Notation 2.1. Whenever we fix, for every $m\in \mathbb {N}$ and $X\subseteq \mathbb {R}^{m}$ subanalytic, a collection $\mathcal {K}\left (X\right )$ of real- or complex-valued functions defined on X, we denote by $\mathcal {K}$ the system of all collections $\mathcal {K}\left (X\right )$ . For instance, $\mathcal {S}$ is the system of collections of all subanalytic functions defined on subanalytic sets:

$$\begin{align*}\mathcal{S}=\left\{ \mathcal{S}\left(X\right):\ X\subseteq\mathbb{R}^{m}\text{ subanalytic},\ m\in\mathbb{N}\right\}. \end{align*}$$

Notation 2.5. The letter $\chi $ will be used for characteristic functions. Thus, if $A\subseteq \mathbb {R}^{n}$ , then $\chi _{A}$ will be the characteristic function of the set A.

We will often work in restriction to subanalytic cells $A\subseteq X\times \mathbb {R}$ , for some $X\subseteq \mathbb {R}^{m}$ subanalytic. If $x\in X$ , then $A_{x}$ denotes the fiber of A over x (i.e., the set $\left \{ y\in \mathbb {R}:\ \left (x,y\right )\in A\right \} $ ). As X serves as a space of parameters (we will never integrate with respect to the variables x ranging in X), for the questions on integration, we consider we are allowed to partition X into subanalytic cells, replace X by one of the cells of the partition and work disjointly in restriction to such a cell. In particular, we may always assume that X is itself a subanalytic cell and that all cells in $X\times \mathbb {R}$ project onto X. Moreover, we will always concentrate on cells A which are open over X (see [Reference Cluckers, Comte, Rolin and ServiCCRS23, Definition 3.1]), as these are the only cells whose fibers give a nonzero contribution when integrating a function defined on $X\times \mathbb {R}$ with respect to its last variable.

Notation 5.1. Let $B\subseteq X\times \mathbb {R}$ be as in (3.1) Consider a cell $D\subseteq B\times \mathbb {R}$ of the form

(5.1) $$ \begin{align} D=\left\{ \left(x,y,t\right):\ \left(x,y\right)\in B,\ \widetilde{a}\left(x,y\right)<t<\widetilde{b}\left(x,y\right)\right\} , \end{align} $$

where $\widetilde {a},\widetilde {b}:B\longrightarrow \mathbb {R}$ are analytic subanalytic functions with $1\leq \widetilde {a}\left (x,y\right )<\widetilde {b}\left (x,y\right )\ \text {for all }\left (x,y\right )\in B$ , or $\widetilde {b}$ is $\equiv +\infty $ . We say that D has bounded t-fibers if $\widetilde {b}<+\infty $ and unbounded t-fibers if $\widetilde {b}\equiv +\infty $ .

Suppose furthermore that $\widetilde {a},\widetilde {b},\widetilde {b}-\widetilde {a}$ have the following prepared form:

(5.2) $$ \begin{align} \begin{aligned} & \widetilde{a}\left(x,y\right)=a_{0}\left(x\right)y^{\frac{\alpha}{d}}u_{a}\left(x,y\right), & \widetilde{b}\left(x,y\right)=b_{0}\left(x\right)y^{\frac{\beta}{d}}u_{b}\left(x,y\right),\\ & \widetilde{b}\left(x,y\right)-\widetilde{a}\left(x,y\right)=d_{0}\left(x\right)y^{\frac{\Delta}{d}}u_{d}\left(x,y\right), \end{aligned} \end{align} $$

where $\alpha ,\beta ,\Delta \in \mathbb {N},\ a_{0},b_{0},d_{0}\in \mathcal {S}\left (X\right )$ are analytic and $u_{a},u_{b},u_{d}\in \mathcal {S}\left (B\right )$ are $\psi $ -prepared subanalytic strong units (for $\psi $ as in (3.2)). If $\widetilde {b}=+\infty $ , we stipulate that $b_{0}=d_{0}=+\infty ,\ u_{b}=u_{d}=1$ and $\beta =\Delta =0$ .

Define

(5.3) $$ \begin{align} \varPsi\left(x,y,t\right)=\left(\psi\left(x,y\right),\left(\frac{a_{0}\left(x\right)y^{\frac{\alpha}{d}}}{t}\right)^{\frac{1}{d}},\left(\frac{t}{b_{0}\left(x\right)y^{\frac{\beta}{d}}}\right)^{\frac{1}{d}}\right), \end{align} $$

where if D has unbounded t-fibers, we omit the last component. Note that $\varPsi $ is a $1$ -bounded subanalytic map on D.

Proof. Write h as a sum of generators of the form (3.8) and apply [Reference Cluckers, Comte, Rolin and ServiCCRS23, Proposition 4.7] simultaneously to all the parametric power-constructible data appearing in the generators: this produces a cell-decomposition of $X\times \mathbb {R}^{2}$ compatible with X, and on each cell D which is open over $X\times \mathbb {R}$ , a prepared form of the data with respect to the last variable t, where the coefficient functions are parametric power-constructible functions depending on the variables $\left (x,y\right )\in X\times \mathbb {R}$ . Now apply [Reference Cluckers, Comte, Rolin and ServiCCRS23, Proposition 4.7] again simultaneously to all the coefficient functions, in order to prepare them with respect to the variable y on suitable cells $B\subseteq X\times \mathbb {R}$ , thus refining the cell-decomposition. This gives the wanted result, up to trivial manipulations to adjust the definition of $\psi $ and $\varPsi $ (see [Reference Cluckers, Comte, Miller, Rolin and ServiCCMRS18, pp. 1268–70] for the details).

