Holomorphic extension of fundamental solutions of elliptic linear partial differential operators of higher order with analytic coefficients (Q1756322)

This paper deals with fundamental solutions of elliptic linear partial differential equation $$P(x,\partial)u= \sum_{|\alpha|\le m} a_\alpha(x)\frac {\partial^\alpha}{\partial x^\alpha}\,u= f$$ in an open set $\Omega\subset\mathbb{R}^n$ and with analytic coefficients. Thus, the distribution $E(x,y)\in D'(\Omega\times\Omega)$ satisfies the equation $PE=\delta(x-y)$, and the principal symbol $a(x,\xi)$ of $P$ is $\ne 0$ on $T^*(\Omega)\setminus 0$, $\frac{\partial}{\partial x_j}\to i\xi_j$. Therefore, the characteristic variety $\{a=0\}$ exists in $T^*(\mathbb{C}^n)$. The complex bicharacteristic associated to $P$ and passing through the point $(y,\xi)$ is defined as in the real case via the Hamiltonian vector field of $a$ and is contained in the complex characteristic variety. The bicharacteristic conoid $\Gamma_y$ is the union of all complex bicharacteristics with initial data $y\in\mathbb{C}^n$. \par This is the main result of the paper under consideration. \par Theorem. Let the elliptic operator $P$ possess simple complex characteristics. Fix $y\in\Omega$. Then there exists a neighborhood $V(y)$ of $y$ such that every fundamental solution $E(x,y)$ of $P$ can be extended (at least locally) in $V(y)\setminus\Gamma_y$ as a multivalued analytic function $x\to E(x,y)$ up to $\mathbb{C}^n\setminus\Gamma_y$. All fundamental solutions belong to the Nilsson class.\par Remark. Let $\Gamma$ be a complex analytic set of codimension in $\mathbb{C}^n$ and let $f$ be multivalued function in $X= \mathbb{C}^n\setminus\Gamma$. In this case $f$ belongs to the Nilsson class iff $f$ may be written as a finite sum of the following functions: $$\varphi_{\sigma,y}(z)\, x^{\sigma_1}_1,\dots, z^{\sigma_1}_n(\log z_1^{\nu_1},\dots, \dots(\log z_n)^{\nu_n},$$ $\sigma_j\in\mathbb{C}^1$, $\nu_j\in\mathbb{N}$ and $\varphi_{\sigma,\nu}(z)$ are meromorphic functions in $\mathbb{C}^n$.