{R}^{4d}$:

$\int_{\mathbb{R}}\bigg{|}\sum_{(x,y)\in\mathbb{Z}^{2d}_{L}}\Phi(s,x,y)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}e(x\cdot\xi+y\cdot\eta+s\langle
x,y\rangle_{\beta})\bigg{|}\mathrm{d}s\lesssim D\lambda^{4d}L^{2d}.$ (6.16)

(2) If $\Phi(s,x,y)$ is supported on the set $|s|\gtrsim L^{1-\upsilon}$, then
the following improved estimate holds uniformly in
$(a,b,\xi,\eta)\in\mathbb{R}^{4d}$:

$\int_{\mathbb{R}}\big{\langle}\frac{s}{\delta
L^{2}}\big{\rangle}^{-2}\bigg{|}\sum_{(x,y)\in\mathbb{Z}^{2d}_{L}}\Phi(s,x,y)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}e(x\cdot\xi+y\cdot\eta+s\langle
x,y\rangle_{\beta})\bigg{|}\mathrm{d}s\lesssim D\lambda^{4d}L^{2d-\upsilon}.$
(6.17)

###### Proof.

Recall that, $\Omega(x,y)=\sum_{j=1}^{d}\beta^{j}x^{j}y^{j}$ where
$x^{j},y^{j}\in\mathbb{Z}_{L}$. We make the change of variables

$L^{-1}p^{j}={x^{j}+y^{j}},\qquad L^{-1}q^{j}={x^{j}-y^{j}},\qquad p^{j}\equiv
q^{j}\pmod{2}.$

The sum in $(x^{j},y^{j})\in\mathbb{Z}_{L}^{2}$ then becomes the linear
combination of four sums, which are taken over
$(p^{j},q^{j})\in\mathbb{Z}^{2}$, or $(p^{j},q^{j})\in
2\mathbb{Z}\times\mathbb{Z}$, or $(p^{j},q^{j})\in\mathbb{Z}\times
2\mathbb{Z}$, or $(p^{j},q^{j})\in(2\mathbb{Z})^{2}$. We will only consider
the first sum, and it will be obvious from the proof that the other sums are
estimated similarly. Define

$\Upsilon(s,z,w)=\Phi\big{(}s,\frac{z+w}{2},\frac{z-w}{2}\big{)}\chi_{0}\big{(}\frac{z+w-2a}{2\lambda}\big{)}\chi_{0}\big{(}\frac{z-w-2b}{2\lambda}\big{)},$

which has all derivatives in $(z,w)$ up to order $10d$ uniformly bounded, and
is supported in the set $\\{g^{j}\leq Lz^{j}\leq g^{j}+2\lambda L,\,h^{j}\leq
Lw^{j}\leq h^{j}+2\lambda L\\}$, where $(g^{j},h^{j})\in\mathbb{Z}^{2}$ are
determined by $(a,b)$.

Now, by possibly redefining $(s,\xi,\eta)$, we need to show that the function

$\displaystyle B(\xi,\eta)$
$\displaystyle=\int_{\mathbb{R}}\bigg{|}\sum_{(p,q)\in\mathbb{Z}^{2d}}\Upsilon\big{(}s;pL^{-1},qL^{-1}\big{)}e\big{[}sL^{-2}(|p|_{\beta}^{2}-|q|_{\beta}^{2})+p\cdot\xi+y\cdot\eta\big{]}\bigg{|}\,\mathrm{d}s$
$\displaystyle=\int_{\mathbb{R}}\bigg{|}\sum_{(p,q)\in\mathbb{Z}^{2d}}\Upsilon\big{(}s,pL^{-1},qL^{-1}\big{)}\prod_{j=1}^{d}e\big{[}sL^{-2}\beta^{j}(p^{j})^{2}+p^{j}\xi^{j}]\cdot
e[-sL^{-2}\beta^{j}(q^{j})^{2}+q^{j}\eta^{j}\big{]}\bigg{|}\,\mathrm{d}s$

satisfies the bounds in (6.16) when $\Upsilon$ is supported on $|s|<L^{2}$,
and that the corresponding integral with $\langle s/\delta L^{2}\rangle^{-2}$
(which we denote by $\widetilde{B}(\xi,\eta)$) satisfies (6.17) when
$\Upsilon$ is supported on $|s|\gtrsim L^{1-\upsilon}$. Note that in the above
sum we must have $p^{j}\in[g^{j},g^{j}+20\lambda L]$ and
$q^{j}\in[h^{j},h^{j}+20\lambda L]$.