Proof. Let

$$\begin{align*}G\left(s,x,y,t\right)=g\left(s,x\right)y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}\chi_{D}\left(x,y,t\right)t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi\left(s,x,y,t\right)\in\mathcal{C}_{\Sigma}^{\mathcal{M}}\left(X\times\mathbb{R}^{2}\right), \end{align*}$$

so that

(5.4) $$ \begin{align} T\left(s,x,y\right)=\int_{\mathbb{R}}G\left(s,x,y,t\right)\text{e}^{\text{i}\left(\varphi\left(x,y\right)+\sigma t\right)}\text{d}t. \end{align} $$

Since $\Phi $ is bounded and extends continuously to $\overline {D}$ , for all $\left (s,x\right )\in \left (\Sigma \setminus P\right )\times X,$ the function ${y\longmapsto \int _{\mathbb {R}}\left |G\right |\text {d}t}$ extends continuously to the closed and bounded interval $\left [a\left (x\right ),b\left (x\right )\right ]$ and is hence integrable on this interval. By Tonelli’s Theorem, for all $\left (s,x\right )\in \left (\Sigma \setminus P\right )\times X,\ \left (y,t\right )\longmapsto \left |G\left (s,x,y,t\right )\right |\in L^{1}\left (\mathbb {R}^{2}\right )$ , so that $G\in \mathcal {C}_{\Sigma }^{\mathcal {M}}\left (X\times \mathbb {R}^{2}\right )_{\text {int}}$ and, by Remark 4.3, T is strongly integrable.

Proof. Apply Proposition 5.3 to h: the proof shows that this produces a cell-decomposition and, for each cell A, a presentation of $h\circ \Pi _{A}$ as a finite sum of prepared generators which are themselves naive in y. By Proposition 5.5, on cells with bounded y-fibers, the generators are strongly integrable. Hence, we may concentrate on a cell $B=B_{A}$ of the form (3.1) with unbounded y-fibers and on a generator T which is prepared on B and naive in y. Thus, T has the form

(5.5) $$ \begin{align} T\left(s,x,y\right)=f\left(s,x\right)y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}\text{e}^{\text{i}\varphi\left(x,y\right)}\Phi\left(s,x,y\right), \end{align} $$

where $f\in \mathcal {C}_{\Sigma }^{\mathcal {M},\mathcal {F}}\left (X\right ),\ \lambda \left (s\right ),\mu $ are as in Definition 5.2 and the prepared forms (with respect to $\psi $ as in (3.2)) of $\varphi \in \mathcal {S}\left (B\right )$ and $\Phi \in \mathcal {A}_{\Sigma }\left (B\right )$ are as in (3.6) and (3.5), respectively. Up to partitioning X into subanalytic cells, we may suppose that $\left |\varphi _{0}\right |$ is either bounded from above or bounded away from zero. If $\left |\varphi _{0}\right |$ is bounded and $\omega <0$ , then $\text {e}^{\text {i}\varphi }$ is a complex-valued $\psi $ -prepared subanalytic strong function. If $\omega \geq 0$ , then write

$$\begin{align*}\varphi\left(x,y\right)=Q\left(x,y\right)+\varphi_{>}\left(x,y\right), \end{align*}$$

where $Q\in \mathcal {S}\left (X\right )\left [y^{\frac {1}{d}}\right ]$ is a polynomial in the variable $y^{\frac {1}{d}}$ with coefficients subanalytic functions of x and

$$\begin{align*}\varphi_{>}\left(x,y\right)=\varphi_{0}\left(x\right)\left(a\left(x\right)\right)^{\frac{\omega}{d}}\left(\frac{a\left(x\right)}{y}\right)^{\frac{1}{d}}\sum_{k\geq0}b_{k+\omega+1}\left(x\right)\left(\frac{a\left(x\right)}{y}\right)^{\frac{k}{d}}. \end{align*}$$

If $\left |\varphi _{0}\right |$ is bounded from above, then clearly, $\text {e}^{\text {i}\varphi _{>}}$ is a complex-valued $\psi $ -prepared subanalytic strong function. If $\left |\varphi _{0}\right |$ is bounded away from zero, then we write

$$\begin{align*}\varphi_{>}\left(x,y\right)=\left(a\left(x\right)\right)^{\frac{\omega}{d}}\left(\frac{a\left(x\right)\varphi_{0}\left(x\right)^{d}}{y}\right)^{\frac{1}{d}}\sum_{k\geq0}b_{k+\omega+1}\left(x\right)\varphi_{0}\left(x\right)^{-k}\left(\frac{a\left(x\right)\varphi_{0}\left(x\right)^{d}}{y}\right)^{\frac{k}{d}}. \end{align*}$$

At the price of creating a new cell with bounded y-fibers (which can be dealt with by Proposition 5.5), we may suppose that $y>a\left (x\right )\varphi _{0}\left (x\right )^{d}$ on B, so that, up to modifying the definition of $\psi $ , it is clear that $\varphi _{>}$ is bounded on B, and hence, $\text {e}^{\text {i}\varphi _{>}}$ is a complex-valued $\psi $ -prepared subanalytic strong function (we deal in the same way with the case when $\left |\varphi _{0}\right |$ is bounded away from zero and $\omega <0$ ).

