2.1 Preliminaries

We denote by

$$ \begin{align*}\overline{\mathbb{C}} = \mathbb{C} \cup \{-\infty\}\end{align*} $$

the logarithmic chart of the Riemann surface of the logarithm, with the additional “origin” of $\overline {\mathbb {C}}$ represented by “ $-\infty $ ,” where we convene that $\operatorname {\mathrm {Re}}(-\infty ) = -\infty $ . For ${\frak r} \in \mathbb {R}$ , we let

$$ \begin{align*} H({\frak r}):= \left\{u+iv \in \overline{\mathbb{C}}:\ u < {\frak r}\right\} \end{align*} $$

be the log-disk of log-radius ${\frak r}$ . For $d,{\frak r} \in \mathbb {R}$ , a log-sector is a set

$$ \begin{align*} S(d,{\frak r},\theta):= \begin{cases} \left\{u+iv \in \mathbb{C}:\ u < {\frak r}, |d-v| < \theta\right\} \cup \{-\infty\}, &\text{if } \theta \in (0,\infty), \\ H({\frak r}), &\text{if } \theta = \infty, \end{cases} \end{align*} $$

and a log-line is a set

$$ \begin{align*}T(d):= \left\{u+iv \in \mathbb{C}:\ v = d\right\} \cup \{-\infty\}.\end{align*} $$

(We shall mainly focus on the direction $d=0$ in this paper.) We extend the standard topology on $\mathbb {C}$ to $\overline {\mathbb {C}}$ by declaring the log-disks as basic open neighborhoods of $-\infty $ . Note that the usual covering map of the Riemann surface of the logarithm is represented in the logarithmic chart by the exponential function, and we extend it to a continuous function on $\overline {\mathbb {C}}$ by setting $e^{-\infty }:= 0$ . For each $d \in \mathbb {R}$ , the restriction of $e^w$ to $S(d,\infty ,\pi ) \setminus \{-\infty \}$ is injective; its inverse is the branch of the logarithm $\log _d$ in the direction d.

We are mostly interested in partial functions on $\overline {\mathbb {C}}$ with values in $\mathbb {C}$ . In this spirit, we call a set $D \subseteq \overline {\mathbb {C}}$ a log-domain if $D \cap \mathbb {C}$ is a domain (in particular, every domain in $\mathbb {C}$ is a log-domain). If $D \subseteq \overline {\mathbb {C}}$ is a log-domain, a log-holomorphic function on D is a continuous function $f:D \longrightarrow \mathbb {C}$ such that the restriction of f to $D \cap \mathbb {C}$ is holomorphic. For example, every holomorphic function on a domain in $\mathbb {C}$ is log-holomorphic, and the exponential function is log-holomorphic on $\overline {\mathbb {C}}$ .

2.2 The logarithmic Borel and Laplace transforms

2.2.1 Logarithmic Borel transform

Let $d,{\frak r} \in \mathbb {R}$ and $\theta> \pi /2$ , and write $S = S(d,{\frak r},\theta )$ . Let $f:S \longrightarrow \mathbb {C}$ be such that $f\!\!\upharpoonright _{\overline S_0}$ is bounded and log-holomorphic, for every closed log-subsector $\overline S_0$ of S. Given a closed log-subsector $\overline S_0 = \operatorname {\mathrm {cl}}(S(d', {\frak r}',\theta '))$ of S with $\theta '>\frac \pi 2$ , denote by $\partial \overline S_0$ the directed path following the boundary of $\overline S_0$ from the “lower left end” to the “upper left end.” We define the logarithmic Borel transform $\mathcal {B}_{d'} f: T(d') \longrightarrow \mathbb {C}$ in the direction $d'$ of f by

$$ \begin{align*} \mathcal{B}_{d'} f(w) := \frac{e^w}{2\pi i} \int_{\partial \overline S_0} e^{e^{w-\eta}} f(\eta) \frac{d\eta}{e^{\eta}}. \end{align*} $$

We leave it as an exercise to check that $\mathcal {B}_{d'} f$ only depends on $d'$ , but not on the other parameters of $\overline S_0$ (as long as they are in the prescribed range). More is true.

Remark 2.1 If $\theta ' < \pi $ and $g(z):= f(\log _d z)$ , for $z \in \exp (\overline S_0)$ , then the change of variables $z = e^w$ gives that

$$ \begin{align*} \left(\mathcal{B}_{d'} f\right)(\log_d z) = z \cdot \left(\mathcal{B}_{d'}g\right) (z), \end{align*} $$

where $\mathcal {B}_{d'} g$ denotes the Borel transform of g in the direction $d'$ as defined in [Reference Loray5] (see also Section 5.2 of [Reference Balser2]). Thus, the following proposition is obtained from Propriétés 1–3 on page 38 of [Reference Loray5].

Proposition 2.2 Set $S':= S\left (d,\infty ,\theta -\frac \pi 2\right )$ .

1. (1) The function $\mathcal {B} f: S' \longrightarrow \mathbb {C}$ defined by $\mathcal {B} f(w):= (\mathcal {B}_{\operatorname {\mathrm {Im}} w} f)(w)$ is log-holomorphic on every closed log-subsector $\overline S_0$ of $S'$ .

2. (2) For every closed log-subsector $\overline S_0$ of $S'$ , there exist $C,D>0$ such that

   $$ \begin{align*}|\mathcal{B} f(w)| \le C e^{De^{\operatorname{\mathrm{Re}} w}} \quad\text{for } w \in \overline S_0.\end{align*} $$

3. (3) Let $\alpha \ge 0$ , and assume that for every closed log-subsector $\overline S_0$ of S, we have $|f(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\overline S_0$ . Then, for every closed log-subsector $\overline S_0$ of $S'$ , we have $|\mathcal {B} f(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\overline S_0$ .

Accordingly, we call the function $\mathcal {B} f$ defined in the proposition above the log-Borel transform of f.

For $D \subseteq \overline {\mathbb {C}}$ and $g:D \longrightarrow \mathbb {C}$ , we set

$$ \begin{align*}\|g\|_D := \sup\left\{|g(z)|:\ z \in D\right\}.\end{align*} $$

For later use, we make the bound in Proposition 2.2(2) more precise.

Lemma 2.3 Let $\overline S_0 = \operatorname {\mathrm {cl}}(S(d', {\frak r}',\theta '))$ be a closed subsector of S with $\theta ' \in \left (\frac \pi 2,\theta \right )$ , and set $S':= S\left (d',r,\theta ' - \frac \pi 2\right )$ and $C:= \sin \left (\frac {\theta -\theta '}2\right )$ . Then

$$ \begin{align*}\left\|\mathcal{B} f\right\|_{S'} \le \begin{cases} \frac{\left\| f\right\|_{\overline S_0}}{C} e, &\text{if } r \le {\frak r}', \\ \frac{\left\| f\right\|_{\overline S_0}}{C} e^{e^{r-{\frak r}'}} e^{r - {\frak r}'}, &\text{if } r \ge {\frak r}'. \end{cases}\end{align*} $$

Proof Let $w \in S'$ ; we compute $\mathcal {B} f(w)$ by computing $\mathcal {B}_d f(w)$ , where $d:= \operatorname {\mathrm {Im}} w$ and the integral is taken along the contour $\delta := \partial S\left (d,\rho ,\alpha \right )$ , where $\alpha := \frac {\theta -\theta '+\pi }2$ and $\rho := \min \{\operatorname {\mathrm {Re}} w, {\frak r}'\}$ . For $\eta \in \delta $ , we distinguish two cases.

Case 1: $|\operatorname {\mathrm {Im}}(w-\eta )| = \alpha $ . Then $\operatorname {\mathrm {Re}}(e^{w-\eta }) = \cos \alpha \cdot e^{\operatorname {\mathrm {Re}} w-\operatorname {\mathrm {Re}}\eta }$ ; since $C = -\cos \alpha $ , we get

$$ \begin{align*} \frac1{2\pi} \left|e^w\int_{|\operatorname{\mathrm{Im}}(w-\eta)| = \pi} e^{e^{w-\eta}} f(\eta) \frac{d\eta}{e^{\eta}}\right| &\le \frac{\left\|f\right\|_{\overline S_0}}{2\pi} \int_{|\operatorname{\mathrm{Im}}(w-\eta)| = \pi} e^{\cos\alpha \cdot e^{\operatorname{\mathrm{Re}} w-\operatorname{\mathrm{Re}}\eta}} e^{\operatorname{\mathrm{Re}} w -\operatorname{\mathrm{Re}}\eta} d\eta \\ &= \frac{\left\|f\right\|_{\overline S_0}}{\pi} \int_{-\infty}^{\rho} e^{\cos\alpha \cdot e^{\operatorname{\mathrm{Re}} w -r}} e^{\operatorname{\mathrm{Re}} w-r} dr \\ &= \frac{\left\|f\right\|_{\overline S_0}}{\pi C} e^{\cos\alpha\cdot e^{\operatorname{\mathrm{Re}} w-r}}\Big|_{-\infty}^{\rho} \\ &\le \frac{\left\|f\right\|_{\overline S_0}}{C}, \end{align*} $$

because $\cos \alpha < 0$ .

Case 2: $\operatorname {\mathrm {Re}}\eta = \rho $ . Then we have

$$ \begin{align*}\operatorname{\mathrm{Re}}(e^{w-\eta}) \le |e^{w-\eta}| = e^{\operatorname{\mathrm{Re}} w-\operatorname{\mathrm{Re}}\eta} = e^{\operatorname{\mathrm{Re}} w-\rho},\end{align*} $$

so that

$$ \begin{align*} \frac1{2\pi} \left|e^w\int_{\operatorname{\mathrm{Re}}\eta = \rho} e^{e^{w-\eta}} f(\eta) \frac{d\eta}{e^{\eta}}\right| \le \left\|f\right\|_{\overline S_0} e^{e^{\operatorname{\mathrm{Re}} w-\rho}} e^{\operatorname{\mathrm{Re}} w - \rho} \le \begin{cases} \|f\|_{\overline S_0} e, &\text{if } \operatorname{\mathrm{Re}} w \le {\frak r}', \\ \|f\|_{\overline S_0} e^{e^{\operatorname{\mathrm{Re}} w-{\frak r}'}} e^{\operatorname{\mathrm{Re}} w - {\frak r}'}, &\text{if } \operatorname{\mathrm{Re}} w \ge {\frak r}'. \end{cases} \end{align*} $$

Combining the two cases, we obtain the lemma.

2.2.2 Logarithmic Laplace transform

We fix an arbitrary direction $d \in \mathbb {R}$ . Let $f:T(d) \longrightarrow \mathbb {C}$ be continuous, and assume that there exist $C,D>0$ such that

$$ \begin{align*}|f(w)| \le Ce^{De^{\operatorname{\mathrm{Re}} w}} \quad\text{for all } w \in T(d).\end{align*} $$

We let

$$ \begin{align*}U(d,D) := \left\{z \in \mathbb{C}\setminus\{0\}:\ \cos(\arg z - d)> D|z|\right\} \cup \{0\}\end{align*} $$

be the Borel disk of diameter $\frac 1D$ touching the origin and centered on the ray in direction d. Correspondingly, we let

$$ \begin{align*}V(d,D):= \left\{w \in \mathbb{C}:\ \cos(\operatorname{\mathrm{Im}} w - d)> De^{\operatorname{\mathrm{Re}} w}\right\} \cup \{-\infty\}\end{align*} $$

the log-Borel disk in the direction d of extent $-\log D$ ; note indeed that $U(d,D) = \exp (V(d,D))$ . We define the log-Laplace transform $\mathcal {L}_df:V(d,D) \longrightarrow \mathbb {C}$ in the direction d of f by

$$ \begin{align*}\mathcal{L}_d f(w):= \int_{T(d)} e^{-e^{\eta-w}} f(\eta) d\eta.\end{align*} $$

Remark 2.4 If $g(z):= f(\log _d z)$ , for $z \in \mathbb {C}$ such that $\arg z = d$ , then the change of variables $z = e^w$ gives that

$$ \begin{align*} \left(\mathcal{L}_{d} f\right)(\log_d z) = \frac{\left(\mathcal{L}_{d}g\right) (z)}z, \end{align*} $$

where $\mathcal {L}_{d} g$ denotes the Laplace transform of g in the direction d as defined in [Reference Loray5] (see also Section 5.1 of [Reference Balser2]). Thus, the following proposition is obtained from Propriétés 1–2 on pages 41 and 42 of [Reference Loray5].

