(Joint work with Tobias Kaiser)
I introduce real domains in $\LL$ and angular level, and I use these notions to describe holomorphic continuations on $\LL$ of one-variable functions definable in $\Ranexp$.
In this post “definable” means “definable in $\Ranexp$”.
Definition
A set $U \subseteq \LL$ is a real domain if there exist $a \gt 0$ and a continuous function $f = f_{U}:(a,+\infty) \into (0,+\infty)$ such that $$U = \set{z \in H_\LL(a):\ \left|\arg z\right| \lt f\left(|z|\right)}.$$ In this situation, we also write $U = U_f$.
Remarks
- The real domain $U_f$ is definable if and only if $f$ is definable.
- Every real domain is simply connected.
Examples
- For $a\gt 0$, the sector $S_\LL(a)$ is a definable real domain, while the half-plane $H_\LL(a)$ is not a real domain (but is definable).
- Of special interest to us are the standard power domains $U^\epsilon_C = (\pi_0)^{-1}\left(\Omega^\epsilon_C\right)$, where $$\Omega^\epsilon_C := \set{z + C (1+z)^\epsilon:\ \re z \gt 0}$$ for some $C \gt 0$ and $\epsilon \in (0,1)$.
Indeed, these standard power domains are examples of the following type of real domains:
Definition
A set $U \subseteq \LL$ is a strictly real domain if there exists a continuous function $f:(a,+\infty) \into (0,+\infty)$ such that $U = (\pi_0)^{-1}(\Omega)$ with $$\Omega = \set{z \in \CC:\ |\im z| \lt f(\re z)}.$$
Below, recall that $\H$ denotes the Hardy field of $\Ranexp$. Similarly to working with germs at $+\infty$ of definable functions, we work with germs at $\infty$ of domains in $\LL$.
Definition
Two sets $X,Y \subseteq \LL$ are called equivalent at $\infty$ if there exists $R>0$ such that $X \cap H_\LL(R) = Y \cap H_\LL(R)$. The corresponding equivalence classes are called the germs at $\infty$ of subsets of $\LL$.
Thus, for $h \in \H_{\gt 0}:= \set{h \in \H:\ h\gt 0}$, we denote by $U_h$ the germ at $\infty$ of any real domain $U_g$ such that $g$ is a representative of $f$.
Remark
If $f,g \in \H_{\gt 0}$ are such that $f \lt g$ and $U_g$ is strictly real, then $U_f$ is strictly real. So we set $$\Hsr := \set{h \in \H_{\gt 0}:\ U_h \text{ is strictly real}},$$ a downward closed subset of $\H_{\gt 0}$.
We are not only interested in the holomorphic continuations of definable functions per se, but also in the image of (definable) real domains under these continuations.
Lemma 1
By direct calculation, we obtain:
- For $r\gt 0$ and a definable real domain $U$, the images $\p_r(U)$ and $\m_r(U)$ are definable real domains.
- For a definable real domain $U$, the image $\llog(U)$ is a definable strictly real domain.
- For a definable strictly real domain $U$, the image $\eexp(U)$ is a definable real domain.
The lemma implies that there are functions $$\nu_{p_r}, \nu_{m_r}:\H_{\gt 0} \into \H_{\gt 0},$$ for $r\gt 0$, and $$\nu_{\log}:\H_{\gt 0} \into \Hsr$$ and $$\nu_{\exp}:\Hsr \into \H_{\gt 0}$$ such that, for $f \in \E$ where $$\E:= \{\log,\exp\} \cup \{p_r:\ r \gt 0\} \cup \{m_r:\ r \gt 0\},$$ and for all appropriate $h \in \H_{\gt 0}$, we have $$\f\left(U_h\right) = U_{\nu_{f}(h)}.$$
However, while $\nu_{p_r}$ and $\nu_{m_r}$ are easy to compute (exercise!), the function $\nu_{\log}$ is a bit harder to figure out (see below). Still, some basic tame calculus observations show the following:
Lemma 2
The maps $\nu_{f}$ are order-preserving bijections and, for $f,g \in \E$ and appropriate $h \in \H_{\gt 0}$, we have $$(\f \circ \g)(U_h) = U_{\nu_{f}(\nu_{g}(h))}.$$
Iterating this observation, we let $\E^\circ$ be the set of all finite words $f_1 \circ \cdots \circ f_n$ with each $f_i \in \E$, and we obtain the following:
Corollary
For each $f \in \E^\circ$, there exist downward closed subsets $\H_1(f), \H_2(f) \subseteq \H_{\gt 0}$ and an order-preserving bijection $\nu_{f}:\H_1(f) \into \H_2(f)$ such that $$\f(U_h) = U_{\nu_{f}(h)}$$ for $h \in \H_1(f)$. Moreover, for $f,g \in \E^\circ$, we have (as germs at $0^+$) $$\nu_{f \circ g} = \nu_{f} \circ \nu_{g}.$$
Remark
A similar statement, with the set $$\I := \set{f:\ \lim_{t \to +\infty} f(t) = +\infty}$$ of all infinitely increasing germs in $\H$ in place of $\E^\circ$, is false. Indeed, the continuation $\t_a$ of the translation $t_a$ maps certain definable real domains to domains that are neither real nor definable.