It follows that $\text {e}^{\text {i}\varphi _{>}}$ can be absorbed into $\Phi $ , and hence,

$$\begin{align*}T\left(s,x,y\right)=y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}\text{e}^{\text{i}Q\left(x,y\right)}\sum_{k}f_{k}\left(s,x\right)y^{-\frac{k}{d}}, \end{align*}$$

where $f_{k}\left (s,x\right )=f\left (s,x\right )\xi _{k}\left (s,x\right )a\left (x\right )^{\frac {k}{d}}\in \mathcal {C}_{\Sigma }^{\mathcal {M},\mathcal {F}}\left (X\right )$ , in the notation of (3.5). Since $\Sigma $ has bounded width, there exists $k_{0}\in \mathbb {N}$ such that setting $\lambda _{k}\left (s\right )=\lambda \left (s\right )-\frac {k}{d}$ , there exists $s\in \Sigma $ such that $\Re \left (\lambda _{k}\left (s\right )\right )\geq -1$ if and only if $k\leq k_{0}$ . It follows that

$$\begin{align*}T\left(s,x,y\right)=\sum_{k\leq k_{0}}f_{k}\left(s,x\right)y^{\lambda_{k}\left(s\right)}\left(\log y\right)^{\mu}\text{e}^{\text{i}Q\left(x,y\right)}+R\left(s,x,y\right), \end{align*}$$

where the terms of the sum are monomial in y and R is strongly integrable.

Proof. Suppose that T has no poles outside some closed discrete set $P\subseteq \mathbb {C}$ . If we write T as in (5.4), then it suffices to prove that $G\in \mathcal {C}_{\Sigma }^{\mathcal {M}}\left (X\times \mathbb {R}^{2}\right )_{\text {int}}$ . By Tonelli’s Theorem, it is enough to prove that for all $\left (s,x\right )\in \left (\Sigma \setminus P\right )\ni X$ , the function

$$\begin{align*}f_{\left(s,x\right)}:y\longmapsto\int_{\mathbb{R}}\left|G\left(s,x,y,t\right)\right|\text{d}t \end{align*}$$

is in $L^{1}\left (\mathbb {R}\right )$ .

1. (1) Let $\widetilde {b}=+\infty $ . Since $\forall s\in \Sigma ,\ \Re \left (\varrho \left (s\right )\right )<-1$ , there is a positive constant M such that

   $$\begin{align*}\int_{1}^{+\infty}\left|t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi\left(s,x,y,t\right)\right|\mathrm{d}t\leq M. \end{align*}$$

   Therefore,

   $$\begin{align*}\left|f_{\left(s,x\right)}\left(y\right)\right|\leq M\left|g\left(s,x\right)\right|\chi_{B}\left(x,y\right)y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}, \end{align*}$$
   and, as $\forall s\in \Sigma ,\ \Re \left (\lambda \left (s\right )\right )<-1$ , we have that $f_{\left (s,x\right )}\in L^{1}\left (\mathbb {R}\right )$ .

2. (2) Let now $\widetilde {b}<+\infty $ . Since $\forall s\in \Sigma ,\ \Re \left (\varrho \left (s\right )\right )\leq 0$ and $\Phi $ is bounded, there is a positive constant $M\left (x\right )$ such that

   $$\begin{align*}\int_{\widetilde{a}\left(x,y\right)}^{\widetilde{b}\left(x,y\right)}\left|t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi\left(s,x,y,t\right)\right|\mathrm{d}t\leq M\left(x\right)y^{\frac{\Delta}{d}}\left(\log y\right)^{\nu}. \end{align*}$$

   Therefore,

   $$\begin{align*}\left|f_{\left(s,x\right)}\left(y\right)\right|\leq M\left(x\right)\left|g\left(s,x\right)\right|\chi_{B}\left(x,y\right)y^{\lambda\left(s\right)+\frac{\Delta}{d}}\left(\log y\right)^{\mu+\nu}, \end{align*}$$
   and, as $\forall s\in \Sigma ,\ \Re \left (\lambda \left (s\right )\right )+\frac {\Delta }{d}<-1$ , we have that $f_{\left (s,x\right )}\in L^{1}\left (\mathbb {R}\right )$ .

Proof. Suppose that T has no poles outside some closed discrete set $P\subseteq \mathbb {C}$ . Write T on B as

$$\begin{align*}g\left(s,x\right)y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}\text{e}^{\text{i}\varphi\left(x,y\right)}\gamma\left(s,x,y\right), \end{align*}$$

where

$$\begin{align*}\gamma\left(s,x,y\right)=\int_{\widetilde{a}\left(x,y\right)}^{\widetilde{b}\left(x,y\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi\left(s,x,y,t\right)\text{e}^{\sigma\text{i}t}\text{d}t. \end{align*}$$

Since B has unbounded y-fibers, $\Phi $ has the following $\varPsi $ -prepared nested form (see [Reference Cluckers, Comte, Rolin and ServiCCRS23, Remark 3.7]) with respect to the last three components of $\varPsi $ :

$$\begin{align*}\Phi\left(s,x,y,t\right)=F\circ\left(s,x,\Xi\left(x,y,t\right)\right), \end{align*}$$

where

is strongly convergent and

$$\begin{align*}\Xi\left(x,y,t\right)=\left(\left(\frac{a\left(x\right)}{y}\right)^{\frac{1}{d}},\left(\frac{a_{0}\left(x\right)y^{\frac{\alpha}{d}}}{t}\right)^{\frac{1}{d}},\left(\frac{t}{b_{0}\left(x\right)y^{\frac{\beta}{d}}}\right)^{\frac{1}{d}}\right) \end{align*}$$

(the variable $Y_{2}$ and the last component of $\Xi $ are missing if D has unbounded t-fibers).