Recall the Gauss sums $G_{h}(s,r,n)$ and $G_{h}(s,r,x)$ defined in (A.8).
Notice that since
$\partial_{x}G_{h}(s,r;x)=\sum_{p\in\mathbb{N}}e(s(h+p)^{2}+r(h+p))\bm{\delta}(x-p)$,
we can write

$\displaystyle B(\xi,\eta)$
$\displaystyle=\int_{\mathbb{R}}\bigg{|}\int_{(u,v)\in\mathbb{R}_{+}^{2d}}\Upsilon\big{(}s,(u+g)L^{-1},(v+h)L^{-1}\big{)}\prod_{j=1}^{d}\partial_{u^{j}}G_{g^{j}}(\beta^{j}sL^{-2},\xi^{j},u^{j})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\times\partial_{v^{j}}G_{h^{j}}(-\beta^{j}sL^{-2},\eta^{j},v^{j})\,\mathrm{d}u\mathrm{d}v\bigg{|}\,\mathrm{d}s$
$\displaystyle\leq
L^{-2d}\int_{\mathbb{R}}\int_{(u,v)\in\mathbb{R}_{+}^{2d}}\big{|}D^{\alpha}\Upsilon\big{(}s,(u+g)L^{-1},(v+h)L^{-1}\big{)}\big{|}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\prod_{j=1}^{d}\big{|}G_{g^{j}}(\beta^{j}sL^{-2},\xi^{j},u^{j})G_{h^{j}}(-\beta^{j}sL^{-2},\eta^{j},v^{j})\big{|}\,\mathrm{d}u\mathrm{d}v\mathrm{d}s,$

where $g=(g^{1},\cdots,g^{d})$ etc., and $D^{\alpha}\Upsilon$ is obtained from
$\Upsilon$ by taking one derivative in each of the variables $u_{j},v_{j}$
(and hence has the same support properties).

(1) We first note that if $\Phi$, and hence $\Upsilon$, is supported on the
set $|s|<L^{2}$, then we have the bound (upon rescaling in $s$ by $L^{2}$)

$\displaystyle|B(\xi,\eta)|\lesssim
DL^{-2d+2}\int_{(u,v)\in\mathbb{R}_{+}^{2d}}\chi_{0}\big{(}\frac{u}{100\lambda
L}\big{)}\chi_{0}\big{(}\frac{v}{100\lambda
L}\big{)}\int_{\mathbb{R}}\chi_{0}(s)\prod_{j=1}^{d}\left|G_{g^{j}}(\beta^{j}s,\xi^{j},u^{j})G_{h^{j}}(-\beta^{j}s,\eta^{j},v^{j})\right|\mathrm{d}u\mathrm{d}v\mathrm{d}s.$

Now, we use Lemma A.5 (and that $2d\geq 6$) to get the needed bound, namely

$\displaystyle|B(\xi,\eta)|\lesssim$ $\displaystyle
DL^{-2d+2}\int_{(u,v)\in\mathbb{R}_{+}^{2d}}\chi_{0}\big{(}\frac{u}{100\lambda
L}\big{)}\chi_{0}(\frac{v}{100\lambda
L})\prod_{j=1}^{d}(u^{j})^{1-1/d}(v^{j})^{1-1/d}\,\mathrm{d}u\mathrm{d}v$
$\displaystyle\lesssim$ $\displaystyle DL^{-2d+2}(\lambda L)^{2d+2d-2}\lesssim
D\lambda^{4d}L^{2d}.$

(2) To obtain the improved bound in (6.17) for $\widetilde{B}(\xi,\eta)$, we
argue a bit differently. Without loss of generality, we can assume
$\beta^{1}=1$ and $\beta^{2}\in[1,2]$. We start by writing the product of
Gauss sums in $\widetilde{B}(\xi,\eta)$ as