Proposition 2.5 Let $\varphi> 0$ and set $S:= S(d,\infty ,\varphi )$ . Let $f:S \longrightarrow \mathbb {C}$ , and assume that for every closed log-subsector $\overline S_0$ of S, the restriction $f\!\!\upharpoonright _{\overline S_0}$ is log-holomorphic and there exist $C,D>0$ such that $|f(w)| \le Ce^{De^{\operatorname {\mathrm {Re}} w}}$ for $w \in \overline S_0$ . Then:

1. (1) For each $\theta \in (0,\varphi )$ , there exists $0 < R(\theta ) \le \frac 1D$ such that $\mathcal {L}_d f$ has a log-holomorphic extension $\mathcal {L} f: V(d,R(\theta )) \longrightarrow \mathbb {C}$ .

2. (2) Let $\alpha \ge 0$ , and assume that for every closed log-subsector $\overline S_0$ of S, we have $|f(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\overline S_0$ . Then, in the situation of part (1), for every closed log-subsector $\overline S_0$ of $V(d,R(\theta ))$ , we have $|\mathcal {L} f(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\overline S_0$ .

In view of the previous proposition, we call the union $V:=\bigcup _{\theta \in (0,\varphi )}V(d,R(\theta ))$ a log-sectorial domain, and we refer to the common extension $\mathcal {L} f:V \longrightarrow \mathbb {C}$ of $\mathcal {L}_0 f$ of $\mathcal {L}_0 f$ given by Proposition 2.5 as the log-Laplace transform of f. Note that, in practice, we shall usually restrict the domain of $\mathcal {L} f$ to a sector $S\left (d,\log R,\theta +\frac \pi 2\right )$ for suitable $\theta \in (0,\varphi )$ and $R>0$ on which it is log-holomorphic.

For $f:S(d,{\frak r},\theta ) \longrightarrow \mathbb {C}$ as in Section 2.2.1, Proposition 2.2 implies that $\mathcal {L}(\mathcal {B} f)$ is defined and log-holomorphic on every closed log-subsector $\overline S_0$ of $S(d,{\frak r},\theta )\cap V$ . Indeed, $\mathcal {L}$ is the inverse operator to $\mathcal {B}$ (see page 44 of [Reference Loray5]).

Proposition 2.6 For $f:S(d,{\frak r},\theta ) \longrightarrow \mathbb {C}$ as in Section 2.2.1, we have $\mathcal {L}(\mathcal {B} f) = f$ on $S(d,{\frak r},\theta )\cap V$ .

Example 2.7 For $\alpha \in \mathbb {R}$ , we set $p_{\alpha }(w):= e^{\alpha w}$ . Then, for $w \in \mathbb {R}$ , we have

$$ \begin{align*} \mathcal{L}_0(p_{\alpha})(w) &= \int_{-\infty}^{\infty} e^{-e^{\eta-w}} e^{\alpha\eta} d\eta & \\ &=\int_0^{\infty} e^{-\zeta/e^w} \zeta^{\alpha-1} d\zeta &\text{(taking } \zeta = e^{\eta}) \\ &= e^{\alpha w} \int_0^{\infty} e^{-\xi} \xi^{\alpha-1} d\xi &\text{(taking } \zeta = e^w\eta) \\ &= \Gamma(\alpha) e^{\alpha w}. \end{align*} $$

It follows, by analytic continuation and Proposition 2.5, that $\mathcal {L}(p_{\alpha }) = \Gamma (\alpha ) p_{\alpha }$ , and hence by Proposition 2.6 that $\mathcal {B}(p_{\alpha }) = \frac {p_{\alpha }}{\Gamma (\alpha )}$ .

2.3 Generalized power series with complex coefficients

Let now $F(X) = \sum _{\alpha \ge 0} a_{\alpha } X^{\alpha } \ \in \mathbb {C}\left \{X^*\right \}$ be such that $\|F\|_r < \infty $ , for some $r>0$ . We explain here how such a series defines a log-holomorphic function on some log-disk. Denoting by $\log $ the principle branch of the logarithm on $\mathbb {C} \setminus (-\infty ,0]$ , we set

$$ \begin{align*} z^{\alpha} := e^{\alpha\log z} \quad\text{for } z \in \mathbb{C} \setminus (-\infty,0]. \end{align*} $$

Then, for $w \in S(0,\infty ,\pi )$ , we have that

$$ \begin{align*} p_{\alpha}(w) = (e^w)^{\alpha}; \end{align*} $$

in other words, the entire function $p_{\alpha }$ extends the function $w \mapsto (e^w)^{\alpha }: S(0,\infty ,\pi ) \longrightarrow \mathbb {C}$ . Since $|p_{\alpha }(w)| = e^{\alpha \operatorname {\mathrm {Re}} w}$ , it follows that the series

$$ \begin{align*} \overline F(w) := \sum a_{\alpha} e^{\alpha w} \end{align*} $$

converges absolutely and uniformly, for $w \in H(\log r)$ . By Weierstrass’s theorem, the function $\overline F:H(\log r)\setminus \{-\infty \} \longrightarrow \mathbb {C}$ is holomorphic, and by the previous remarks, we have

$$ \begin{align*}\overline F(w) = F(e^w) \quad\text{for } w \in S(0,\log r,\pi).\end{align*} $$

It follows, in particular, from [Reference Van den Dries and Speissegger11, Lemma 5.5] that $\overline F$ extends continuously to $-\infty $ and satisfies $\overline F(-\infty ) = F(0)$ . Below, we refer to the log-holomorphic function $\overline F$ thus defined on $H(\log r)$ as the log-sum of $F(X)$ .

2.3.1 Logarithmic Borel transform of convergent generalized power series with natural support

We again fix $F(X) = \sum a_{\alpha } X^{\alpha } \in \mathbb {C}\left \{X^*\right \}$ and $r> 0$ such that $\|F\|_r < \infty .$ In addition, we assume that the support of $F(X)$ —a subset of $[0,\infty )$ by definition—is natural. Since $\overline F$ is defined on $H(\log r) = S(0,\log r,\infty )$ , we obtain from Proposition 2.2 that its log-Borel transform $\mathcal {B}\overline F$ is log-holomorphic on $\overline {\mathbb {C}}$ .

In view of Example 2.7, we set

$$ \begin{align*}\mathcal{B} F(X):= \sum \frac{a_{\alpha}}{\Gamma(\alpha)} X^{\alpha},\end{align*} $$

called the formal Borel transform of $F(X)$ . Note that, for $\sigma>0$ , we have by Binet’s second formula (see [Reference Whittaker and Watson14]) that

$$ \begin{align*}C(\sigma):= \max_{\alpha \ge 0} \frac{\sigma^{\alpha}}{\Gamma(\alpha)} < \infty.\end{align*} $$

Thus, for any $\sigma> 0$ , we have that

(2.1) $$ \begin{align} \|\mathcal{B} F\|_{\sigma} = \sum \frac{(\sigma/r)^{\alpha}}{\Gamma(\alpha)} |a_{\alpha}| r^{\alpha} \le C(\sigma/r) \|F\|_{r}. \end{align} $$

Since F has natural support, the sum is finite for all $\sigma $ ; so the series $\mathcal {B} F(X)$ has infinite radius of convergence, and its log-sum $\overline {\mathcal {B} F}$ is also log-holomorphic on $\overline {\mathbb {C}}$ . In summary:

Proposition 2.8 Let $F(X)$ be a convergent generalized power series with natural support. Then both $\mathcal {B}\overline {F}$ and $\overline {\mathcal {B} F}$ are log-holomorphic on $\overline {\mathbb {C}}$ , and we have $\overline {\mathcal {B} F} = \mathcal {B}\overline F$ .

Proof Since $F(X)$ has natural support, we write $F(X)= \sum _{n=0}^N a_n r^{\alpha _n}$ with either $N \in \mathbb {N}$ , or $N=\infty $ and $\lim _{n \to \infty } \alpha _n = +\infty $ . For $w \in \mathbb {C}$ , let $\overline S$ be the closure of the $\log $ -sector $S(\operatorname {\mathrm {Im}} w,\log r,\pi )$ , and define $K:\partial \overline S \longrightarrow \mathbb {C}$ by

$$ \begin{align*}K(\eta):= \frac1{2\pi i} e^{w-\eta} e^{e^{w-\eta}}.\end{align*} $$

For $n \in \mathbb {N}$ with $n \le N$ , let $u_n: \partial \overline S \longrightarrow \mathbb {C}$ be defined by

$$ \begin{align*}u_n(\eta):= a_n e^{\alpha_n \eta} K(\eta) = a_n p_{\alpha_n}(\eta) K(\eta),\end{align*} $$

where $p_{\alpha _n}$ is defined as in Example 2.7. Proceeding as in the proof of Lemma 2.3, we obtain a $C>0$ such that

$$ \begin{align*}\int_{\partial \overline S} |u_n(\eta)| d\eta \le C|a_n|r^{\alpha_n}, \quad\text{for each } n.\end{align*} $$

Since $\|F\|_r < \infty $ , it follows that $\sum _n \int _{\partial \overline S} |u_n(\eta )| d\eta < \infty $ . If follows from analysis that the functions $u_n$ , for each n, as well as $\sum _n u_n$ and $\sum _n |u_n|$ are integrable on $\partial \overline S$ and that

$$ \begin{align*} \overline{\mathcal{B} F}(w) &= \sum_n \frac{a_n}{\Gamma(\alpha_n)} e^{\alpha_n w}= \sum_n a_n (\mathcal{B} p_{\alpha_n})(w) \\ &= \sum_n \int_{\partial \overline S} u_n(\eta) d\eta = \int_{\partial \overline S} \left(\sum_n u_n(\eta)\right) d\eta \\ &= \int_{\partial \overline S} \overline F(\eta) K(\eta) d\eta = (\mathcal{B} \overline F)(w), \end{align*} $$

as claimed.

2.4 Generalized multisummable functions

We now define generalized multisummable functions inspired by Tougeron’s characterization of multisummable functions [Reference Tougeron9] and by their presentation in [Reference Van den Dries and Speissegger12]. However, while it was possible in [Reference Van den Dries and Speissegger12] to refer to the existing literature for summability, it is not the case in our setting. More precisely, our aim is to show a quasianalyticity result for our functions analogous to that in [Reference Van den Dries and Speissegger12, Proposition 2.18]. To this end, we need to introduce suitable Borel and Laplace transforms adapted to the generalized multisummable framework (see Section 2.5). The presentation turns out to be more readable in this setting by replacing the usual “Gevrey order” k by $1/k$ . This leads to the following definitions.

For $R,k \ge 0$ , $\theta>\pi /2$ and $p \in \mathbb {N}$ , we set

$$ \begin{align*}\rho^{R,k}_p:= \frac R{(1+p)^{k}} \quad\text{and}\quad S^{R,k}_p:= \operatorname{\mathrm{cl}}\left(S\left(0,\log R, {\theta k}\right) \cup H\left(\log\rho^{R,k}_p\right)\right).\end{align*} $$

Let $K \subseteq [0,\infty )$ be a nonempty finite set and $r> 1$ (note that the situation studied in [Reference Van den Dries and Speissegger12] corresponds, in the current notation, to $\pi /2<\theta <\pi $ and $K\subseteq [0,1]$ , in order to avoid dealing with the logarithmic chart), and set

(2.2) $$ \begin{align}M_K:= \max K,\ \mu_K:= \min K.\end{align} $$

Moreover, we fix a natural set $\Delta \subseteq [0,\infty )$ and set $\tau := (K,R,r,\theta ,\Delta )$ (note that $\Delta = \mathbb {N}$ in [Reference Van den Dries and Speissegger12]). We define

$$ \begin{align*}S^{\tau}:= \bigcap_{k \in K \setminus \{0\}} S\left(0,\log R, {\theta k}\right) \text{ if } K \ne \{0\},\ \quad\text{and}\quad S^{\tau}:= H(\log R) \text{ if } K=\{0\}\end{align*} $$

and, for $p \in \mathbb {N}$ ,

$$ \begin{align*}\rho^{\tau}_p:= \min_{k \in K}\rho^{R,k}_p = \rho^{R,M_K}_p\end{align*} $$

and

$$ \begin{align*}S^{\tau}_{p}:= \bigcap_{k \in K} S^{R,k}_p.\end{align*} $$

Definition 2.10 For each $p \in \mathbb {N}$ , let $f_p:S^{\tau }_p \longrightarrow \mathbb {C}$ be log-holomorphic, that is, there exists a log-domain $D_p \supseteq S^{\tau }_p$ and a log-holomorphic $g_p:D_p \longrightarrow \mathbb {C}$ such that $f_p = g_p\!\!\upharpoonright _{S^{\tau }_p}$ . Moreover, we assume that there are generalized power series $F_p(X) \in \mathbb {C}\left \{X^*\right \}$ with support contained in $\Delta $ such that $\|F_p\|_{\rho ^{\tau }_p} < \infty $ and

$$ \begin{align*}f_p(w) = \overline{F_p}(w) \quad\text{for } w \in H(\log\rho^{\tau}_p).\end{align*} $$