Fix $\left (s,x,y\right )\in \left (\Sigma \setminus P\right )\times B$ and apply integration by parts to the transcendental element $\gamma $ , where we integrate $\mathrm {e}^{\mathrm {i}\sigma t}$ and derivate $f\left (t\right ):=t^{\varrho \left (s\right )}\left (\log t\right )^{\nu }\Phi \left (s,x,y,t\right )$ . For this, write

$$\begin{align*}f'\left(t\right)=t^{-1}\left[\varrho\left(s\right)t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi+\nu t^{\varrho\left(s\right)}\left(\log t\right)^{\nu-1}\Phi+t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\widetilde{\Phi}\right], \end{align*}$$

where

$$\begin{align*}\widetilde{\Phi}\left(s,x,y,t\right)=\widetilde{F}\circ\left(s,x,\Xi\left(x,y,t\right)\right)\text{ with }\widetilde{F}=-\frac{1}{d}Y_{1}\frac{\partial F}{\partial Y_{1}}+\frac{1}{d}Y_{2}\frac{\partial F}{\partial Y_{2}}. \end{align*}$$

In particular, $\widetilde {F}$ is strongly convergent and $\widetilde {\Phi }$ is a $\varPsi $ -prepared parametric strong function. Notice that each of the terms of $f'\left (t\right )\frac {\mathrm {e}^{\mathrm {i}\sigma t}}{\mathrm {i}\sigma }$ gives rise to a prepared generator such that the exponent of t in the transcendental element is $\varrho \left (s\right )-1$ . The other terms produced by integration by parts are of the form

(5.6) $$ \begin{align} -\sigma\text{i}\widetilde{c}\left(x,y\right)^{\varrho\left(s\right)}\left(\log\widetilde{c}\left(x,y\right)\right)^{\nu}\mathrm{e}^{\sigma\mathrm{i}\widetilde{c}\left(x,y\right)}\Phi\left(s,x,y,\widetilde{c}\left(x,y\right)\right), \end{align} $$

where $\widetilde {c}$ is either $\widetilde {a}$ or $\widetilde {b}$ (or the whole term is replaced by zero, if $\widetilde {b}=+\infty $ , since then in this case, for all $s\in \Sigma ,\ \Re \left (\varrho \left (s\right )\right )<-1<0$ ). Now, if $\widetilde {c}=\widetilde {b}$ , then

$$\begin{align*}\Xi\left(x,y,\widetilde{b}\left(x,y\right)\right)=\left(\left(\frac{a\left(x\right)}{y}\right)^{\frac{1}{d}},\left(\frac{a_{0}\left(x\right)y^{\frac{\alpha}{d}}}{b_{0}\left(x\right)y^{\frac{\beta}{d}}u_{b}\left(x,y\right)}\right)^{\frac{1}{d}},\left(u_{b}\left(x,y\right)\right)^{\frac{1}{d}}\right) \end{align*}$$

and

$$\begin{align*}\frac{a_{0}\left(x\right)y^{\frac{\alpha}{d}}}{b_{0}\left(x\right)y^{\frac{\beta}{d}}u_{b}\left(x,y\right)}=\left[\frac{a_{0}\left(x\right)}{b_{0}\left(x\right)a\left(x\right)^{\frac{\beta-\alpha}{d}}}\right]\left(u_{b}\left(x,y\right)\right)^{-1}\left(\frac{a\left(x\right)}{y}\right)^{\frac{\beta-\alpha}{d}}. \end{align*}$$

The term between square brackets is bounded on B because all the other terms are, so we can add it to the list of functions $c\left (x\right )$ in the definition of $\psi $ (see 3.2). The unit $u_{b}$ is $\psi $ -prepared; hence, $\Phi \left (s,x,y,\widetilde {b}\left (x,y\right )\right )$ is a $\psi $ -prepared parametric strong function. A similar calculation shows that so is $\Phi \left (s,x,y,\widetilde {a}\left (x,y\right )\right )$ . Thus, the terms 5.6 are generators which are naive in y and prepared with respect to $\psi $ .

We iterate the process to further reduce the exponent of t: since $\Sigma $ has bounded width, ${\displaystyle \sup _{s\in \Sigma }}\left (\Re \left (\varrho \left (s\right )\right )\right )\in \mathbb {R}$ . Let $M:={\displaystyle \left \lfloor {\displaystyle \sup _{s\in \Sigma }}\left (\Re \left (\varrho \left (s\right )\right )\right )\right \rfloor }$ . By integrating by parts $M+k_{0}+1$ times, we can rewrite T as a finite sum of generators which are either naive in y and prepared with respect to $\psi $ , or such that the exponent of t in the transcendental element $\gamma $ is $\varrho \left (s\right )-M-k_{0}-1$ , whose real part is $<-k_{0}$ .

Proof. Recall Notation 5.1. Note that $0\leq \alpha \leq \beta $ . There are three cases:

1. (1) $\alpha =\beta =0$

2. (2) $\alpha>0$

3. (3) $\alpha =0,\beta>0$ .

Define $C:={\displaystyle \sup _{s\in \Sigma }}\left (\Re \left (\lambda \left (s\right )\right )\right )$ .

1. (1) The case $\beta =0$ also includes the case $\widetilde {b}\left (x,y\right )=b_{0}\left (x\right )=+\infty $ . We claim that, at the price of creating a new cell with bounded y-fibers (which can be handled using Proposition 5.5), we may suppose that for all $\left (x,y\right )\in B$ ,

   $\bullet $ $\widetilde {a}\left (x,y\right )\leq a_{0}\left (x\right )$ and $b_{0}\left (x\right )\leq \widetilde {b}\left (x,y\right )$ ;

   $\bullet $ $\left |\widetilde {a}\left (x,y\right )-a_{0}\left (x\right )\right |\leq 1$ and $\left |\widetilde {b}\left (x,y\right )-b_{0}\left (x\right )\right |\leq 1$ .

