Holomorphic continuation of a formal series along analytic curves

This talk is devoted to a curvilinear analogue of the well-known Forelli theorem [1]: if a function $f$ is infinitelly smooth in a neighborhood of the origin $0 \in \mathbb C^{n} $ $f\in C^{\infty }\{0\},$ and for every complex line $l$ passing through the origin the restriction $f|_{l} $ continues holomorphically into the unit disk $l\bigcap B\left(0,1\right),$ then $f$ continues holomorphically into the unit ball $B\left(0,1\right) \subset {\mathbb C}^{n}.$
An example of a function
$$ f\left(z_{1} ,z_{2} \right)=\frac{z_{1}^{k+1} \bar{z}_{2} }{z_{1} \bar{z}_{1} +z_{2} \bar{z}_{2} } \in C^{k} \left({\mathbb C}^{2} \right) $$
shows that the condition of infinite smoothness in Forelli's Theorem is essential. The restrictions $f|_{l} $ to complex lines $l \ni 0$ are polynomials, but $f\left(z_{1} ,z_{2} \right)$ is not holomorphic.
The following takes place
Theorem 1. Let the unit ball $B(0,1) \subset\mathbb{C}^{n} $ be fibered by a smooth family of analytic curves $A_{\lambda } =\left\{z=p_{\lambda } \left(\xi \right)\right\}, \lambda \in \mathbb{P}^{n-1},$ at the point $0$, where $p_{\lambda } \left(\xi \right)=\left(p_{\lambda }^{1} \left(\xi \right),p_{\lambda }^{2} \left(\xi \right),...,p_{\lambda }^{n} \left(\xi \right)\right)$ is a holomorphic vector function in the unit disk $U= \left \{\left|\xi \right|<1 \right \}:$ $p_{\lambda } \left(\xi \right)=a_{1} \left(\lambda \right)\xi +a_{2} \left(\lambda \right)\xi ^{2} +..., a_{k} \left(\lambda \right)\in C^{1} \left(\mathbb{C}^{n}\right),\, \, k=1,2,...,\, \, \, \, B(0,1) =\bigcap _{\lambda }A_{\lambda }^{}.$ If a function $f\in C^{\infty }\{0\}$ has the property that each restriction $f|_{A_{\lambda } } ,\, \, \lambda \in \mathbb{P}^{n-1} ,$ that is defined in the neighborhood of $0,$ holomorphically continues to the whole $A_{\lambda },$ then $f$ continues holomorphically to $B(0,1).$
Theorem 1 in the following version is also true under a weaker requirements.
Theorem 2. Under the conditions of Theorem 1, if each restriction $f|_{A_{\lambda } } ,\, \, \lambda \in W \subset \mathbb{P}^{n-1} ,$ holomorphically continues to the whole $A_{\lambda },$ then $f$ continues holomorphically to the domain $\hat {O}=\left\{z\in {\mathbb C}^{n} :\, \, \, \left| z\right| \exp V^{*}\left(\frac{z}{| z | } ,\, O\right)<1\right\}.$ Here $W \neq \emptyset$ is an open subset of $\mathbb{P}^{n-1},$ ${O} =\bigcap _{\lambda \in W }A_{\lambda }^{}, $ $V^{*}\left(\omega ,\, {O} \right)$ is the Green's function in ${\mathbb C}^{n}.$
In the work [2] Chirka showed the validity of the curvilinear analogue of Forelli’s theorem for $n=2$. Further advances on variations of the Forelli's theorem, were obtained in the works Kim et al. [3-5].