Assume also that

$$ \begin{align*}\sum_p \|F_p\|_{\rho^{\tau}_p}r^p < \infty \quad\text{and}\quad \sum_p \|f_p\|_{S^{\tau}_p} r^p < \infty,\end{align*} $$

where $\|f_p\|_{S^{\tau }_p}:= \sup _{w \in S^{\tau }_p} |f_p(w)|$ denotes the sup norm of $f_p$ on $S^{\tau }_p$ . The second of these finiteness assumptions implies that $\sum _p f_p$ converges uniformly on $S^{\tau } \setminus \{-\infty \}$ to a holomorphic function $g:S^{\tau } \setminus \{-\infty \} \longrightarrow \mathbb {C}$ , while the first implies that this g extends continuously to $-\infty $ , so that the resulting $g:S^{\tau } \longrightarrow \mathbb {C}$ is log-holomorphic. From now on, we abbreviate this situation by writing

$$ \begin{align*}g =_{\tau} \sum_p f_p.\end{align*} $$

Thus, for a log-holomorphic function $f:S^{\tau } \longrightarrow \mathbb {C}$ , we set

$$ \begin{align*}\|f\|_{\tau}:= \inf\left\{\max\left\{\sum_p \|F_p\|_{\rho^{\tau}_p}r^p,\sum_p \|f_p\|_{S^{\tau}_p} r^p\right\}:\ f =_{\tau} \sum_p f_p\right\} \in [0,\infty];\end{align*} $$

note that $\|f\|_{\tau } < \infty $ if and only if there exists a sequence $f_p$ such that $f =_{\tau } \sum _p f_p$ .

We set

$$ \begin{align*}\mathcal{G}_{\tau}:= \left\{f: S^{\tau} \longrightarrow \mathbb{C}:\ f \text{ is log-holomorphic and } \|f\|_{\tau} < \infty\right\}.\end{align*} $$

It is immediate from this definition that $\mathcal {G}_{\tau }$ is a $\mathbb {C}$ -vector space under pointwise addition; moreover, if $\Delta $ is closed under addition, then $\mathcal {G}_{\tau }$ is closed under multiplication of functions, making $\mathcal {G}_{\tau }$ a $\mathbb {C}$ -algebra.

Let $f \in \mathcal {G}_{\tau }$ with associated functions $f_p$ and series $F_p$ be as in Definition 2.10 such that $\sum _p \|F_p\|_{\rho ^{\tau }_p}r^p < 2\|f\|_{\tau }$ and $\sum _p \|f_p\|_{S^{\tau }_p} r^p < 2\|f\|_{\tau }$ . In Proposition 2.16, we show that f has asymptotic expansion $F(e^w)$ at $-\infty $ , for the generalized power series $F(X)$ with support contained in $\Delta $ (and hence natural), defined in (2.3). To do so, say $F_p(X) = \sum a_{p,\alpha } X^{\alpha }$ for each p, where each $a_{p,\alpha } \in \mathbb {R}$ , and write $\rho _p$ for $\rho _p^{\tau }$ . Then, for each p and $\alpha $ , and for arbitrary $s \in (1,r)$ , we have

$$ \begin{align*}|a_{p,\alpha}| \le \frac{\|F_p\|_{\rho_p}}{\rho_p^{\alpha}} = \frac{\|F_p\|_{\rho_p}}{R^{\alpha}} (p+1)^{\alpha M_K} \le \frac{\|F_p\|_{\rho_p}}{R^{\alpha}} s^p (p+1)^{\alpha M_K}.\end{align*} $$

Therefore, for each $\alpha $ and arbitrary $s \in (1,r)$ , we get

$$ \begin{align*} \sum_p |a_{p,\alpha}| &\le \frac1{R^{\alpha}} \sum_p (p+1)^{\alpha M_K} \|F_p\|_{\rho_p} s^p \\ &= \frac1{R^{\alpha}} \sum_p \|F_p\|_{\rho_p} r^p (p+1)^{\alpha M_K} \left(\frac sr\right)^p \\ &\le \frac{C(s,\alpha)}{R^{\alpha}} \sum_p \|F_p\|_{\rho_p} r^p < \infty, \end{align*} $$

where $C(s,\alpha ):= \max _p (p+1)^{\alpha M_K} (s/r)^p < \infty $ . So we set

$$ \begin{align*}a_{\alpha} := \sum_p a_{p,\alpha},\end{align*} $$

for each $\alpha $ , and

(2.3) $$ \begin{align} F(X) := \sum a_{\alpha} X^{\alpha}, \end{align} $$

which has support contained in $\Delta $ .

Lemma 2.15 There exist $D,E> 0$ such that for all $p \in \mathbb {N}$ and all $\beta \ge 0$ , we have

$$ \begin{align*}\left|f_p(w) - \sum_{\alpha < \beta} a_{\alpha,p} e^{\alpha w}\right| \le CD^{\beta} \frac{(p+1)^{\beta M_K}}{r^p} \left|e^{\beta w}\right| \quad\text{for } w \in S_p^{\tau}.\end{align*} $$

Proof Fix $p \in \mathbb {N}$ and $\beta \ge 0$ , and let $w \in S_p^{\tau }$ . We distinguish two cases:

Case 1: $\operatorname {\mathrm {Re}} w < \log \rho _p$ . Then

$$ \begin{align*} \left|f_p(w) - \sum_{\alpha < \beta} a_{p,\alpha} e^{\alpha w}\right| &= \left|\sum_{\alpha \ge \beta} a_{p,\alpha} e^{\alpha w}\right| \\ &\le \left|e^{\beta w}\right| \sum_{\alpha \ge \beta} |a_{p,\alpha}| \left|e^{(\alpha-\beta) w}\right| \\ &\le \left|e^{\beta w}\right| \sum_{\alpha \ge \beta} |a_{p,\alpha}| (\rho_p)^{\alpha-\beta} \\ &\le \frac{\left|e^{\beta w}\right|}{(\rho_p)^{\beta}} \|F_p\|_{\rho_p} \\ &\le 2\left|e^{\beta w}\right| \frac{(p+1)^{\beta M_K}}{R^{\beta} r^p} \|f\|_{\tau}, \end{align*} $$

which proves the estimate in this case.

Case 2: $\operatorname {\mathrm {Re}} w \ge \log \rho _p$ . Then

$$ \begin{align*} \left|f_p(w) - \sum_{\alpha < \beta} a_{p,\alpha} e^{\alpha w}\right| \le \left|f_p(w)\right| + \sum_{\alpha < \beta} |a_{p,\alpha}|\left|e^{\alpha w}\right|, \end{align*} $$

so we further split up the estimate:

$$ \begin{align*} |f_p(w)| &\le \|f_p\|_{S_p^{\tau}} \\ &\le \|f_p\|_{S_p^{\tau}} \frac{|e^{\beta w}|}{(\rho_p)^{\beta}} &\text{as } |e^{w}| \ge \rho_p \\ &= \frac{(p+1)^{\beta M_K}}{R^{\beta}} \|f_p\|_{S_p^{\tau}} \left|e^{\beta w}\right| \\ &\le \frac{(p+1)^{\beta M_K}}{R^{\beta} r^p} 2\|f\|_{\tau} \left|e^{\beta w}\right|, \end{align*} $$

while

$$ \begin{align*} \sum_{\alpha < \beta} |a_{p,\alpha}| \left|e^{\alpha w}\right| &\le \left|e^{\beta w}\right| \sum_{\alpha < \beta} |a_{p,\alpha}| (\rho_p)^{\alpha-\beta} &\text{as } \alpha-\beta < 0 \text{ and } |e^{ w}| \ge \rho_p \\ &= \frac{\left|e^{\beta w}\right|}{(\rho_p)^{\beta}} \sum_{\alpha < \beta} |a_{p,\alpha}|(\rho_p)^{\alpha} \\ &\le \left|e^{\beta w}\right| \frac{(p+1)^{\beta M_K}}{R^{\beta}} \|F_p\|_{\rho_p} \\ &\le \left|e^{\beta w}\right| \frac{(p+1)^{\beta M_K}}{R^{\beta} r^p} \cdot 2\|f\|_{\tau}. \end{align*} $$

This completes the proof of Case 2 and therefore of the lemma.

Proposition 2.16 (Gevrey estimates)

For every closed log-subsector $\overline S_0$ of $S^{\tau }$ , there exist $D,E>0$ such that, for each $\beta \ge 0$ ,

$$ \begin{align*}\left|f(w) - \sum_{\alpha < \beta} a_{\alpha} e^{\alpha w}\right| \le DE^{\beta} \Gamma\left(\beta M_K\right) \left|e^{\beta w}\right| \quad\text{for } w \in \overline S_0.\end{align*} $$

Proof Let $D,E>0$ be obtained from Lemma 2.15, and let $\beta \ge 0$ . Since $\Delta \cap [0,\beta )$ is finite we have, for $w \in \overline S_0$ ,

$$ \begin{align*} \left|f(w) - \sum_{\alpha < \beta} a_{\alpha} e^{\alpha w}\right| &= \left|\left(\sum_p f_p(w)\right) - \sum_{\alpha < \beta} \left(\sum_p a_{\alpha,p}\right) e^{\alpha w}\right| \\ &\le \sum_p \left|f_p(w) - \sum_{\alpha < \beta} a_{\alpha,p} e^{\alpha w}\right| \\ &\le CD^{\beta} \left(\sum_p \frac{(p+1)^{\beta M_K}}{r^p}\right) \left|e^{\beta w}\right| &\text{by Lemma 2.15}. \end{align*} $$

Since $\sum _p \frac {(p+1)^{\beta M_K}}{r^p} \le C'(D')^{\beta } \beta ^{\beta M_K}$ for some $C',D'>0$ (see, for instance, the proof of [Reference Van den Dries and Speissegger12, Lemma 2.6]), the proposition now follows from Stirling’s formula for $\Gamma $ (see [Reference Artin1]).

Proposition 2.16 implies that $F(e^w)$ is an asymptotic expansion of f at $-\infty $ ; hence, it is uniquely determined by f (and is, in particular, independent of the particular sequence $\{f_p\}$ ), and we write $Tf(X) := F(X)$ . The map $T:\mathcal {G}_{\tau } \longrightarrow \mathbb {C}\left [\!\left [X^*\right ]\!\right ]$ is a $\mathbb {C}$ -algebra homomorphism.

Example 2.18 Assume that $K=\{0\}$ . If $f \in \mathcal {G}_{\tau }$ , then $Tf$ converges, $\|Tf\|_R < \infty $ and $f = \overline {Tf}$ . To see this, let $f =_{\tau } \sum _p f_p$ with $Tf_p = \sum _{\alpha } a_{p,\alpha } X^{\alpha }$ for each p; then $Tf = \sum _{\alpha } a_{\alpha }X^{\alpha }$ with $a_{\alpha } = \sum _p a_{p,\alpha }$ as above. Since $\rho _p^{\tau } = R$ for each p, the assumption $\sum _p \|Tf_p\|_R r^p < \infty $ implies that $\sum _{p,\alpha } |a_{p,\alpha }| R^{\alpha } < \infty $ . This implies that the family $\{a_{p,\alpha } e^{\alpha w}\}$ is summable on $\operatorname {\mathrm {cl}} H(\log R)$ ; in particular, the order of summation can be changed. Thus, we have for $w \in \operatorname {\mathrm {cl}} H(\log R)$ that

$$ \begin{align*} f(w) = \sum_p f_p(w) = \sum_p \sum_{\alpha} a_{p,\alpha} e^{\alpha w} = \sum_{\alpha} \left(\sum_p a_{p,\alpha}\right) e^{\alpha w} = \sum_{\alpha} a_{\alpha} e^{\alpha w} = \overline{Tf}(w), \end{align*} $$

as claimed.

Conversely, if $F \in \mathbb {R}\left [\!\left [X^*\right ]\!\right ]$ has natural support and satisfies $\|F\|_R < \infty $ , then (the appropriate restriction of) the function $\overline {F}$ belongs to $\mathcal {G}_{\tau }$ , where $\tau = \left (K,R, r, \theta ,\Delta \right )$ with K, r, $\theta $ and $\Delta \supseteq \operatorname {\mathrm {supp}}(F)$ arbitrary. To see this, simply take $f_0 := \overline F$ and $f_p:= 0$ , for $p> 0$ .