   The proof of the claim can be found in [Reference Cluckers, Comte, Miller, Rolin and ServiCCMRS18, p. 1277] and only uses basic o-minimal properties of subanalytic sets and functions.

   Therefore, we may write the transcendental element $\gamma $ as the sum of three integrals, with integration bounds, respectively, $\left (\widetilde {a}\left (x,y\right ),a_{0}\left (x\right )\right )$ , $\left (a_{0}\left (x\right ),b_{0}\left (x\right )\right )$ and $\left (b_{0}\left (x\right ),\widetilde {b}\left (x,y\right )\right )$ (if $\widetilde {b}=+\infty $ , then the second integral has $+\infty $ as upper integration bound and the third integral is missing). The integral with bounds $\left (\widetilde {a}\left (x,y\right ),a_{0}\left (x\right )\right )$ can be written as

   (5.7) $$ \begin{align} \mathrm{e}^{\mathrm{i}\sigma a_{0}\left(x\right)}\int_{a_{0}\left(x\right)}^{a_{0}\left(x\right)u_{a}\left(x,y\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi\left(s,x,y,t\right)\mathrm{e}^{\sigma\mathrm{i}\left(t-a_{0}\left(x\right)\right)}\text{d}t, \end{align} $$
   where, thanks to the claim, $\mathrm {e}^{\sigma \mathrm {i}\left (t-a_{0}\left (x\right )\right )}$ is a complex-valued subanalytic function on B. Hence, the integrand is in $\mathcal {C}_{\Sigma }^{\mathcal {M}}\left (B\right )$ , and we can invoke Theorem 2.18 to obtain that the term containing (5.7) can be written, outside some closed discrete set $P'$ containing P, as a generator of $\mathcal {C}_{\Sigma }^{\mathcal {M},\mathcal {F}}\left (X\times \mathbb {R}\right )$ which is naive in y. The integral with bounds $\left (b_{0}\left (x\right ),\widetilde {b}\left (x,y\right )\right )$ , if present, is handled similarly. For the integral with bounds $\left (a_{0}\left (x\right ),b_{0}\left (x\right )\right )$ , notice that now the variable y only appears in the parametric strong function $\Phi $ , which we can write in nested form with respect to the last component of $\psi $ as
   (5.8) $$ \begin{align} \sum_{k\geq0}\widetilde{\xi}_{k}\left(s,x,t\right)\left(\frac{a\left(x\right)}{y}\right)^{\frac{k}{d}}, \end{align} $$
   for some $\widetilde {\xi }_{k}\in \mathcal {A}_{\Sigma }\left (E\right )$ , where $E=\left \{ \left (x,t\right ):\ x\in X,\ a_{0}\left (x\right )<t<b_{0}\left (x\right )\right \} $ . Let $k_{0}=\lceil d\left (C+1\right )\rceil +1$ , so that for all $s\in \Sigma ,\ \Re \left (\lambda \left (s\right )\right )-\frac {k_{0}}{d}<-1$ , and let
   $$\begin{align*}\Phi_{>k_{0}}=\Phi-\sum_{k\leq k_{0}}\widetilde{\xi}_{k}\left(s,x,t\right)\left(\frac{a\left(x\right)}{y}\right)^{\frac{k}{d}}. \end{align*}$$

   Setting

   $$\begin{align*}f_{k}\left(s,x\right)=g\left(s,x\right)\left(a\left(x\right)\right)^{\frac{k}{d}}\int_{a_{0}\left(x\right)}^{b_{0}\left(x\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\widetilde{\xi}_{k}\left(s,x,t\right)\mathrm{e}^{\sigma\mathrm{i}t}\text{d}t\in\mathcal{C}_{\Sigma}^{\mathcal{M},\mathcal{F}}\left(X\right), \end{align*}$$
   write the term
   $$\begin{align*}g\left(s,x\right)y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}\mathrm{e}^{\mathrm{i}\varphi\left(x,y\right)}\int_{a_{0}\left(x\right)}^{b_{0}\left(x\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\mathrm{e}^{\sigma\mathrm{i}t}\Phi\left(s,x,y,t\right)\text{d}t \end{align*}$$
   as the following sum of generators which are naive in y,
   $$\begin{align*}\sum_{k\leq k_{0}}f_{k}\left(s,x\right)y^{\lambda\left(s\right)-\frac{k}{d}}\left(\log y\right)^{\mu}\mathrm{e}^{\mathrm{i}\varphi\left(x,y\right)}, \end{align*}$$
   plus the term
   (5.9) $$ \begin{align} g\left(s,x\right)\left(a\left(x\right)\right)^{\frac{k_{0}}{d}}y^{\lambda\left(s\right)-\frac{k_{0}}{d}}\left(\log y\right)^{\mu}\mathrm{e}^{\mathrm{i}\varphi\left(x,y\right)}\int_{a_{0}\left(x\right)}^{b_{0}\left(x\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi_{>k_{0}}\left(s,x,y,t\right)\mathrm{e}^{\sigma\mathrm{i}t}\text{d}t, \end{align} $$
   which is strongly integrable on B by definition of $k_{0}$ and since $\Phi _{>k_{0}}$ is bounded.

