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My question relates to the extensibility of a real analytic function of several variables to a specific complex domain.

In order to formulate the question, let me define the following complex domains:

$$\Delta_\theta := \{t = \rho e^{i\gamma} \in \mathbb{C} : \rho > 0 \text{ and } \gamma \in (-\theta,\theta)\}$$ (an open sector),

$$\Delta_\theta^{T} := \Delta_\theta \cap \{t\in \mathbb{C} : 0 < \text{Re}\ t < T\}$$ (an open triangle),

$$\Delta_\theta^{T',T} := \bigcup_{0\leq \xi \leq T-T'} (\xi + \Delta_\theta^{T'})$$ (an open pencil shape),

$$\mathcal{X}^{r} := \{z=x+iy \in \mathbb{C}^n : |y|_\infty < r\}$$ (a strip).

Suppose $f : [0,1]\times \mathbb{R}^n \mapsto \mathbb{R}$ and $f\in C¹(\overline{(0,T)\times\mathbb{R}^n})$ and $f$ is (jointly) real analytic on $(0,1)\times \mathbb{R}^n$.

Let $\Omega := \mathcal{X}^{r_0}\times \Delta^{T_0,T}_{\theta_0}$. Then I am interested in conditions on $f$ (e.g. on the coefficients of its power series expansion), considered as a real-valued function on $[0,1]\times \mathbb{R}^n$, which allow it to be extended to $\Omega$, such that:

(i) $f$ is holomorphic on $\Omega$,

(ii) $f$ is bounded on $\Omega$,

(iii) $f\in C^1(\overline{\Omega})$,

for some (arbitrary) $r_0\in(0,\infty)$, $0<T_0\leq T < \infty$ and $\theta_0\in (0,\pi/2)$.

I know that every real analytic function can be extended into a neighborhood of the real axis in the complex domain. However, I am unclear about conditions, which allow me to conclude that the extension actually includes $\Omega$ and further, that this extension is bounded and in $C^1(\overline{\Omega})$.

Are there any well known ways of approaching these type of extension problems and / or any literature recommendations?

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