$\displaystyle\prod_{j=1}^{d}G_{g^{j}}(\beta^{j}sL^{-2},\xi^{j},u^{j})G_{h^{j}}(-\beta^{j}sL^{-2},\eta^{j},v^{j})$
$\displaystyle=\prod_{j=1}^{2}G_{g^{j}}(\beta^{j}sL^{-2},\xi^{j},u^{j})\prod^{*}G_{g^{j}}(\cdots)G_{h^{j}}(\cdots)$
$\displaystyle=\mathcal{M}(s,\xi,\eta,u)\prod^{*}G_{g^{j}}(\cdots)G_{h^{j}}(\cdots)$

where
$\mathcal{M}(s,\xi,\eta,u)=G_{g^{1}}(sL^{-2},\xi^{1},u^{1})G_{g^{2}}(\beta^{2}sL^{-2},\xi^{2},u^{2})$
and $\prod^{*}$ is the product of the remaining $2d-2$ Gauss sums. We claim
that, for $L^{1-\upsilon}\lesssim|s|\leq L^{2+5\upsilon}$, the following
estimate holds for $\mathcal{M}$:

$\sup_{s,\xi,\eta,u}|\mathcal{M}(s,\xi,\eta,u)|\lesssim(\lambda
L)^{2-10\upsilon}.$ (6.18)

Before proving this claim, let us see why it implies (6.17). Using (6.18), we
have

$\displaystyle|\widetilde{B}(\xi,\eta)|\lesssim$ $\displaystyle
DL^{-2d+2}\int_{(u,v)\in\mathbb{R}_{+}^{2d}}\chi_{0}\big{(}\frac{u}{100\lambda
L}\big{)}\chi_{0}\big{(}\frac{v}{100\lambda L}\big{)}\bigg{(}(\lambda
L)^{2-10\upsilon}\int_{L^{-1-\upsilon}\lesssim|s|\leq
L^{5\upsilon}}\prod^{*}|G_{g^{j}}(\beta^{j}s,\xi^{j},u^{j})|$
$\displaystyle\times$
$\displaystyle|G_{h^{j}}(-\beta^{j}s,\eta^{j},v^{j})|\,\mathrm{d}s+\int_{|s|\geq
L^{5\upsilon}}\langle\delta^{-1}s\rangle^{-2}\prod_{j=1}^{d}\left|G_{g^{j}}(\beta^{j}s,\xi^{j},u^{j})G_{h^{j}}(-\beta^{j}s,\eta^{j},v^{j})\right|\mathrm{d}s\bigg{)}\mathrm{d}u\mathrm{d}v.$

Splitting the $s$ region into intervals of length 1, and using Lemma A.5 again
on each subinterval we obtain that

$\displaystyle|\widetilde{B}(\xi,\eta)|$ $\displaystyle\lesssim
DL^{2}L^{-2d}\int_{(u,v)\in\mathbb{R}_{+}^{2d}}\chi_{0}\big{(}\frac{u}{100\lambda
L}\big{)}\chi_{0}\big{(}\frac{v}{100\lambda L}\big{)}$
$\displaystyle\qquad\times\bigg{[}(\lambda
L)^{2-10\upsilon}L^{6\upsilon}(\lambda L)^{2d-4}+\sum_{|k|\geq
L^{5\upsilon}}\langle\delta^{-1}k\rangle^{-2}(\lambda
L)^{2d-2}\bigg{]}\mathrm{d}u\mathrm{d}v\lesssim D\lambda^{4d}L^{2d-\upsilon}.$

Thus, it remains to prove (6.18). We may assume without loss of generality
that $L^{-1-\upsilon}\lesssim sL^{-2}\leq L^{5\upsilon}$ (since
$G(-s,r;n)=\overline{G(s,-r;n)}$ and the estimates we shall use for $G$ are
independent of $r$). To bound the Gauss sums, we write

$sL^{-2}=n+\tau^{1},\qquad\beta^{2}sL^{-2}=m+\tau^{2};\qquad
n,m\in\mathbb{N}\cup\\{0\\},\tau^{1},\tau^{2}\in[0,1),$

and use Dirichlet’s approximation to find, for $j\in\\{1,2\\}$, integers
$0\leq a^{j}<q^{j}\leq u^{j}$ such that $(a^{j},q^{j})=1$ and

$\bigg{|}\tau_{j}-\frac{a^{j}}{q^{j}}\bigg{|}<\frac{1}{q^{j}u^{j}},\qquad
j=1,2.$

By periodicity of sum $G(s,r,x)$ in $s$ and the Gauss lemma for such sums we
have

$\left|G_{g^{1}}(sL^{-2},\xi^{1},u^{1})G_{g^{2}}(\beta^{2}sL^{-2},\xi^{2},u^{2})\right|\leq\frac{u^{1}u^{2}}{\sqrt{q^{1}q^{2}}(1+u^{1}|\tau^{1}-\frac{a^{1}}{q^{1}}|^{1/2})(1+u^{2}|\tau^{2}-\frac{a^{2}}{q^{2}}|^{1/2})}.$
(6.19)