Recall from 2.2 that $\mu _K= \min K \in [0,\infty )$ . If $\mu _K \ge 1$ , then ${\theta \mu _K}> \frac \pi 2$ , so by Proposition 2.2, the function $\mathcal {B} f$ is defined on the sector $S\left (0, \infty , {\theta \mu _K} - \frac \pi 2\right )$ . The next proposition explains in more detail what happens when we apply the Borel transform to f.

Proof For $p \in \mathbb {N}$ , we set $\rho _p:= \rho _p^{\tau }$ and $\sigma _p := \rho _p^{\tau '}$ . Then, for $\alpha \ge 0$ , we have

$$ \begin{align*}\frac{(\sigma_p/\rho_p)^{\alpha}}{\Gamma(\alpha)} = \left(\frac {R'}R\right)^{\alpha} \frac{(p+1)^{\alpha}}{\Gamma(\alpha)}.\end{align*} $$

Claim: There exists $C = C(r,r',R,R')> 0$ such that

$$ \begin{align*}\left(\frac {R'}R\right)^{\alpha}\frac{(p+1)^{\alpha}}{\Gamma(\alpha)}\left(\frac {r'}r\right)^p \le C,\end{align*} $$

for all $p \in \mathbb {N}$ and $\alpha \ge 0$ .

To see the claim, note that the function $x \mapsto f_{\alpha }(x):= (x+1)^{\alpha } \left (\frac {r'}r\right )^x$ attains its maximum at $x_{\alpha } = \frac \alpha {\log (r/r')}-1$ ; so this maximum is

$$ \begin{align*}f_{\alpha}(x_{\alpha}) = \frac r{r'} A^{\alpha} \alpha^{\alpha},\end{align*} $$

with $A:= \frac {(r'/r)^{1/\log (r/r')}}{\log (r/r')} = (e\log (r/r'))^{-1}$ independent of $\alpha $ . From Binet’s second formula, we get a constant $C">0$ such that, for all $\alpha \ge 0$ ,

$$ \begin{align*} \alpha^{\alpha} \le \frac{\sqrt{\alpha}e^{\alpha}}{C" e^{\phi(\alpha)}}\Gamma(\alpha), \end{align*} $$

where $\phi (x)$ is the Stirling function. Since the latter is bounded at $\infty $ , there is a constant $C'>0$ such that

$$ \begin{align*} \alpha^{\alpha} \le C'\sqrt{\alpha}e^{\alpha}\Gamma(\alpha) \le C' e^{2\alpha} \Gamma(\alpha). \end{align*} $$

Therefore,

$$ \begin{align*} \left(\frac {R'}R\right)^{\alpha} (p+1)^{\alpha}\left(\frac {r'}r\right)^p \le \frac r{r'} \left(\frac {R'}R\right)^{\alpha} A^{\alpha} \alpha^{\alpha} \le C' \frac r{r'} \left(\frac {R'}R K e^2\right)^{\alpha} \Gamma(\alpha) \le C \Gamma(\alpha), \end{align*} $$

where $C:= C'\frac r{r'}$ , because our assumptions on $r'$ and $R'$ imply that $Ae^2 \le \frac R{R'}$ . This proves the claim.

It follows from the claim that

(2.4) $$ \begin{align} \|\mathcal{B} F_p\|_{\sigma_p}\left(\frac {r'}r\right)^p = \sum \frac{(\sigma_p/\rho_p)^{\alpha}}{\Gamma(\alpha)}\left(\frac {r'}r\right)^p |a_{p,\alpha}|\rho_p^{\alpha} \le C\|F_p\|_{\rho_p} \end{align} $$

for each $p \in \mathbb {N}$ , so that

(2.5) $$ \begin{align} \sum_p \|\mathcal{B} F_p\|_{\sigma_p} (r')^p \le C \sum_p \|F_p\|_{\rho_p} r^p < \infty. \end{align} $$

Next, we show $\sum \|\mathcal {B} f_p\|_{S^{\tau '}_p} (r')^p < \infty $ . If $|K| = \mu _K = 1$ , then (2.5) also proves that $\mathcal {B} F$ is convergent and that $\sum \|\mathcal {B} f_p\|_{H(\log R')} (r')^p < \infty $ ; this settles the assertion in this case.

So assume that $\mu _K> 1$ or $|K|> 1$ . By (2.5), it suffices to show that $\sum \|\mathcal {B} f_p\|_{S^{\tau '}} (r')^p < \infty $ and, for $k \in K\setminus \{M_K\}$ , that $\sum \|\mathcal {B} f_p\|_{S^k_p} (r')^p < \infty $ , where we set

$$ \begin{align*}\sigma^k_p:= \rho_p^{R',k-1} \quad\text{and}\quad S^k_p := S\left(0, \log\sigma^k_p, {\theta(k-1)}\right).\end{align*} $$

For the first estimate: since ${\theta (\mu _K-1)}< {\theta \mu _K} - \frac \pi 2$ , Lemma 2.3 shows that $\left \|\mathcal {B} f_p\right \|_{S^{\tau '}} \le C\left \|f_p\right \|_{S^{\tau }}$ , for some constant $C>0$ that only depends on $\theta $ and $\theta '$ .

For the second estimate: fix $k \in K\setminus \{M_K\}$ and set $\rho ^k_p:= \rho _p^{R,k}$ , $k' := \min (K \setminus [\mu _K,k])$ and $T^k_p:= S\left (0,\log \rho ^k_p,{\theta k'}\right )$ . Since ${\theta (k'-1)} < {\theta k'} - \frac \pi 2$ , Lemma 1.3 shows that

$$ \begin{align*}\left\|\mathcal{B} f_p\right\|_{S^k_p} \le C \left\|f_p\right\|_{T^k_p} e^{\sigma^k_p / \rho^k_p} \frac{\sigma^k_p}{\rho^k_p} = C\|f_p\|_{T^k_p} \left(e^{R'/R}\right)^{1+p} \frac{R'}R (1+p),\end{align*} $$

for some constant $C>0$ independent of p. Now note that our assumptions on $r'$ and $R'$ imply that $e^{R'/R}\cdot \frac {r'}r < 1$ ; therefore, $D:= \max _p (1+p) \left (e^{R'/R}\right )^{(1+p)}\left (\frac {r'}r\right )^p < \infty $ . It follows that

$$ \begin{align*} \sum_p \|\mathcal{B} f_p\|_{S^k_p} (r')^p \le D \sum_p \left\|f_p\right\|_{T^k_p} r^p < \infty, \end{align*} $$

as claimed.

Finally, it remains to show that $\mathcal {B} f =_{\tau '} \sum _p \mathcal {B} f_p$ and $T(\mathcal {B} f) = \mathcal {B}(Tf)$ : set $g:= \sum _p \mathcal {B} f_p$ . The above estimates show that $g \in \mathcal {G}_{\tau '}$ with $Tg = \sum _p \mathcal {B} F_p = \mathcal {B}(Tf)$ , so we need to show that $\mathcal {B} f = g$ . Since $f = \sum _p f_p$ uniformly in $S^{\tau }$ , it follows from integration theory that $\mathcal {B} f = \mathcal {B}\left (\sum _p f_p\right ) = \sum _p \mathcal {B} f_p = g$ , as required.

For later purposes, we note the following special case of Proposition 2.20:

2.5 Ramified logarithmic transforms

Let $\lambda>0$ . In the classical situation, the ramified Borel transform $\mathcal {B}^{\lambda } f$ of a function f is obtained from $\mathcal {B} f$ by the change of variables $z \mapsto z^{\lambda }$ . In the logarithmic chart, this means that $\mathcal {B}^{\lambda } f$ is obtained from $\mathcal {B} f$ by the change of variables $w \mapsto \lambda w$ .

2.5.1 Ramified logarithmic Borel transform

Let $d,{\frak r} \in \mathbb {R}$ and $\theta> \pi $ , and write $S = S(d,{\frak r},\theta \lambda )$ . We denote by $\mathbf {m}_{\lambda }:\mathbb {C} \longrightarrow \mathbb {C}$ the logarithmic ramification map defined by $\mathbf {m}_{\lambda }(w):= \lambda w$ .

Let $f:S \longrightarrow \mathbb {C}$ be such that $f\!\!\upharpoonright _{\widetilde S}$ is bounded and log-holomorphic, for every closed log-subsector $\widetilde S$ of S. Then the map $f \circ \mathbf {m}_{\lambda }: S(d/\lambda ,{\frak r}/\lambda ,\theta ) \longrightarrow \mathbb {C}$ has logarithmic Borel transform $\mathcal {B}(f \circ \mathbf {m}_{\lambda }):S\left (d/\lambda ,\infty ,\theta -\frac \pi 2\right ) \longrightarrow \mathbb {C}$ . We define the log- $\lambda $ -Borel transform $\mathcal {B}^{\lambda } f: S\left (d,\infty ,{\theta \lambda } - \frac {\pi \lambda }{2}\right ) \longrightarrow \mathbb {C}$ of f by

$$ \begin{align*} \mathcal{B}^{\lambda} f := \left(\mathcal{B} (f \circ \mathbf{m}_{\lambda})\right) \circ \mathbf{m}_{1/\lambda}. \end{align*} $$

We immediately obtain the following from Proposition 2.2 and Example 2.7.

Corollary 2.22 Let $f:S \longrightarrow \mathbb {C}$ be such that $f\!\!\upharpoonright _{\widetilde S}$ is bounded and log-holomorphic, for every closed log-subsector $\widetilde S$ of S, and set $S':= S\left (d,\infty ,{\theta \lambda } -\frac {\pi \lambda }{2}\right )$ .

1. (1) For every closed log-subsector $\widetilde S$ of $S'$ , the function $(\mathcal {B}^{\lambda } f)\!\!\upharpoonright _{\widetilde S}$ is log-holomorphic and there exist $C,M>0$ such that

   $$ \begin{align*}|(\mathcal{B}^{\lambda} f)(w)| \le C e^{De^{(\operatorname{\mathrm{Re}} w)/\lambda}} \quad\text{for } w \in \widetilde S.\end{align*} $$

2. (2) Let $\alpha \ge 0$ , and assume that for every closed log-subsector $\widetilde S$ of S, we have $|f(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\widetilde S$ . Then, for every closed log-subsector $\widetilde S$ of $S'$ , we have $|(\mathcal {B}^{\lambda } f)(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\widetilde S$ .

3. (3) For $\alpha \ge 0$ , we have $\mathcal {B}^{\lambda } p_{\alpha } = \frac {p_{\alpha }}{\Gamma (\alpha \lambda )}$ .

2.5.2 Ramified logarithmic Laplace transform

Let $\varphi> 0$ and set $S:= S(d,\infty ,\varphi \lambda )$ . Let $f:S \longrightarrow \mathbb {C}$ be a function, and assume that for every closed log-subsector $\widetilde S$ of S, the restriction $f\!\!\upharpoonright _{\widetilde S}$ is log-holomorphic and there exist $C,D>0$ such that $|f(w)| \le Ce^{De^{(\operatorname {\mathrm {Re}} w)/\lambda }}$ for $w \in \widetilde S$ .

Let also $\theta \in (0,\varphi )$ ; then the map $f \circ \mathbf {m}_{\lambda }:S\left (d/\lambda ,\infty ,\theta \right ) \longrightarrow \mathbb {C}$ has logarithmic Laplace transform $\mathcal {L}(f \circ \mathbf {m}_{\lambda }): S\left (d/\lambda ,{\frak r}/\lambda ,\theta + \frac \pi 2\right ) \longrightarrow \mathbb {C}$ , for some ${\frak r} \le \log (D)\lambda $ . We define the log- $\lambda $ -Laplace transform $\mathcal {L}^{\lambda } f:S\left (d,{\frak r},{\theta \lambda } + \frac {\pi \lambda }{2}\right ) \longrightarrow \mathbb {C}$ of f by

$$ \begin{align*}\mathcal{L}^{\lambda} f:= \left(\mathcal{L}(f \circ \mathbf{m}_{\lambda})\right) \circ \mathbf{m}_{1/\lambda}.\end{align*} $$

We immediately obtain the following from Proposition 2.5 and Example 2.7.

Corollary 2.23 Let $f:S \longrightarrow \mathbb {C}$ be a function, and assume that for every closed log-subsector $\widetilde S$ of S, the restriction $f\!\!\upharpoonright _{\widetilde S}$ is log-holomorphic and there exist $C,D>0$ such that $|f(w)| \le Ce^{De^{(\operatorname {\mathrm {Re}} w)/\lambda }}$ for $w \in \widetilde S$ . Let also $\theta \in (0,\varphi )$ , and let ${\frak r} \le \log (D)\lambda $ be as above and set $S':= S\left (d,{\frak r},{\theta \lambda } + \frac {\pi \lambda }{2}\right )$ .