2. (2) Let $N_{0}=\left \lceil \frac {dC+\Delta +d}{\alpha }\right \rceil +1$ . By Lemma 5.9, we may suppose that for all $s\in \Sigma ,\ \Re \left (\varrho \left (s\right )\right )<-N_{0}$ . Write

   $$\begin{align*}t^{\varrho\left(s\right)}=t^{\varrho\left(s\right)+N_{0}}\left(a_{0}\left(x\right)y^{\frac{\alpha}{d}}\right)^{-N_{0}}\left[\left(\frac{a_{0}\left(x\right)y^{\frac{\alpha}{d}}}{t}\right)^{\frac{1}{d}}\right]^{dN_{0}}. \end{align*}$$

   The rightmost term in the above formula can be absorbed into $\Phi $ , and the central term can be factored out of the integral defining the transcendental element $\gamma $ . By the choice of $N_{0}$ , for all $s\in \Sigma ,$

   (5.10) $$ \begin{align} \Re\left(\lambda\left(s\right)\right)-\frac{N_{0}\alpha}{d}+\frac{\Delta}{d}<-1, \end{align} $$
   so by Lemma 5.8 (either of the two conditions, depending on the nature of the t-fibers of the cell D), T is strongly integrable on B.

3. (3) Let $N_{0}=\left \lceil \frac {dC+\Delta +d}{\beta }\right \rceil +1$ and $k_{0}=d\left (N_{0}-1\right )$ . By Lemma 5.9, we may suppose that for all $s\in \Sigma ,\ \Re \left (\varrho \left (s\right )\right )<-N_{0}$ . This implies, in particular, that for all $s\in \Sigma $ ,

   (5.11) $$ \begin{align} & \Re\left(\varrho\left(s\right)\right)+\frac{k_{0}}{d}<-1,\qquad\ \ \kern0.1pt\qquad \end{align} $$
   (5.12) $$ \begin{align} & \Re\left(\varrho\left(s\right)\right)+\frac{k_{0}+1}{d}\leq0,\qquad\ \ \kern2pt\quad \end{align} $$
   (5.13) $$ \begin{align} & \Re\left(\lambda\left(s\right)\right)-\frac{\beta\left(k_{0}+1\right)}{d^{2}}+\frac{\Delta}{d}<-1. \end{align} $$

   First, we split $\Phi \left (s,x,y,t\right )$ into the sum of two series by separating the positive and the negative powers of t:

   $$\begin{align*}\Phi=\sum_{k>0}\xi_{k}^{<}\left(s,x,y\right)\left(\frac{a_{0}\left(x\right)}{t}\right)^{\frac{k}{d}}+\sum_{k\geq0}\xi_{k}^{>}\left(s,x,y\right)\left(\frac{t}{b_{0}\left(x\right)y^{\frac{\beta}{d}}}\right)^{\frac{k}{d}}. \end{align*}$$

   Next, write

   $$ \begin{align*} \gamma_{\leq k_{0}}\left(s,x,y\right) & =\int_{a_{0}\left(x\right)u_{a}\left(x,y\right)}^{b_{0}\left(x\right)y^{\frac{\beta}{d}}u_{b}\left(x,y\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\Phi_{\leq k_{0}}\left(s,x,y,t\right)\mathrm{e}^{\sigma\mathrm{i}t}\text{d}t,\\[6pt] \gamma_{>k_{0}}\left(s,x,y\right) & =\int_{a_{0}\left(x\right)u_{a}\left(x,y\right)}^{b_{0}\left(x\right)y^{\frac{\beta}{d}}u_{b}\left(x,y\right)}t^{\varrho\left(s\right)}\left(\log t\right)^{\nu}\left(\frac{t}{b_{0}\left(x\right)y^{\frac{\beta}{d}}}\right)^{\frac{k_{0}+1}{d}}\Phi_{>k_{0}}\left(s,x,y,t\right)\mathrm{e}^{\sigma\mathrm{i}t}\text{d}t, \end{align*} $$
   where
   $$ \begin{align*} \Phi_{\leq k_{0}}= & \sum_{k>0}\xi_{k}^{<}\left(s,x,y\right)\left(\frac{a_{0}\left(x\right)}{t}\right)^{\frac{k}{d}}+\sum_{k=0}^{k_{0}}\xi_{k}^{>}\left(s,x,y\right)\left(\frac{t}{b_{0}\left(x\right)y^{\frac{\beta}{d}}}\right)^{\frac{k}{d}},\\[6pt] \Phi_{>k_{0}}= & \sum_{k\geq0}\xi_{k+k_{0}+1}^{>}\left(s,x,y\right)\left(\frac{t}{b_{0}\left(x\right)y^{\frac{\beta}{d}}}\right)^{\frac{k}{d}}. \end{align*} $$

   By (5.11) and linearity, we may write $\gamma _{\leq k_{0}}$ as the sum of two integrals with upper integration bound equal to $+\infty $ and the lower integration bounds equal to, respectively, $a_{0}\left (x\right )u_{a}\left (x,y\right )$ and $b_{0}\left (x\right )y^{\frac {\beta }{d}}u_{b}\left (x,y\right )$ . The first integral falls within the scope of the first part of this proof, whereas the second integral falls within the scope of the second part of this proof.