We start by dealing with the case $n=0$ (i.e. when $sL^{-2}=\tau^{1}<1$). Note
that this implies $m\in\\{0,1\\}$. Here we will use the fact that
$L^{-1-\upsilon}\lesssim\tau^{1}<1$. First note that if $a^{1}=0$, then we
have

$\left|G_{g^{1}}(sL^{-2},\xi^{1},u^{1})G_{g^{2}}(\beta^{2}sL^{-2},\xi^{2},u^{2})\right|\leq\frac{u^{1}u^{2}}{1+u^{1}|\tau^{1}|^{1/2}}\lesssim\frac{u^{2}}{|\tau^{1}|^{1/2}}\lesssim\lambda
LL^{\frac{1+\upsilon}{2}}\ll\lambda L^{7/4},$

which satisfies (6.18). Therefore, we now consider the case $a^{1}\neq 0$.
Computing

$\displaystyle
q^{1}q^{2}\big{(}\beta^{2}\big{|}\tau^{1}-\frac{a^{1}}{q^{1}}\big{|}+\big{|}\tau^{2}-\frac{a^{2}}{q^{2}}\big{|}\big{)}=$
$\displaystyle
q^{1}q^{2}\big{(}\big{|}\beta^{2}sL^{-2}-\frac{\beta^{2}a^{1}}{q^{1}}\big{|}+\big{|}\beta^{2}sL^{-2}-m-\frac{a^{2}}{q^{2}}\big{|}\big{)}\geq
q^{1}q^{2}\big{|}\frac{\beta^{2}a^{1}}{q^{1}}-m-\frac{a^{2}}{q^{2}}\big{|}$
$\displaystyle=$
$\displaystyle|\beta^{2}a^{1}q^{2}-(mq^{1}q^{2}+a^{2}q^{1})|\gtrsim\frac{\log^{-4}(2q^{1}q^{2})}{a^{1}q^{2}+mq^{1}q^{2}+a^{2}q^{1}}\gtrsim\frac{1}{(q^{1}q^{2})^{1.01}},$

where we have used the diophantine condition (A.1) and the fact that
$0<a^{j}<q^{j}$ and $m\in\\{0,1\\}$. Therefore, we obtain

$\max\big{(}\big{|}\tau^{1}-\frac{a^{1}}{q^{1}}\big{|},\big{|}\tau^{2}-\frac{a^{2}}{q^{2}}\big{|}\big{)}\gtrsim\frac{1}{(q^{1}q^{2})^{2.01}},$

which when plugged into (6.19) gives the bound

$\lesssim\frac{u^{1}u^{2}}{\sqrt{q^{1}q^{2}}[1+\min(u^{1},u^{2})\frac{1}{(q^{1}q^{2})^{1.01}}]}\lesssim\min\bigg{(}\frac{u^{1}u^{2}}{\sqrt{q^{1}q^{2}}},\max(u^{1},u^{2})(q^{1}q^{2})^{0.51}\bigg{)}\lesssim(\lambda
L)^{\frac{7}{4}},$ (6.20)

since $q^{j}\leq u^{j}\lesssim\lambda L$, which is better than (6.18).

It remains to consider the case when $1\leq n\leq L^{5\upsilon}$. Here, we
argue similar to the above, to obtain

$\displaystyle
q^{1}q^{2}\big{(}\beta^{2}\big{|}\tau^{1}-\frac{a^{1}}{q^{1}}\big{|}+|\tau^{2}-\frac{a^{2}}{q^{2}}|\big{)}$
$\displaystyle=q^{1}q^{2}\big{(}\big{|}\beta^{2}sL^{-2}-\beta^{2}n-\beta^{2}\frac{a^{1}}{q^{1}}\big{|}+\big{|}\beta^{2}sL^{-2}-m-\frac{a^{2}}{q^{2}}\big{|}\big{)}$
$\displaystyle\geq
q^{1}q^{2}\big{|}\beta^{2}\frac{a^{1}}{q^{1}}+\beta^{2}n-m-\frac{a^{2}}{q^{2}}\big{|}=|\beta_{2}(nq^{1}q^{2}+a^{1}q^{2})-(mq^{1}q^{2}+a^{2}q^{1})|$
$\displaystyle\gtrsim\frac{\log^{-4}(2nq^{1}q^{2})}{nq^{1}q^{2}+a^{1}q^{2}+mq^{1}q^{2}+a^{2}q^{1}}\gtrsim\frac{1}{(nq^{1}q^{2})^{1.01}}.$

Since $n\leq L^{5\upsilon}$, we can repeat the same estimates as above and
obtain the needed bound (notice the room in (6.20) compared to needed bound in
(6.18)). ∎

######  Lemma 6.5.

Suppose that $\Phi(x,y)$ is a function satisfying (6.9) without $s$. Let
$\Omega(x,y)=\langle x,y\rangle_{\beta}$.