1. (1) Let $\alpha \ge 0$ , and assume that for every closed log-subsector $\widetilde S$ contained in S, we have $|f(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\widetilde S$ . Then, for every closed log-subsector $\widetilde S$ contained in $S'$ , we have $|(\mathcal {L}^{\lambda } f)(w)| = O\left (e^{\alpha \operatorname {\mathrm {Re}} w}\right )$ as $w \to -\infty $ in $\widetilde S$ .

2. (2) For $\alpha \ge 0$ , we have $\mathcal {L}^{\lambda } p_{\alpha } = \Gamma (\alpha \lambda )p_{\alpha }$ .

For $f:S(d,{\frak r},\theta \lambda ) \longrightarrow \mathbb {C}$ as in Section 2.5.1, Corollary 2.22 implies that $\mathcal {L}^{\lambda }(\mathcal {B}^{\lambda } f)$ is defined and log-holomorphic on every closed log-subsector $\widetilde S$ contained in $S(d,{\frak r},\theta \lambda )$ . From Proposition 2.6, we therefore obtain the following corollary.

Corollary 2.24 For $f:S(d,{\frak r},\theta \lambda ) \longrightarrow \mathbb {C}$ as in Section 2.5.1, we have $\mathcal {L}^{\lambda }(\mathcal {B}^{\lambda } f) = f$ .

2.5.3 Formal Borel and Laplace transforms

Let now $F(X) = \sum a_{\alpha } X^{\alpha } \in \mathbb {C}\left \{X^*\right \}$ . In view of the above, we define the formal $\lambda $ -Borel transform

$$ \begin{align*}(\mathcal{B}^{\lambda} F)(X) := \sum \frac{a_{\alpha}}{\Gamma(\alpha\lambda)} X^{\alpha}\end{align*} $$

and the formal $\lambda $ -Laplace transform

$$ \begin{align*}(\mathcal{L}^{\lambda} F)(X):= \sum \Gamma(\alpha\lambda) a_{\alpha} X^{\alpha}.\end{align*} $$

We get the following corollary from Proposition 2.8.

Corollary 2.25 Let F be a convergent generalized power series with natural support. Then, both $\mathcal {B}^{\lambda }\overline F$ and $\overline {\mathcal {B}^{\lambda } F}$ are log-holomorphic on $\overline {\mathbb {C}}$ , and we have $\mathcal {B}^{\lambda } \overline F = \overline {\mathcal {B}^{\lambda } F}$ .

2.5.4 Ramified Borel transforms of generalized multisummable functions

Let $K \subseteq (0,\infty )$ be finite and nonempty and $\lambda \le \mu _K$ , and define

$$ \begin{align*}K(\lambda):= \left\{k\lambda:\ k \in K\right\} \quad\text{and}\quad \tau(\lambda):= \left(K(\lambda),R^{1/\lambda}, r, \theta \right);\end{align*} $$

note that $f \in \mathcal {G}_{\tau }$ if and only if $f \circ \mathbf {m}_{\lambda } \in \mathcal {G}_{\tau (\lambda )}$ .

Let $f \in \mathcal {G}_{\tau }$ ; by Corollary 2.22, taking $\theta $ there equal to $\theta \cdot \frac {\mu _K}\lambda $ here, the function $\mathcal {B}^{\lambda } f$ is defined on the log-sector $S\left (0, \infty , {\theta \mu _K} - \frac {\pi \lambda }{2}\right )$ . Thus, we obtain the following corollary from Proposition 2.20 and Corollary 2.21.

Corollary 2.26 Let $f \in \mathcal {G}_{\tau }$ , and assume that $\mu _K \ge \lambda $ (in particular, $\mathcal {B}^{\lambda } f$ is well defined). Let $R' \in (0,R)$ and $r' \in (1,r)$ be such that $(R')^{1/\lambda } \le \frac {R^{1/\lambda }} e \log (r/r')$ , and set

$$ \begin{align*}K':= \left\{k-\lambda:\ k \in K,\ k> \lambda\right\}\end{align*} $$

and $\tau ':= \left (K',R',r',\theta ,\Delta \right )$ . Then $\mathcal {B}^{\lambda } f$ belongs to $\mathcal {G}_{\tau '}$ and satisfies $T(\mathcal {B}^{\lambda } f) = \mathcal {B}^{\lambda }(Tf)$ .

2.6 Summation and quasianalyticity

Let $K \subseteq [0,\infty )$ be nonempty and finite, and let $R>0$ , $r> 1$ , $\theta> \pi /2$ , and $\Delta \subseteq [0,\infty )$ be natural and closed under addition, and set $\tau = (K,R,r,\theta ,\Delta )$ . The goal of this section is to establish the quasianalyticity of the algebra $\mathcal {G}_{\tau }$ (Theorem 2.28). The key ingredient is the following summation method.

Proposition 2.27 (Summation)

Assume that $K = \{k_1, \dots , k_l\}$ with $0 < k_1 < \cdots < k_l < \infty $ and $l \ge 1$ . Let $f \in \mathcal {G}_{\tau }$ , and set $\kappa _1:= k_1$ and $\kappa _i:= k_i - k_{i-1}$ for $i=2, \dots , l$ . Then the series $\left (\mathcal {B}^{\kappa _1} \circ \cdots \circ \mathcal {B}^{\kappa _{l}}\right )(Tf)$ converges, and we have

$$ \begin{align*}f = \left(\mathcal{L}^{\kappa_1} \circ \cdots \circ \mathcal{L}^{\kappa_l}\right) \left(\overline{\left(\mathcal{B}^{\kappa_l} \circ \cdots \circ \mathcal{B}^{\kappa_1}\right) (Tf)}\right).\end{align*} $$

Proof By induction on l. If $l=1$ , then

$$ \begin{align*} \mathcal{L}^{\kappa_1} \left(\overline{\mathcal{B}^{\kappa_1}(Tf)}\right) &= \mathcal{L}^{\kappa_1} \left(\overline{T(\mathcal{B}^{\kappa_1} f)}\right) &\text{by Corollary 2.26} \\ &= \mathcal{L}^{\kappa_1} \left(\mathcal{B}^{\kappa_1} f\right) &\text{by Example 2.18} \\ &= f &\text{by Corollary 2.24}. \end{align*} $$

So we assume that $l>1$ and the proposition holds for lower values of l. Then by Corollary 2.26, the function $\mathcal {B}^{\kappa _1} f$ belongs to $\mathcal {G}_{\tau '}$ and satisfies $T\left (\mathcal {B}^{\kappa _1} f\right ) = \mathcal {B}^{\kappa _1}(Tf)$ , where $\tau ' = (K',R',r',\theta ,\Delta )$ for $K' := \left (k_2-k_1, \dots , k_l-k_1\right )$ and some appropriate $R'>0$ and $r'>1$ . From the inductive hypothesis applied to $\mathcal {B}^{k_1} f$ , we get that $\left (\mathcal {B}^{\kappa _l} \circ \cdots \circ \mathcal {B}^{\kappa _{2}}\right ) \left (T\left (\mathcal {B}^{\kappa _1}f\right )\right )$ converges, so that

$$ \begin{align*} f &= \mathcal{L}^{\kappa_1}(\mathcal{B}^{\kappa_1} f) &\text{(Proposition 2.6)}\\ &= \mathcal{L}^{\kappa_1}\left[\left(\mathcal{L}^{\kappa_{2}} \circ \cdots \circ \mathcal{L}^{\kappa_l}\right) \left(\overline{\left(\mathcal{B}^{\kappa_l} \circ \cdots \circ \mathcal{B}^{\kappa_{2}}\right) (T(\mathcal{B}^{\kappa_1}f))}\right)\right] &(\text{ind. case applied to } \mathcal{B}^{\kappa_1} f) \\ &= \mathcal{L}^{\kappa_1}\left[\left(\mathcal{L}^{\kappa_{2}} \circ \cdots \circ \mathcal{L}^{\kappa_l}\right)\left(\overline{\left(\mathcal{B}^{\kappa_l} \circ \cdots \circ \mathcal{B}^{\kappa_{2}} \circ \mathcal{B}^{\kappa_1}\right)(Tf)}\right)\right] \\ &= \left(\mathcal{L}^{\kappa_1} \circ \mathcal{L}^{\kappa_{2}} \circ \cdots \circ \mathcal{L}^{\kappa_l}\right) \left(\overline{\left(\mathcal{B}^{\kappa_l} \circ \cdots \circ \mathcal{B}^{\kappa_{2}}\circ \mathcal{B}^{\kappa_1}\right) (Tf)}\right), \end{align*} $$

as claimed.

2.7 Open questions

1. (1) Let f be generalized K-summable in the real direction, where $K = (k_1, \dots , k_l)$ . Do there exist generalized $(k_i)$ -summable functions $g_i$ , for $i=1, \dots , l$ , such that $f = g_1 + \cdots + g_l$ ?

   This question is motivated by the following: let f be a K-summable function in the positive real direction (in the classical sense, at the origin; to avoid branching, let’s assume $k_1> \frac 12$ ). Then by Example 2.14, the function $f \circ \exp $ belongs to $\mathcal {G}_{\tau }$ for some $\tau = (K,R,r,\theta ,\mathbb {N})$ ; indeed, Tougeron’s characterization implies that $f \circ \exp =_{\tau } \sum _p f_p \circ \exp $ for functions $f_p$ that are holomorphic at the origin. This property of being holomorphic at the origin can be used, via Cauchy integration, to show that there exist $(k_i)$ -summable functions $g_i$ , for $i=1, \dots , l$ , such that $f = g_1 + \cdots + g_l$ . However, for general $g \in \mathcal {G}_{\tau }$ with $g =_{\tau } \sum _p g_p$ , the functions $g_p \circ \log $ have essential singularities at the origin, so the Cauchy integration argument used for f does not work for $g \circ \log $ to write g as a sum of a generalized $(k_i)$ -summable functions.

2. (2) From the point of view of multisummability, there is nothing special about the real direction chosen here. (We are only interested in the real direction here, because we are aiming to construct algebras of real functions.) Indeed, one can similarly define generalized K-summable functions in any direction d. However, it is not clear to us what the right generalization of “generalized multisummable” (without specified direction) should be: in the classical case, all multisummable functions are $2\pi i$ -periodic in the logarithmic chart, and they are defined to be multisummable if they are multisummable in all but finitely many directions in $\mathbb {R}/2\pi \mathbb {Z}$ . In contrast, the logarithmic sums of generalized convergent power series are not $ai$ -periodic for any $a>0$ in general (take, for instance, the function $e^{\alpha w} + e^{\beta w}$ with $\alpha $ and $\beta $ linearly independent over $\mathbb {Q}$ ), so generalized multisummable functions in the real direction aren’t either. Possibly, the right way to define “generalized multisummable” would be to look for something like Stokes phenomena in differential equations over (quotients of) convergent generalized power series.

3. (3) Is there a Ramis–Sibuya theorem (see [Reference Ramis and Sibuya6]) for ordinary differential equations involving quotients of convergent generalized power series? As hinted at in Question 2, such a theorem might inform the correct definition of the term “generalized multisummable.”

3.1 Convergent generalized power series

Let $X = (X_1, \dots , X_m)$ . Similar to the one-variable case, we denote by $\mathbb {C}\left [\!\left [X^*\right ]\!\right ]$ the set of all generalized power series of the form $F(X) = \sum _{\alpha \in [0,\infty )^m} a_{\alpha } X^{\alpha }$ , where each $a_{\alpha } \in \mathbb {C}$ and the support

$$ \begin{align*}\operatorname{\mathrm{supp}}(F):= \left\{\alpha \in [0,\infty)^m:\ a_{\alpha} \ne 0\right\}\end{align*} $$

is contained in a Cartesian product of well-ordered subsets of $[0,\infty )$ (see [Reference Van den Dries and Speissegger11, Section 4] for details). The series $F(X)$ converges if there exists a polyradius $r \in (0,\infty )^m$ such that $\|F\|_r:= \sum _{\alpha } |a_{\alpha }|r^{\alpha } < \infty $ ; we denote by $\mathbb {C}\left \{X^*\right \}$ the set of all convergent generalized power series [Reference Van den Dries and Speissegger11, Section 5].