   It remains to consider

   (5.14) $$ \begin{align} & T_{>k_{0}}\left(s,x,y\right):=g\left(s,x\right)y^{\lambda\left(s\right)}\left(\log y\right)^{\mu}\mathrm{e}^{\mathrm{i}\varphi\left(x,y\right)} \gamma_{>k_{0}}\left(s,x,y\right)\\ & =g\left(s,x\right)y^{\lambda\left(s\right)-\frac{\beta\left(k_{0}+1\right)}{d^{2}}}\left(\log y\right)^{\mu}\mathrm{e}^{\mathrm{i}\varphi\left(x,y\right)}\left(b_{0}\left(x\right)\right)^{-\frac{\beta\left(k_{0}+1\right)}{d^{2}}}\cdot\nonumber\\ & \cdot\int_{a_{0}\left(x\right)u_{a}\left(x,y\right)}^{b_{0} \left(x\right)y^{\frac{\beta}{d}}u_{b}\left(x,y\right)}t^{\varrho\left(s\right)+\frac{k_{0}+1}{d}}\left(\log t\right)^{\nu}\Phi_{>k_{0}}\left(s,x,y,t\right)\mathrm{e}^{\sigma\mathrm{i}t}\text{d}t.\nonumber \end{align} $$

   By (5.12) and (5.13), $T_{\geq k_{0}}$ satisfies the second condition in Lemma 5.8.

Notation 6.3. Given a subanalytic set $X\subseteq \mathbb {R}^{m}$ and functions $Q_{1},Q_{2} \in \mathcal {S}\left (X\right )\left [y^{\frac {1}{d}}\right ]$ which are polynomials in $y^{\frac {1}{d}}$ with coefficients subanalytic functions of x, it is clearly possible to partition X into finitely many subanalytic cells such that for each cell $X'$ , either for all $x\in X',\ Q_{1}\left (x,\cdot \right )$ and $Q_{2}\left (x,\cdot \right )$ , define the same polynomial function, or for all $x\in X',\ Q_{1}\left (x,\cdot \right )$ and $Q_{2}\left (x,\cdot \right )$ , define different polynomial functions. In this case, we will say, respectively, that $Q_{1}=Q_{2}$ on $X'$ or $Q_{1}\not =Q_{2}$ on $X'$ , as functions.

Proof. Apply Proposition 5.3 to h. This produces d and a d-cell-decomposition of $\mathbb {R}^{m+1}$ such that on each cell A open over X, $h\circ \Pi _{A}$ is a finite sum of prepared generators T. Collect the grid data of all the prepared generators and generate the corresponding grid $\mathcal {G}$ with denominator $d^{2}$ . For each cell A, apply Propositions 5.10 and 5.7 to each prepared generator T on $B_{A}$ . This produces a refinement of the d-cell-decomposition and rewrites T on each cell as a finite sum of prepared generators $T'$ which are either strongly integrable (and satisfying condition (1)) or monomial in y. Up to absorbing $\text {e}^{\text {i}Q_{j}\left (x,0\right )}$ into $f_{j}\left (s,x\right )$ and up to partitioning X into subanalytic cells, we may suppose that item (2.a) in the statement of the proposition is satisfied. Summing like terms, we may also suppose that item (2.b) is satisfied. Revisiting the proofs of Propositions 5.10 and 5.7, which are based on integration by parts of the transcendental elements and series expansion of parametric strong functions on cells with unbounded y-fibers, we see that if the exponents of y and t in the original prepared generator T are $\lambda \left (s\right )$ and $\varrho \left (s\right )$ , respectively, then the exponents of y in the newly created monomial generators $T'$ have the form $\lambda \left (s\right )-\frac {k}{d}+\frac {\delta }{d}\left (\varrho \left (s\right )-k'\right )$ , for some $k,k'\in \mathbb {N}$ and $\delta \in \left \{ 0,\alpha ,\beta \right \} $ . In particular, the grid generated by the grid data of the new monomial generators does not create any new cell. By Remark 5.4, we may rename $d^{2}$ as d and adapt accordingly the definitions of $\ell _{j},\eta _{j},\alpha _{j},\beta _{j}$ , so that, by Remark 6.2 (2), item (2.c) in the statement of the proposition is also satisfied.

Proof. Apply Proposition 6.4 to h: this produces a closed discrete set $P'\subseteq \mathbb {C}$ containing P and such that $P'\setminus P$ is contained in a finitely generated $\mathbb {Z}$ -lattice, $d\in \mathbb {N}\setminus \left \{ 0\right \} $ , finite sets $J_{\mathrm {int}},J_{\mathrm {mon}}\subseteq \mathbb {N}$ , a grid $\mathcal {G}$ and a d-cell-decomposition, such that the conclusion of the proposition holds. By linearity of the integral, it suffices to prove the statement of the theorem for $h\restriction A$ , where A is a cell of the decomposition which is open over X. Recall Notation 3.5 and note that

$$\begin{align*}\frac{\partial\Pi_{A}}{\partial y}\left(x,y\right)=\sigma_{A}\tau_{A}y^{\tau_{A}-1}. \end{align*}$$

Thus, up to multiplying each $T_{j}$ in (6.1) by $\frac {\partial \Pi _{A}}{\partial y}\left (x,y\right )$ , we may write that for all $\left (s,x\right )\in \text {Int}\left (h\restriction A;\left (\Sigma \setminus P'\right )\times X\right ),$

$$\begin{align*}\int_{A_{x}}h\left(s,x,y\right)\text{d}y=\int_{a_{A}\left(x\right)}^{b_{A}\left(x\right)}\left(\sum_{j\in J_{\mathrm{int}}}T_{j}\left(s,x,y\right)+\sum_{j\in J_{\mathrm{mon}}}T_{j}\left(s,x,y\right)\right)\text{d}y. \end{align*}$$

If $B_{A}$ has bounded y-fibers, then we are done by Proposition 5.5 and Corollary 4.6.