(1) Suppose $\psi$ is a function such that $\|\psi\|_{L^{1}(\mathbb{R})}\leq
D$, then

$\left|\mu^{-1}\int_{\mathbb{R}^{2d}}\psi(\mu\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(x,y)e(x\cdot\xi+y\cdot\eta)\,\mathrm{d}x\mathrm{d}y\right|\lesssim
D\lambda^{2d}$ (6.21)

uniformly in $(a,b,\xi,\eta)\in\mathbb{R}^{4d}$. The same holds if
$\psi(\mu\Omega)\Phi(x,y)$ is replaced by $\Psi(\mu\Omega,x,y)$ where
$\Psi=\Psi(u,x,y)$ satisfies
$\big{\|}\sup_{x,y}|\partial_{x}^{\alpha}\partial_{y}^{\beta}\Psi|\big{\|}_{L_{u}^{1}}\leq
D$ for all multi-indices $|\alpha|,|\beta|\leq 10d$.

(2) Suppose further that $\|\langle
y\rangle^{\frac{1}{8}}\psi\|_{L^{1}(\mathbb{R})}\leq D$, then

$\displaystyle\left|\mu^{-1}\int_{\mathbb{R}^{2d}}\psi(\mu\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(x,y)e(x\cdot\xi+y\cdot\eta)\,\mathrm{d}x\mathrm{d}y-\right.$
(6.22)
$\displaystyle\qquad\qquad\left.\left(\int\psi\right)\int_{\mathbb{R}^{2d}}\bm{\delta}(\Omega)\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{y-b}{\lambda}\big{)}\Phi(x,y)e(x\cdot\xi+y\cdot\eta)\,\mathrm{d}x\mathrm{d}y\right|\lesssim
D\lambda^{2d}\mu^{-\frac{1}{9}}(1+|\xi|+|\eta|),$

uniformly in $(a,b)\in\mathbb{R}^{2d}$.

###### Proof.

Using a smooth partition of unity, it is enough to consider the region when
$|x^{1}|\sim\max(|x|,|y|)$ (other regions are treated symmetrically). In this
case, we do the same change of variables as in the proof of Lemma 6.3,
replacing the variable $y^{1}$ by $u=\Omega$ to write the corresponding
integral in (6.21) as

$\displaystyle\int_{\mathbb{R}^{d}}|\beta^{1}x^{1}|^{-1}\chi_{0}\big{(}\frac{x-a}{\lambda}\big{)}\chi_{0}\big{(}\frac{x^{\prime}}{|x^{1}|}\big{)}e(x\cdot\xi)\int_{\mathbb{R}^{d-1}}\chi_{0}\big{(}\frac{y^{\prime}-b^{\prime}}{\lambda}\big{)}\chi_{0}\big{(}\frac{y^{\prime}}{|x^{1}|}\big{)}e(y^{\prime}\cdot\eta^{\prime})\,\mathrm{d}y^{\prime}\mathrm{d}x$
$\displaystyle\qquad\qquad\qquad\qquad\int_{\mathbb{R}}\mu^{-1}\psi(\mu
u)\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Phi}(x,u,y^{\prime})e\bigg{(}\frac{u\eta^{1}-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}\eta^{1}}{\beta^{1}x^{1}}\bigg{)}\mathrm{d}u,$

where
$\widetilde{\Phi}(x,u,y^{\prime})=\Phi(x,y^{1},y^{\prime})\chi_{0}(\frac{y}{|x^{1}|})$
evaluated at $y^{1}=\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}}{\beta^{1}x^{1}}$. From this one
can directly obtain (6.21), as well as the extension with $\psi\cdot\Phi$
replaced by $\Psi$. To obtain (6.22), we look at the difference