For ${\frak r} \in \mathbb {R}^m$ , we let

$$ \begin{align*}H({\frak r}) := \left\{w \in \overline{\mathbb{C}}^m:\ \operatorname{\mathrm{Re}} w< {\frak r}\right\} = H({\frak r}_1) \times \cdots \times H({\frak r}_m)\end{align*} $$

be the log-disk of log-polyradius $\frak r$ . For an set $D \subseteq \overline {\mathbb {C}}^m$ , we set

$$ \begin{align*}D^{\infty}:= D \setminus \mathbb{C}^m.\end{align*} $$

We call set $D \subseteq \overline {\mathbb {C}}^m$ a log-domain if $D \cap \mathbb {C}^m$ is a domain. If $D \subseteq \overline {\mathbb {C}}^m$ is a log-domain, a log-holomorphic function on D is a continuous function $f:D \longrightarrow \mathbb {C}$ such that the restriction of f to $D \cap \mathbb {C}^m$ is holomorphic.

Let $F(X) = \sum _{\alpha \in [0,\infty )^m} a_{\alpha } X^{\alpha } \in \mathbb {C}\left [\!\left [X^*\right ]\!\right ]$ such that $\|F\|_r < \infty $ . Similar to Section 2.3, there is a log-holomorphic function $\overline F: H(\log r) \longrightarrow \mathbb {C}$ , called the log-sum of F, such that $\overline F(w) := F(e^w)$ whenever $|\operatorname {\mathrm {Im}} w| < \pi /2$ .

3.2 Logarithmic polydomains

We define here the logarithmic versions of the domains discussed in [Reference Van den Dries and Speissegger12, Section 2]. Let ${\frak r} \in \mathbb {R}^m$ , $\theta>\pi /2 $ and $k \in [0,\infty )^m$ , and put

$$ \begin{align*} S^k({\frak r}, \theta) &:= \left\{w \in H({\frak r}):\ k \cdot \|\operatorname{\mathrm{Im}} w\| < \theta\right\} &(log\text{-}k\text{-}polysector). \end{align*} $$

Note that if $m=1$ , then $S^k({\frak r}, \theta ) = S(0,{\frak r}, \theta /k)$ , where the latter is the sector defined in Section 2.1. The reason for allowing $k_i=0$ is that we need our class of log-polysectors to be closed under taking Cartesian products with log-disks; for instance, if $m>1$ and $k = (k', 0)$ with $k' \in [0,\infty )^{m-1}$ , then $S^k({\frak r}, \theta ) = S^{k'}({\frak r}', \theta ) \times H({\frak r}_m)$ . Finally, our polystrips are “in the real multidirection”; they can easily be defined in any multidirection,Footnote ¹ but we shall not do this here as we do not need to for our purposes.

Next, for $p \in \mathbb {N}$ we put, by adapting [Reference Van den Dries and Speissegger12, Section 2] to the logarithmic chart,

$$ \begin{align*} H_p^k({\frak r}) &:= \left\{w \in H({\frak r}):\ k \cdot \operatorname{\mathrm{Re}} w < k \cdot {\frak r} - \log(1+p))\right\} &(\textit{log}\text{-}k{\text{-}\textit{polydisk}}), \\ S_p^{k}({\frak r}, \theta) &:= S^k({\frak r}, \theta) \cup H_p^k({\frak r}). \end{align*} $$

Note that

$$ \begin{align*}\exp\left(H^k_p({\frak r})\right) = D^k_p\left(e^{\frak r}\right) = \left\{z \in D\left(e^{\frak r}\right):\ \|z\|^k < \frac{e^{k\cdot {\frak r}}}{1+p}\right\},\end{align*} $$

corresponding to [Reference Van den Dries and Speissegger12, Definition 2.1]. Thus, for a nonempty finite $K \subseteq [0,\infty )^m$ , we set

$$ \begin{align*}S^K({\frak r},\theta):= \bigcap_{k \in K} S^k({\frak r},\theta),\end{align*} $$

and for $p \in \mathbb {N}$ ,

$$ \begin{align*}S^K_p({\frak r},\theta):= \bigcap_{k \in K} S^k_p({\frak r},\theta).\end{align*} $$

Note that if $z \in S_p^K({\frak r}, \theta )$ and $t \le 0$ , then $t+z \in S_p^K({\frak r}, \theta )$ ; in particular, $S_p^K({\frak r}, \theta )$ is connected.

3.3 Generalized multisummable functions

We now fix $R \in (0,\infty )^m$ , $r>1$ , $\theta> \pi /2$ , a nonempty finite $K \subseteq [0,\infty )^m$ and a natural $\Delta \subseteq [0,\infty )^m$ that is closed under addition, and we set $\tau := (K,R,r,\theta ,\Delta )$ . In this situation, we write

$$ \begin{align*}S^{\tau}:= S^K(\log R,\theta) \quad\text{and}\quad S^{\tau}_p:= S^K_p(\log R,\theta).\end{align*} $$

We need to introduce the following norms for generalized power series: let $U \subseteq \mathbb {C}^m$ be an open neighborhood of the origin such that $|z| \in U$ for every $z \in U$ . For a generalized power series $F \in \mathbb {C}\left \{X^*\right \}$ , we set

$$ \begin{align*}\|F\|_{U} := \sup\left\{\|F\|_s:\ s \in \operatorname{\mathrm{cl}}(U) \cap (0,\infty)^n\right\}.\end{align*} $$

It follows from the previous section that, if $\|F\|_{U} < \infty $ , then F is convergent and the log-sum of F extends to a log-holomorphic function $\overline F: U \longrightarrow \mathbb {C}$ such that $\left \|\overline F\right \|_U \le \|F\|_U$ , where $\left \|\overline F\right \|_U$ denotes the sup norm of $\overline F$ on U.

Similar to Section 2.4, we now define generalized multisummable functions in several variables. The role of the usual norm $\|\cdot \|_{\rho }$ on generalized power series there is taken on here by the norm $\|\cdot \|_U$ as defined above, where $U = D^k_p\left (R\right )$ ; note indeed that $z \in D^k_p\left (R\right )$ implies $|z| \in D^k_p\left (R\right )$ , as required. Below, we denote this particular norm $\|\cdot \|_U$ by $\|\cdot \|_{R,k,p}$ .

Thus, let $f_p:S^{\tau }_p \longrightarrow \mathbb {C}$ be log-holomorphic and bounded, and assume that there is a natural set $\Delta \subseteq [0,\infty )^m$ and, for each p, a convergent generalized power series $T(f_p)(X) \in \mathbb {C}\left \{X^*\right \}$ with support contained in $\Delta $ such that $\|T(f_p)\|_{R,k,p} < \infty $ and

$$ \begin{align*}f_p(w) = \overline{ T(f_p)}(w) \quad\text{for } w \in H^k_p\left(\log R\right).\end{align*} $$

Assuming that

$$ \begin{align*}\sum_p \|T(f_p)\|_{R,k,p}\ r^p < \infty \quad\text{and}\quad \sum_p \|f_p\|_{S^{\tau}_p}\ r^p < \infty,\end{align*} $$

we have that $\sum _p f_p$ converges uniformly on $S^{\tau }$ to a log-holomorphic function $f:S^{\tau } \longrightarrow \mathbb {C}$ . As before, we abbreviate this situation by writing

$$ \begin{align*}f =_{\tau} \sum_p f_p,\end{align*} $$

and we set

$$ \begin{align*}\|f\|_{\tau}:= \inf\left\{\max\left\{\sum_p \|F_p\|_{R,k,p}\ r^p,\ \sum_p \|f_p\|_{S^{\tau}_p}\ r^p\right\}:\ f =_{\tau} \sum_p f_p\right\}.\end{align*} $$

Thus, as in Section 2.4, we obtain a Banach algebra

$$ \begin{align*}\mathcal{G}_{\tau}:= \left\{f: S^{\tau} \longrightarrow \mathbb{C}:\ \text{there exist } f_p \text{ such that } f =_{\tau} \sum_p f_p\right\}\end{align*} $$

over $\mathbb {C}$ of functions in m variables with norm $\|\cdot \|_{\tau }$ . Note that if $m=1$ , $\mathcal {G}_{\tau }$ as defined here is the same as $\mathcal {G}_{\tau '}$ defined in Section 2.4, where $\tau '=(K',R,r,\theta ,\Delta )$ with $K'$ obtained from K by replacing all nonzero $k\in K$ by $1/k$ (see the introductory remarks at the beginning of Section 3).

Arguing as in the one-variable case, if $T(f_p)(X) = \sum a_{\alpha ,p} X^{\alpha }$ for each p, we obtain a generalized power series $Tf(X) = \sum a_{\alpha } X^{\alpha }$ , where $a_{\alpha } = \sum _p a_{\alpha ,p}$ for each $\alpha $ .

3.4 Quasianalyticity

In view of proving our o-minimality result in Section 5, we show in this section that the map $T:\mathcal {G}_{\tau } \longrightarrow \mathbb {C}\left [\!\left [X^*\right ]\!\right ]$ is injective. First, we explain the reason why we define the log-k-polysectors and the log-k-polydisks via scalar products rather than just taking Cartesian products of the corresponding objects in one variable. This is done because scalar product behaves well with respect to fibers, as shown in Remark 3.2.

We set $\mu _{K,i} := \min \left \{k_i:\ k \in K\right \}$ for $i=1, \dots , m$ , $\mu _K:= (\mu _{K,1}, \dots , \mu _{K,m})$ and

$$ \begin{align*}\rho^{\tau}_p := \left(\frac{R_1}{(1+p)^{1/\mu_{K,1}}}, \dots, \frac{R_m}{(1+p)^{1/\mu_{K,m}}}\right).\end{align*} $$

The next lemma shows that, although the log-k-polydisks are not themselves log-disks, the set $S^{\tau }_p$ contains a suitable log-disk.

Proof The first and third inclusions are straightforward. For the second, let $k,l \in (0,\infty )^m$ be such that $l \le k$ , $p \in \mathbb {N}$ and $\rho \in \mathbb {R}$ ; it suffices to show that $H^k_p(\rho ) \subseteq H^k_p(\rho )$ . To see this, let $w \in H^k_p(\rho )$ . Then

$$ \begin{align*}k \cdot \operatorname{\mathrm{Re}} w = l \cdot \operatorname{\mathrm{Re}} w + (k-l) \cdot \operatorname{\mathrm{Re}} w < l \cdot \rho + (k-l) \cdot \operatorname{\mathrm{Re}} w - \log(1+p).\end{align*} $$

Since $\operatorname {\mathrm {Re}} w < \rho $ as well, it follows that $k \cdot \operatorname {\mathrm {Re}} w < l \cdot \rho - \log (1+p)$ , as required.

For the rest of this section, we set $x':= (x_1, \dots , x_{m-1})$ .

Remark 3.2 Let $a \in \overline {\mathbb {C}}^{m-1}$ . Set $\theta (a):= \theta - \max \{k' \cdot \|\operatorname {\mathrm {Im}} a\|:\ k \in K\}$ and

$$ \begin{align*}\tau(a):= (\{k_m:\ k \in K\},R_m,r,\theta(a), \Pi_m(\Delta)),\end{align*} $$

where $\Pi _m:\mathbb {R}^m \longrightarrow \mathbb {R}$ is the projection on the last coordinate. Note that if $|\operatorname {\mathrm {Im}} a|$ is sufficiently small, then $\theta (a)> \pi /2$ .

If $\theta (a)> 0$ , then $S^{\tau (a)}$ is contained in the fiber $(S^{\tau })_a$ of $S^{\tau }$ over a. Moreover, if $\operatorname {\mathrm {Re}} a < R'$ then, for each $p \in \mathbb {N}$ , the set $H^{k_m}_p\left (\log R_m\right )$ is contained in the fiber of the set $H^k_p(\log R)$ over a. Therefore, if $\theta (a)> 0$ and $\operatorname {\mathrm {Re}} a < \log R'$ , then $S^{\tau (a)}_p$ is contained in the fiber $\left (S^{\tau }_p\right )_a$ , for each p.

Proof Choose $f_p$ , for $p \in \mathbb {N}$ , such that $f =_{\tau } \sum f_p$ . For $p \in \mathbb {N}$ , let $f_{a,p}:S^{\tau (a)}_p \longrightarrow \mathbb {C}$ be defined by $f_{a,p}(w):= f_p(a,w)$ ; these functions are well defined by Remark 3.2, and $f_a(w) = \sum _p f_{a,p}(w)$ for every $w \in S^{\tau (a)}$ .