If $B_{A}$ has unbounded y-fibers, then for all $j\in J_{\text {mon}},\ T_{j}$ has the form $f_{j}\left (s,x\right )y^{\lambda _{j}\left (s\right )}\left (\log y\right )^{\mu _{j}}\text {e}^{\text {i}Q_{j}\left (x,y\right )}$ , with $\lambda _{j}\left (s\right )=\frac {\ell _{j}s+\eta _{j}}{d}$ , and for all $\mathcal {G}$ -cell S, there is a set $J_{S}\subseteq J_{\text {mon}}$ such that for all $j\in J_{S},\ \text {Int}\left (T_{j}\chi _{B_{A}};\left (S\setminus P'\right )\times X\right )=\left \{ \left (s,x\right )\in \left (S\setminus P'\right )\times X:\ f_{j}\left (s,x\right )=0\right \} $ , whereas for all $j\in J_{\text {mon}}\setminus J_{S},\ \text {Int}\left (T_{j}\chi _{B_{A}};\left (S\setminus P'\right )\times X\right )=\left (S\setminus P'\right )\times X$ . Thus, the set

$$\begin{align*}E:=\bigcap_{j\in J_{\text{int}}\cup J_{\text{mon}}}\text{Int}\left(T_{j}\chi_{B_{A}};\left(\Sigma\setminus P'\right)\times X\right) \end{align*}$$

is of the form of the right-hand side of (6.2) and, applying either Corollary 4.6 or Proposition 4.8 to $T_{j}\chi _{B_{A}}$ and possibly enlarging $P'$ , we find $H\in \mathcal {C}_{\Sigma }^{\mathcal {M},\mathcal {F}}\left (X\right )$ without poles outside $P'$ which interpolates the integral of $h\restriction A$ for all $\left (s,x\right )\in E$ .

Note that $E\subseteq \mathrm {Int}\left (h\restriction A;\left (\Sigma \setminus P'\right )\times X\right )$ . It remains to show that, up to possibly enlarging $P'$ , the set E coincides with $\mathrm {Int}\left (h\restriction A;\left (\Sigma \setminus P'\right )\times X\right )$ . Let

$$\begin{align*}P_{A}=P'\cup\left\{ s\in\mathbb{C}:\ \exists i,j\in J_{\text{mon}}\text{ such that}\ i\not=j,\ \lambda_{i}\left(s\right)=\lambda_{j}\left(s\right),\ \mu_{i}=\mu_{j},\ Q_{i}=Q_{j}\right\}. \end{align*}$$

By item (2.b) in Proposition 6.4, if $s\in \Sigma $ is such that $\lambda _{i}\left (s\right )=\lambda _{j}\left (s\right )$ for some $i\not =j$ such that $\mu _{i}=\mu _{j},\ Q_{i}=Q_{j}$ , then necessarily, $\ell _{i}\not =\ell _{j}$ , so $P_{A}\setminus P'$ is finite.

By definition of $P_{A}$ , if $s\in \Sigma \setminus P_{A}$ and $i,j\in J_{\text {mon}}$ are such that $i\not =j,\ \mu _{i}=\mu _{j}$ and $\Re \left (\lambda _{i}\left (s\right )\right )=\Re \left (\lambda _{j}\left (s\right )\right )$ , then $Q_{i}=Q_{j}\Longrightarrow \Im \left (\lambda _{i}\left (s\right )\right )\not =\Im \left (\lambda _{j}\left (s\right )\right )$ .

Let $\left (s_{0},x_{0}\right )\in \mathrm {Int}\left (h\restriction A;\left (\Sigma \setminus P_{A}\right )\times X\right )$ and let S be the $\mathcal {G}$ -cell to which $s_{0}$ belongs. Define

$$\begin{align*}\rho_{j}=\Re\left(\lambda_{j}\left(s_{0}\right)\right),\ \sigma_{j}=\Im\left(\lambda_{j}\left(s_{0}\right)\right),\ p_{j}\left(y\right)=Q_{j}\left(x_{0},y\right)\in\mathbb{R}\left[y^{\frac{1}{d}}\right]. \end{align*}$$

Let $\left (r_{0},\nu _{0}\right )$ be the lexicographic maximum of the set $\left \{ \left (\rho _{j},\mu _{j}\right ):\ j\in J_{S}\right \} $ and let $J_{0}=\{j\in J_{S}:\ \left (\rho _{j},\mu _{j}\right )=\left (r_{0},\nu _{0}\right )\}$ . Then,

$$\begin{align*}\sum_{j\in J_{S}}T_{j}\left(s_{0},x_{0},y\right)=y^{r_{0}}\left(\log y\right)^{\nu_{0}}\sum_{j\in J_{0}}f_{j}\left(s_{0},x_{0}\right)y^{\text{i}\sigma_{j}}\mathrm{e}^{\mathrm{i}p_{j}\left(y\right)}+\sum_{j\in J_{S}\setminus J_{0}}f_{j}\left(s_{0},x_{0}\right)y^{\rho_{j}+\mathrm{i}\sigma_{j}}\left(\log y\right)^{\mu_{j}}\mathrm{e}^{\mathrm{i}p_{j}\left(y\right)}. \end{align*}$$

Since $\left (s_{0},x_{0}\right )\in \mathrm {Int}\left (h\restriction A;\left (\Sigma \setminus P_{A}\right )\times X\right )$ , it follows from [Reference Cluckers, Comte, Rolin and ServiCCRS23, Proposition 3.4 (1)] that $\bigwedge _{j\in J_{0}}f_{j}\left (s_{0},x_{0}\right )=0$ . By repeating this procedure with the index set $J_{S}\setminus J_{0}$ , we end up proving that $\left (s_{0},x_{0}\right )\in E$ .