Angular level
Next, I introduce a rough measure of size for a real domain $U_f$, based on the level of its boundary function $f$.
Let $\I$ be the set of all infinitely increasing $\,f \in \H$ and set $\bo:= \H_{\gt 0} \setminus \I$.
By a theorem of Marker and Miller, every $f \in \I$ has level, that is, there exist $k,l \in \ZZ$ such that $$\log_l \circ f \sim \log_{l-k},$$ where $h_1 \sim h_2$ if and only if $h_1(t)/h_2(t) \to 1$ as $t \to +\infty$. In this situation, $k$ is unique and called the level of $\,f$, and $\log_s \circ f \sim \log_{s-k}$ for $s \ge l$.
We extend the level to all of $\H_{>0}$ as follows: we set $$\D := \set{1/f:\ f \in \I},$$ and for $f \in \D$, we set the $$\level(f):= \level(1/f).$$ Furthermore, for $\,f \in \H_{\gt 0} \setminus \{\I \cup \D\}$, we set $\level(f):= -\infty$.
Fact (Marker and Miller)
Let $\,f,g \in \H_{\gt 0}$.
- If $\,f,g \in \I$ and $f \le g$, then $\level(f) \le \level(g)$.
- If $\,f,g \in \bo$ and $f \le g$, then $\level(f) \ge \level(g)$.
- If $\,g \in \I$, then $\level(f \circ g) = \level(f) + \level(g)$.
- $\level(fg) \le \max\{\level(f),\level(g)\}$; equality holds whenever $\level(f) \ne \level(g)$.
Exercise
For $f \in \I$, show that $\level(f) = \eh(\lm(f)) \le \eh(f)$, where $\lm(f)$ denotes the leading monomial of the transseries $T(f)$.
Given $\,f \in \H_{\gt 0}$ and a word $w \in \E^\circ$, how do the levels of $f$ and of $\nu_w(f)$ compare?
Example
If $w = p_r$ or $w = m_r$, for $r>0$, then $\level(f) = \level(\nu_w(f))$.
Things get a little more interesting for $w = \log$:
Lemma
Let $f \in \D \cdot \log$. Then there exists $u \in \H_{>0}$ such that $u \sim 1$ and $$\nu_{\log}(f) \sim \frac{f}{\log} \circ e^{xu}.$$
Proof. Since $\frac f\log \in \D$ and $\arctan(x) -= x+o(x)$ as $x \to 0$, we get $$\theta:= \theta(f) \sim \frac f\log \quad\text{and}\quad \rho:= \rho(f) \sim \log.$$ So there exists $v \in \H_{>0}$ such that $v \sim 1$ and $\rho = \log \cdot v$. Composing oin the right with $\rho^{-1}$ (the compositional inverse of $\rho$) gives $$x = \left(\log \circ \rho^{-1}\right) \cdot \frac1u,$$ where $u \in \H_{>0}$ is such that $u \sim 1$. So $\rho^{-1} = e^{xu}$; it follows from the exercise that $$\nu_{\log}(f) = \theta \circ \rho^{-1} \sim \frac f\log \circ \rho^{-1} = \frac f\log \circ e^{xu},$$ as claimed. $\qed$
Corollary
For $\,f \in \H_{\gt 0}$, we have $$\level(\nu_{\log}(f)) \begin{cases} = \level(f/\log)+ 1 &\text{if } f \in \D\cdot \log, \\ -\infty &\text{otherwise.} \end{cases}$$
Proof. If $f > \D\cdot\log$, then $\arctan(f/\log) \in \H_{>0}\setminus\D$, so $\nu_{\log}(f) \in \bo\setminus\D$ by the exercise. $\qed$
In view of this corollary, we make the following
Definition
For $k \in \ZZ$, set $$\D_k:= \set{f \in \D:\ \level(f) = k}$$ and $$\D_{\ge k}:= \set{f \in \D:\ \level(f) \ge k}.$$ We also set $$\J:= \H_{>0} \setminus (\D_{\ge 1}\cdot \log).$$
Exercise
- $\D_{-1} \subseteq \D_{-1} \cdot \log$ and $\D_0 \cap \D_{-1}\cdot\log = \emptyset$; in particular, the sets $\J$, $\D_{-1}\cdot\log$ and $\D_{\ge 0}$ form a partition of $\H_{>0}$.
