# Real domains

(Joint work with Tobias Kaiser)

I introduce here the types of domains in $\LL$ used later to describe holomorphic extensions 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

1. The real domain $U_f$ is definable if and only if $f$ is definable.
2. Every real domain is simply connected.

Examples

1. 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).
2. 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 coinitial subinterval of $\H_{\gt 0}$.

We are not only interested in the holomorphic extensions of definable functions per se, but also in the image of (definable) real domains under these extensions.

Lemma 1
By direct calculation, we obtain:

1. For $r\gt 0$ and a definable real domain $U$, the images $\p_r(U)$ and $\m_r(U)$ are definable real domains.
2. For a definable real domain $U$, the image $\llog(U)$ is a definable strictly real domain.
3. 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 (more on that in another post). 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 coinitial subintervals $\H_1(f), \H_2(f) \subseteq \H_{\gt 0}$ 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 $$\nu_{f \circ g} = \nu_{f} \circ \nu_{g}.$$

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. The main difficulty here appears to be that addition does not extend to $\LL$.

In fact, by the Final Remark in this post, the extension $\t_a$ of the translation $t_a$ maps certain definable real domains to domains that are neither real nor definable.

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