For each p, we set $F_{a,p}(X_m):= T(f_p)(e^a,X_m)$ , a generalized power series in the indeterminate $X_{m}$ . Since $\|T(f_p)\|_{R,k,p} < \infty $ and since, by Remark 3.2, the polyradius $\rho (b):= \left ( |e^a|,b \right )$ belongs to $\operatorname {\mathrm {cl}}\left (D^k_p\left (R\right )\right )$ for every $b \in \operatorname {\mathrm {cl}}\left (D^{k_m}_p\left ({R_m}\right )\right )$ , we have that

$$ \begin{align*}\|F_{a,p}\|_b = \|T(f_p)\|_{\rho(b)} \le \|T(f_p)\|_{R,k,p};\end{align*} $$

in particular, $\|F_{a,p}\|_{R_m,k_m,p} \le \|T(f_p)\|_{R,k,p}$ and $f_{a,p} = \overline {F_{a,p}}$ , for each p. Therefore, $f_a =_{\tau (a)} \sum f_{a,p}$ , that is, $f_a \in \mathcal {G}_{\tau (a)}$ . Since the inequality $\sum \|F_{a,p}\|_{R_m,k_m,p}\ r^p \le \sum \|T(f_p)\|_{R,k,p}\ r^p$ holds for all choices of $\{f_p\}$ , we also get $\|f_a\|_{\tau (a)} \le \|f\|_{\tau }$ .

Claim The series $T(f)(e^a,X_m)$ belongs to $\mathbb {R}\left [\!\left [X_m^*\right ]\!\right ]$ and is equal to $T(f_a)(X_m)$ .

To see this claim, since $\sum \|T(f_p)\|_{R,k,p}\ r^p < \infty $ and $\left (|e^a|,\rho ^{\tau (a)}_p\right ) \in \operatorname {\mathrm {cl}}\left (D^k_p\left (R\right )\right )$ for each p, it follows from Lemma 3.1 that for each $\alpha _m \in [0,\infty )$ and each p,

$$ \begin{align*} \sum_{\alpha' \in [0,\infty)^{m-1}} |a_{(\alpha',\alpha_m),p}|\ |e^{\alpha'\cdot a}| &\le \frac{\|Tf_p\|_{R,k,p}}{\left(\rho_p^{\tau(a)}\right)^{\alpha_m}} \\ &= \frac{\|Tf_p\|_{R,k,p}}{R_m^{\alpha_m}}\ (1+p)^{\alpha_m/k_m} \\ &\le \frac{\|Tf_p\|_{R,k,p}}{R_m^{\alpha_m}}\ s^p\ (1+p)^{\alpha_m/k_m}, \end{align*} $$

for any $s> 1$ . In particular, for $s \in (1,r)$ we have, for each $\alpha _m \in [0,\infty )$ , that

(3.1) $$ \begin{align} \qquad\sum_p \left(\sum_{\alpha'}|a_{(\alpha',\alpha_m),p}|\ |e^{\alpha'\cdot a}|\right) \le \frac1{R_m^{\alpha_m}} \sum_p \|Tf_p\|_{R,k,p}\ r^p \left(\frac sr\right)^p (1+p)^{\alpha_m/k_m} < \infty, \end{align} $$

which proves that $T(f)(e^a,X_m)$ belongs to $\mathbb {R}\left [\!\left [X_m^*\right ]\!\right ]$ . Moreover, since $f_a \in \mathcal {G}_{\tau (a)}$ , we have

$$ \begin{align*}T(f_a)(X_m) = \sum_p T(f_{a,p})(X_m) = \sum_p F_{a,p}(X_m).\end{align*} $$

It follows from (3.1) that for all $\alpha _m$ ,

(3.2) $$ \begin{align} \sum_{\alpha'}a_{(\alpha',\alpha_m)}e^{\alpha'\cdot a}=\sum_{\alpha'}\left(\sum_p a_{(\alpha',\alpha_m),p}\right)e^{\alpha'\cdot a}=\sum_p \sum_{\alpha'}a_{(\alpha',\alpha_m),p}e^{\alpha'\cdot a}. \end{align} $$

Hence, $T(f)(e^a,X_m) = \sum _p F_{a,p}(X_m) = T(f_a)(X_m)$ , as claimed.

As in [Reference Van den Dries and Speissegger12, Corollary 2.19], we now obtain the following corollary.

3.5 Monomial division

Let $F = \sum _{a \in [0,\infty )^m} a_{\alpha } X^{\alpha } \in \mathbb {C}\left [\!\left [X^*\right ]\!\right ]$ . Recall from [Reference Van den Dries and Speissegger11, Section 4] that

$$ \begin{align*}\operatorname{\mathrm{ord}}(F)= \begin{cases} \min\{|\alpha|:\ a_{\alpha} \ne 0\}, &\text{if } f \ne 0, \\ \infty, &\text{if } f = 0. \end{cases}\end{align*} $$

For $i \in \{1, \dots , m\}$ , we also consider F as an element of $\mathbb {C}\left [\!\left [X_1^*, \dots , X_{i-1}^*, X_{i+1}^*, \dots , X_m^*\right ]\!\right ]\left [\!\left [X_i^*\right ]\!\right ]$ , and we denote by $\operatorname {\mathrm {ord}}_i(F)$ the corresponding order of F in the indeterminate $X_i$ . Note that $\operatorname {\mathrm {ord}}_i(F)>0$ implies $\operatorname {\mathrm {ord}}(F)>0$ , for each i.

Lemma 3.6 Let $f \in \mathcal {G}_{\tau }$ and assume that $\gamma := \operatorname {\mathrm {ord}}_m(Tf)> 0$ . Let also $s \in (1,r)$ and set $\tau ':= (K,R,s,\theta ,\Delta )$ . Then there exist $g \in \mathcal {G}_{\tau '}$ and $C = C(s/r)> 0$ depending only on $\frac sr$ such that $\|g\|_{\tau '} \le C \|f\|_{\tau '}$ and

$$ \begin{align*}f(w) = e^{\gamma w_m} g(w) \quad\text{for } w \in S^{\tau'}.\end{align*} $$

Proof For simplicity, we write $\rho _p = \rho _p^{\tau } = \rho _p^{\tau '}$ and $S_p = S_p^{\tau } = S_p^{\tau '}$ , for each p; recall that $\rho _p \in \operatorname {\mathrm {cl}}\left (D^k_p(R)\right )$ for each p. Say $f =_{\tau } \sum _p f_p$ with $\sum _p \|Tf_p\|_{R,k,p}\ r^p \le 2\|f\|_{\tau }$ and $\sum _p \|f_p\|_{S_p}\ r^p \le 2\|f\|_{\tau }$ ; since each $Tf_p$ is convergent we may assume, after replacing each $f_p$ by $f_p - \overline {(Tf_p)_{\gamma }}$ if necessary,Footnote ² that $\operatorname {\mathrm {ord}}_m(Tf_p) \ge \gamma $ for each p. So there are $G_p \in \mathbb {C}\left [\!\left [X^*\right ]\!\right ]$ with support contained in $\operatorname {\mathrm {supp}}(Tf_p)$ (and hence natural) such that $Tf_p = X_m^{\gamma } G_p$ ; since $\|Tf_p\|_{\rho _p} = \rho _{p,m}^{\gamma } \|G_p\|_{\rho _p}$ , it follows that $G_p$ has radius of convergence at least $\rho _p$ , and that

$$ \begin{align*}f_p(w) = e^{\gamma w_m} g_p(w) \quad\text{for } w \in H(\log\rho_p),\end{align*} $$

where $g_p:= \overline {G_p}$ for each p. We extend $g_p$ to all of $S_p$ by setting $g_p(w):= f_p(w)/e^{\gamma w_m}$ for $w \in S_p\setminus H(\log \rho _p)$ .

Since $\|Tf_p\|_{\rho _p} = \rho _{p,m}^{\gamma } \|Tg_p\|_{\rho _p}$ , for each p, we get

$$ \begin{align*} \sum_p \|Tg_p\|_{\rho_p} s^p = \sum_p \frac{\|Tf_p\|_{\rho_p}}{\rho_{p,m}^{\gamma}} s^p \le \frac1{R_m^{\gamma}} \sum_p \|Tf_p\|_{\rho_p} r^p (1+p)^{k_m} \left(\frac sr\right)^p \le C \|g\|_{\tau} \end{align*} $$

for some $C = C(s/r)>0$ depending only on $\frac sr$ . It follows, on the one hand, that

$$ \begin{align*}\sum_p \|g_p\|_{H(\log \rho_p)} s^p \le C \|g\|_{\tau}\end{align*} $$

as well. On the other hand, since $\|Tg_p\|_{S_p \setminus H(\log \rho _p)} \le \rho _{p,m}^{-\gamma }\|Tf_p\|_{S_p\setminus H(\log \rho _p)}$ , the same argument as above also proves that $\sum _p \|g_p\|_{S_p \setminus H(\log \rho _p)} s^p \le C \|g\|_{\tau }$ . Therefore, the function $g:S^{\tau '} \longrightarrow \mathbb {C}$ defined by $g:= \sum _p g_p$ belongs to $\mathcal {G}_{\tau '}$ and satisfies $\|g\|_{\tau '} \le C \|g\|_{\tau }$ and $f(w) = e^{\gamma w_m} g(w)$ for $w \in S^{\tau '}$ , as claimed.

3.6 Generalized multisummable germs

Similar to Section 2 of [Reference Van den Dries and Speissegger12], we let $(\mathcal {T}_m, \le )$ be the directed set of all tuples $\tau = (K,R,r,\theta , \Delta )$ as above, where $\tau = (K,R,r,\theta ,\Delta ) \le \tau ' = (K',R',r',\theta ',\Delta ')$ if and only if $K \supseteq K'$ , $R \le R'$ , $r \le r'$ , $\theta \le \theta '$ and $\Delta \supseteq \Delta '$ . Then $S^{\tau '} \subseteq S^{\tau }$ whenever $\tau ' \le \tau $ and in this situation, for $f \in \mathcal {G}_{\tau }$ , the restriction $f\!\!\upharpoonright _{S^{\tau '}}$ belongs to $\mathcal {G}_{\tau '}$ . The directed limit of the directed system $\left (\mathcal {G}_{\tau }:\ \tau \in \mathcal {T}_m\right )$ under these restrictions is the set $\mathcal {G}_m$ of germs at $-\infty $ of functions in $\mathcal {G}_{\tau }$ , as $\tau $ ranges over $\mathcal {T}_m$ . This $\mathcal {G}_m$ is a $\mathbb {C}$ -algebra containing the germs at $-\infty $ of the functions $e^{\gamma z_i}$ , for $i=1, \dots , m$ and $\gamma \ge 0$ , and we extend each norm $\|\cdot \|_{\tau }$ to all of $\mathcal {G}_m$ by setting $\|f\|_{\tau } := \infty $ whenever $g \notin \mathcal {G}_{\tau }$ .

Below, we write $-\infty :=(-\infty , \dots , -\infty )$ . Note that $f(-\infty ) = (Tf)(0)$ , since $Tf$ is the asymptotic expansion of f at $-\infty $ ; hence $f(-\infty ) = 0$ if and only if $\operatorname {\mathrm {ord}}(Tf)>0$ .

Similar to [Reference Van den Dries and Speissegger12, Section 3], we set $\mathbb {C}\left \{X^*\right \}_{\tau } := T(\mathcal {G}_{\tau })$ , for $\tau \in \mathcal {T}_m$ , and $\mathbb {C}\left \{X^*\right \}_{\mathcal {G}} := T(\mathcal {G}_m)$ . We refer to the latter as the $\mathbb {C}$ -algebra of all generalized multisummable series in the indeterminates X; note that $\mathbb {C}\left \{X^*\right \}_{\mathcal {G}} = \bigcup _{\tau \in \mathcal {T}_m} \mathbb {C}\left \{X^*\right \}_{\tau }$ . We make each $\mathbb {C}\left \{X^*\right \}_{\tau }$ into an isomorphic copy of $\mathcal {G}_{\tau }$ by setting $\|Tf\|_{\tau } := \|f\|_{\tau }$ , for $f \in \mathcal {G}_{\tau }$ ; and we extend each norm $\|\cdot \|_{\tau }$ to all of $\mathbb {C}\left \{X^*\right \}_{\mathcal {G}}$ by setting $\|F\|_{\tau }:= \infty $ for $F \notin \mathbb {C}\left \{X^*\right \}_{\tau }$ .

Extending our notation of log-sum of convergent generalized power series, we also shall write $\overline F:= T^{-1}(F)$ , for $F \in \mathbb {C}\left \{X^*\right \}_{\tau }$ .