- $\nu_{\log}(\J) \subseteq \D_{-1}\cdot\log$.
- $\nu_{\log}(\D_{-1}\cdot\log) = \D_0$.
- $\nu_{\log}(\D_k) = \D_{k+1}$ for $k \in \NN$.
Based on this exercise, the angular level $\alevel:\H_{\gt 0} \into \{-1, 0, \dots \}$ is defined as $$\alevel(f):= \max\{-1,\level(\nu_{\log}(f)\}.$$
Combining the Fact and the Lemma, we obtain:
Proposition
The angular level is decreasing and, for $\,f \in \H_{\gt 0}$, we have
- $\alevel(\nu_{p_r}(f)) = \alevel(\nu_{m_r}(f)) = \alevel(f)$, for $r>0$;
- $\alevel(\nu_{\log}(f)) = \alevel(f) + 1$;
- if $\,f \in \Hsr$, then $\alevel(\nu_{\exp}(f)) = \alevel(f)\ – 1$. $\qed$
Corollary
Let $\,w \in \E^\circ$, and let $f$ be in the domain of $\nu_w$. Then $$\alevel(\nu_w(f)) = \alevel(f)\ – \level(w). \qed$$
The general continuation statement
In order to describe the continuation properties of general $f \in \H$, we need to relax the notion of (definable) real domain, while keeping in mind the scale provided by the angular level.
Definition
Let $\lambda \in \ZZ$, $U,V \subseteq \LL$ be domains and $\phi:U \into V$.
- $U$ is a $k$-domain if there exist $\,f,g \in \H_{>0}$ of angular level $k$ such that $U_f \subseteq U \subseteq U_g$.
- $\phi$ has angular level $\lambda$ if $U$ is an $l$-domain for some $l \ge -1$ and, for $k \ge \max\{\lambda -1,l\}$ and every $k$-domain $U’ \subseteq U$, the image $\phi(U’)$ is a $(k-\lambda)$-domain.
Examples
- $\m_r$ and $\p_r$ hacve angular level 0, $\eexp$ has angular level $-1$ and $\llog$ has angular level $1$.
- $\t_a$ has angular level $0$.
- Let $w \in \E^\circ$ of level $\lambda = \eh(w)$, and set $\eta:= \max\{\lambda,0\}$. Then, by the earlier observations, it follows that there exists an $(\eta-1)$-domain $U \subseteq \LL$ and an injective holomorphic continuation $\w:U \into \LL$ such that $\w$ has angular level $\lambda$.
It is this way of describing the extension properties of words in $\E^\circ$ that works for general definable $f$:
Continuation Theorem
Let $\,f \in \I$ and set $\eta:=\max\{\eh(f),0\}$ and $\lambda:= \level(f) \le \eta$. Then there exist an $(\eta-1)$-domain $U$, and $(\eta-\lambda-1)$-domain $V$ and a biholomorphic continuation $\f: U \into V$ of $\,f$ of angular level $\lambda$.
How optimal are these holomorphic extensions? Using the Uniqueness Principle and the Continuation Theorem, we obtain a converse of the latter:
Proposition
Let $\,f \in I$ and $\,\eta \in \NN$, and assume that $\,f$ has a holomorphic continuation $\,\f:U \into H_{\LL}(a)$, where $U \subseteq \LL$ is an $(\eta-1)$-domain and $a > 0$. Then $\eh(f) \le \eta$.
Corollary
Let $f \in \I$ and set $\eta:= \max\{\eh(f),0\}$ and $\lambda:= \level(f)$. Then $$-\lambda \le \eh(f^{-1}) \le \eta-\lambda.$$
Proof. Let $\f:U \into V$ be a biholomorphic continuation of $f$ of angular level $\lambda$ obtained from the Continuation Theorem. Then $\f^{-1}$ is a continuation of $f^{-1}$, and since $U \subseteq H_{\LL}(a)$ for some $a>0$ and $U$ is an $(\eta-\lambda-1)$-domain, the corollary follows from the proposition. $\qed$