$\displaystyle\int_{\mathbb{R}}\mu^{-1}\psi(\mu
u)\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Phi}(x,u,y^{\prime})e\bigg{(}\frac{u\eta^{1}-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}\eta^{1}}{\beta^{1}x^{1}}\bigg{)}\mathrm{d}u$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\int_{\mathbb{R}}\mu^{-1}\psi(\mu
u)\chi_{0}\bigg{(}\frac{-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Phi}(x,0,y^{\prime})e\big{(}\frac{-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}\eta^{1}}{\beta^{1}x^{1}}\big{)}\mathrm{d}u,$

which can be written as $\int_{\mathbb{R}}\mu^{-1}\psi(\mu
u)\widetilde{\Delta}(u;x,y^{\prime})\mathrm{d}u$, where

$\displaystyle\widetilde{\Delta}(u;x,y^{\prime})$
$\displaystyle=\chi_{0}\bigg{(}\frac{u-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Phi}(x,u,y^{\prime})e\bigg{(}\frac{u\eta^{1}-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}\eta^{1}}{\beta^{1}x^{1}}\bigg{)}$
$\displaystyle\qquad\qquad\qquad\qquad-\chi_{0}\bigg{(}\frac{-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}-\beta^{1}x^{1}b^{1}}{\beta^{1}x^{1}\lambda}\bigg{)}\widetilde{\Phi}(x,0,y^{\prime})e\big{(}\frac{-\langle
x^{\prime},y^{\prime}\rangle_{\beta^{\prime}}\eta^{1}}{\beta^{1}x^{1}}\big{)}.$

It is easy to see that $|\widetilde{\Delta}|\lesssim
D\cdot\min\big{[}1,u\cdot\big{(}\frac{|\eta^{1}|}{\beta^{1}|x^{1}|}+\frac{1}{\beta^{1}|x^{1}|\min(1,|x^{1}|)}\big{)}\big{]}$.
Using this, we split the $u$ integral into two regions. If
$|u|\leq\mu^{-1/9}$, we can use the second of the two bounds on
$\widetilde{\Delta}$ to obtain a contribution
$D\mu^{-1/9}|\eta^{1}|\lambda^{2d}$ to (6.22); if $u\geq\mu^{-1/9}$, we can
use the first bound and the weighted norm $\|\langle
y\rangle^{1/8}\psi\|_{L^{1}}$ to obtain a contribution
$\mu^{-1/9}D\lambda^{2d}$ to (6.22). This finishes the proof. ∎

With the help of Lemmas 6.2–6.5, we can now prove Proposition 6.1.

###### Proof of Proposition 6.1.

We start with some simplifying notation and reductions. Set
$\bm{x}:=(x_{1},\ldots,x_{n})\in\mathbb{R}^{dn}$,
$\bm{y}:=(y_{1},\ldots,y_{n})\in\mathbb{R}^{dn}$,
$\bm{s}:=(s_{1},\ldots,s_{n})\in\mathbb{R}^{n}$,
$\bm{\Omega}=(\Omega_{1},\ldots,\Omega_{n})\in\mathbb{R}^{n}$ where
$\Omega_{j}=\langle x_{j},y_{j}\rangle_{\beta}$, and $\mu=L^{2}\delta$. We
will also use the notations like $\bm{x}_{\leq
j}:=(x_{1},\ldots,x_{j})\in\mathbb{R}^{dj}$ and similarly for the other
variables $\bm{y}$ and $\bm{\Omega}$.

Write $W(x,y)=W(\bm{x},\bm{y})\Upsilon(\bm{x},\bm{y})$ where $\Upsilon$ is a
smooth function supported in the set described in (6.3), namely

$\Upsilon(\bm{x},\bm{y})=\prod_{j=1}^{n}\chi_{0}\big{(}\frac{\widetilde{x_{j}}-a_{j}}{\lambda_{j}}\big{)}\chi_{0}\big{(}\frac{\widetilde{y_{j}}-b_{j}}{\lambda_{j}}\big{)}$
(6.23)

where $\widetilde{x_{j}}$ and $\widetilde{y_{j}}$ are as in (6.3), and we may
use a different $\chi_{0}$ as said in Section 2.3.1. Also note that we can
assume (by rearranging the indices) that if $j^{\prime}\prec j$ then
$j^{\prime}<j$.

Set $K:=(10d\max{\lambda_{j}})^{-1}L^{1-\upsilon}$, and write (with
$\widehat{\Psi}$ being the Fourier transform on $\mathbb{R}^{n}$)

$\displaystyle S$