3.7 Mixed series

We now consider additional indeterminates $Y = (Y_1, \dots , Y_n)$ , and we defined mixed series similar to [Reference Van den Dries and Speissegger12, Section 3]. Thus, for $\tau \in \mathcal {T}_m$ and $\rho \in (0,\infty )^n$ , and for $F = \sum _{\beta \in \mathbb {N}^n} F_{\beta }(X) Y^{\beta } \in \mathbb {C}\left \{X^*\right \}_{\tau } [\![Y]\!]$ , we set

$$ \begin{align*} \|F\|_{\tau,\rho}:= \sum_{\beta} \|F_{\beta}\|_{\tau} \rho^{\beta} \end{align*} $$

and

$$ \begin{align*} \mathbb{C}\left\{X^*;Y\right\}_{\tau,\rho} := \mathbb{C}\left\{X^*\right\}_{\tau} \left\{Y\right\}_{\rho} = \left\{F \in \mathbb{C}\left\{X^*\right\}_{\tau} [\![Y]\!]:\ \|F\|_{\tau,\rho} < \infty\right\}. \end{align*} $$

The latter is a Banach $\mathbb {C}$ -algebra with respect to the former norm. We sometimes refer to the indeterminates $X_i$ as the generalized Gevrey variables and to the indeterminates $Y_j$ as the convergent variables. Each $F = \sum _{\beta } F_{\beta }(X)Y^{\beta } \in \mathbb {C}\left \{X^*;Y\right \}_{\tau ,\rho }$ defines a log-holomorphic function $\overline F:S^{\tau } \times D(\rho ) \longrightarrow \mathbb {C}$ by setting

$$ \begin{align*}\overline F(w,y):= \sum_{\beta} \overline{F_{\beta}}(w)y^{\beta}. \end{align*} $$

As in Section 3.6, we set

$$ \begin{align*}\mathbb{C}\left\{X^*;Y\right\}_{\mathcal{G}} := \bigcup_{\tau \in \mathcal{T}_m, \rho \in (0,\infty)^n} \mathbb{C}\left\{X^*;Y\right\}_{\tau,\rho},\end{align*} $$

and we extend each norm $\|\cdot \|_{\tau ,\rho }$ to all of $\mathbb {C}\left \{X^*;Y\right \}_{\mathcal {G}}$ by setting $\|F\|_{\tau ,\rho }:= \infty $ whenever $F \notin \mathbb {C}\left \{X^*;Y\right \}_{\tau ,\rho }$ . Ordering the product $\mathcal {T}_m \times (0,\infty )^n$ by the product order, we obtain the following generalization of Lemma 3.7.

Recall that $F \in \mathbb {C}\left [\!\left [X^*;Y\right ]\!\right ]$ is regular in $Y_n$ of order d if $F(0,0,Y_n) = uY_n +$ terms in $Y_n$ of higher degree, with $u \in \mathbb {C}$ nonzero. Using Lemma 3.8, the proof of [Reference Tougeron9, Proposition 4.1] now establishes the following proposition, where $Y' = (Y_1, \dots , Y_{n-1})$ .

4.1 Permutations

Let $\sigma $ be a permutation of $\{1, \dots , m\}$ and, for $x \in \overline {\mathbb {C}}^m$ , we denote by $\sigma (x)$ the tuple $ \left (x_{\sigma (1)}, \dots , x_{\sigma (m)}\right )$ . For $\tau \in \mathcal {T}_m$ , we set

$$ \begin{align*}\sigma(\tau):= \left(K,\sigma(R),r,\theta,\sigma(\Delta)\right);\end{align*} $$

note that $\sigma :\mathcal {T}_m \longrightarrow \mathcal {T}_m$ is a bijection.

4.2 Blow-up charts

We let $1 \le j < i \le m$ and $\lambda> 0$ and consider the regular blow-up chart $\pi ^{\lambda }_{i,j}$ ; the singular blow-up charts are handled similarly, but are actually easier to deal with and are left to the reader. Permuting the Gevrey variables if necessary, we assume that $j = m-1$ and $i = m$ ; to simplify notation, we write $\sigma = \pi ^{\lambda }_{m,m-1}$ .

Fix $\tau \in \mathcal {T}_m$ and $\rho \in (0,\infty )^n$ . Similar to [Reference Van den Dries and Speissegger12, Lemma 4.7], we now choose $\theta ' \in (\pi /2,\theta )$ and $\rho _0> 0$ such that $k_m |\arg (\lambda + v)| < \theta - \theta '$ for all $v \in D(2\rho _0)$ and all $k \in K$ . We set $k':= (k_1, \dots , k_{m-2}, k_{m-1}+k_m)$ for $k \in K$ , $l:= \max \{k_m:\ k \in K\}$ and $R_{m-1}':= \min \left \{R_{m-1}, R_m, \frac {R_m}{(\lambda + 2\rho _0)^l}\right \}$ , as well as

$$ \begin{align*}K':= \left\{k':\ k \in K\right\} \quad\text{and}\quad R':= (R_1, \dots, R_{m-2}, R_{m-1}'),\end{align*} $$

$\Delta ':= \Pi _{m-1}(\Delta )$ and, finally, $\tau ':= (K',R',r,\theta ',\Delta ')$ and $\rho ':= (\rho _0,\rho )$ . By Claims 1 and 2 in the proof of [Reference Van den Dries and Speissegger12, Lemma 4.7], we have $\widetilde \sigma \left (S^{\tau '} \times D(\rho ')\right ) \subseteq S^{\tau } \times D(\rho )$ and, for each p, that $\widetilde \sigma \left (S^{\tau '}_p \times D(\rho ')\right ) \subseteq S^{\tau }_p \times D(\rho )$ .

Proof Let $F \in \mathbb {C}\left \{X^*,Y\right \}_{\tau ,\rho }$ ; again, we distinguish two cases.

Case 1: $n=0$ . Then $F \in \mathbb {C}\left \{X^*\right \}_{\tau }$ , and we choose convergent $F_p \in \mathbb {C}\left \{X^*\right \}$ such that $\overline F =_{\tau } \sum _p \overline {F_p}$ . As in the proof of [Reference Van den Dries and Speissegger12, Lemma 4.7], for each $p, \nu \in \mathbb {N}$ , we define $f_{p,\nu }:S^{\tau '}_p \longrightarrow \mathbb {C}$ and $f_{\nu }:S^{\tau '} \longrightarrow \mathbb {C}$ by

$$ \begin{align*}f_{p,\nu} (w') := \frac 1{\nu!} \frac {\partial^{\nu} \left(\overline{F_p} \circ \widetilde\sigma\right)} {\partial v^{\nu}} (w', 0) \quad\text{and}\quad f_{\nu} (w') := \frac 1{\nu!} \frac {\partial^{\nu} \left(\overline F \circ \widetilde\sigma\right)} {\partial v^{\nu}} (w', 0).\end{align*} $$

The argument there shows that, for each $\nu $ , we have $f_{\nu } = \sum _p f_{p,\nu }$ on $S^{\tau '}$ with $\|f_{\nu }\|_{S^{\tau '}} \le \left \|\overline F\right \|_{S^{\tau }}/(2\rho _0)^{\nu }$ . Hence, $\sum _{\nu } \|f_{\nu }\|_{S^{\tau '}} \rho _0^{\nu } \le \left \|\overline F\right \|_{S^{\tau }}$ , and it follows from Taylor’s theorem that

$$ \begin{align*}\left(\overline F \circ \widetilde\sigma\right)(w',y) = \sum_{\nu} f_{\nu}(w') y^{\nu} \quad\text{ on } S^{\tau'} \times D(\rho_0).\end{align*} $$

So it remains to show that each $f_{\nu }$ belongs to $\mathcal {G}_{\tau '}$ ; to do so, we define $k'$ for $K'$ as k was defined for K, and we establish the following claim.

Claim Each $f_{p,\nu }$ is given by a generalized power series $F_{p,\nu }$ with support in $\Delta '$ such that $\|F_{p,\nu }\|_{R',k',p} \le C \cdot \|F_p\|_{R,k,p}/(2\rho _0)^{\nu }$ , where $C>0$ is independent of F, p or $\nu $ .

To see the claim, we use [Reference Van den Dries and Speissegger11, Lemma 6.5]—or more precisely, the following modification of it: the stated hypotheses there, namely, that $\tau \le \rho $ , $\tau _m < \lambda $ and $\tau _{m-1}^{\gamma }(\lambda + \tau _m) < \rho _m$ , were sufficient for the purposes of that paper, but not quite necessary to obtain the same conclusion from the proof of that lemma. Indeed, it suffices to assume that $\tau _i \le \rho _i$ for all $i \ne m$ , that $\tau _m < \lambda $ and that $\tau _{m-1}^{\gamma }(\lambda + \tau _m) \le \rho _m$ to obtain the same conclusion, and we shall verify these weaker hypotheses below (with $\gamma = 1)$ in order to apply that lemma here, without further mention of this discrepancy.

On the one hand, by Taylor’s theorem, we have for each p that

$$ \begin{align*}\left(\overline{F_p} \circ \widetilde\sigma\right)(w',y) = \sum_{\nu} f_{p,\nu}(w') y^{\nu} \quad\text{on } S^{\tau'}_p \times D(\rho_0).\end{align*} $$

On the other hand, using the binomial formula, we have for each p that

$$ \begin{align*}\sigma F_p = \sum_{\nu} F_{p,\nu}(X') \cdot X_m^{\nu} \quad\text{with}\quad F_{p,\nu}(X') = \frac1{\nu!} \frac{\partial^{\nu} (\sigma F_p)}{\partial X_m^{\nu}}(X',0);\end{align*} $$

note that each $F_{p,\nu }$ has support contained in $\Delta '$ .

We now fix an arbitrary $s' \in \operatorname {\mathrm {cl}}\left (D^{k'}_p(\log R')\right ) \cap (0,\infty )^{m-1}$ and set $s:= (s',2\rho _0)$ and $t:= \sigma (s)$ . By Claim 2 of [Reference Van den Dries and Speissegger12, Lemma 4.7], we have $t \in \operatorname {\mathrm {cl}}\left (D^k_p(\log R)\right )$ . Since $s_i = t_i$ , for $i=1, \dots , m-1$ , $s_m = 2\rho _0 < \lambda $ , and $s_{m-1}(\lambda + s_m) = t_m$ , we get from [Reference Van den Dries and Speissegger11, Lemma 6.5] a constant $C\ge 1$ , independent of F, p, or $\nu $ , such that

$$ \begin{align*}\|\sigma F_p\|_s \le C \|F_p\|_t \le C \|F_p\|_{R,k,p}.\end{align*} $$

Since $\|\sigma F_p\|_s = \sum _{\nu } \|F_{p,\nu }\|_{s'} s_m^{\nu }$ by definition of $\|\cdot \|_s$ , it follows that

$$ \begin{align*}\|F_{p,\nu}\|_{s'} \le \frac C{(2\rho_0)^{\nu}} \|F_p\|_{R,k,p}, \quad\text{for each } p \text{ and } \nu.\end{align*} $$

Since $s' \in \operatorname {\mathrm {cl}}\left (D^{k'}_p(\log R')\right ) \cap (0,\infty )^{m-1}$ was arbitrary, we finally get

$$ \begin{align*}\|F_{p,\nu}\|_{R',k',p} \le \frac C{(2\rho_0)^{\nu}} \|F_p\|_{R,k,p}, \quad\text{for each } p \text{ and } \nu.\end{align*} $$

Finally, we get from [Reference Van den Dries and Speissegger11, Lemmas 5.9(2,3) and 6.3(4)] that $f_{p,\nu } = \overline {F_{p,\nu }}$ , for each p and $\nu $ . This finishes the proof of the claim.

It follows from the claim that $f_{\nu } =_{\tau '} \sum _p f_{p,\nu }$ . Moreover, since the claim holds for all sequences $F_p$ such that $\overline F =_{\tau } \sum _p \overline {F_p}$ , we also get that $\|f_{\nu }\|_{\tau '} \le C \cdot \|F\|_{\tau }/(2\rho _0)^{\nu }$ for each $\nu $ . It follows that $\|\sigma F\|_{\tau ',\rho _0} = \sum _{\nu } \|f_{\nu }\|_{\tau '} \rho _0^{\nu } \le C \|F\|_{\tau }$ , which finishes the proof of the proposition in Case 1.

Case 2: $n>0$ . Let $F = \sum _{\beta \in \mathbb {N}^n} F_{\beta }(X) Y^{\beta }$ . Then $\|\sigma F\|_{\tau ',\rho '} = \sum _{\beta \in \mathbb {N}^n} \|\sigma F_{\beta }\|_{\tau ',\rho _0}\ \rho ^{\beta }$ , so the proposition in Case 2 follows from Case 1 as in the proof of Lemma 4.2.