Microlocal analysis of partial differential operators with irregular singularities

概要

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We denote the variables in $R^{n + 1}$ by $x=(x_0 ,x^')$, where $x_0 \in R$ and $x^' \in R^n$. We consider partial differential operators of the form $P(x,\partial /\partial x) = \sum\limits_{\left｜ \alpha \right｜ \leqslant m} {a_\alpha } (x)x_0^{\kappa (\left｜ \alpha \right｜)} (\partial /\partial x)^\alpha $, where $\kappa (j)$ is some integer $ \geqslant 0$, $a_\alpha (x)$ is real analytic in a neighborhood of $x=0$, and $a_{(m,0, \ldots ,0)}=1$. We define the irregularity $\iota \in \left[ {1,\infty )} \right.$ and the characteristic exponents $\lambda _1 , \ldots ,\lambda _{\kappa (m)} \in C$ of the operator P at the point $\mathop x\limits^ \circ ^* = (0;\sqrt { - 1} ,0, \ldots ,0) \in \sqrt { - 1} T*R^{n + 1}$. It will be proved that if $\iota >1$ and all the characteristic exponents of P are distinct, then P is equivalent microlocally to the operator $x_0^{k(m)} :C_{R^{n + 1} } \to C_{R^{n + 1} }$ $u \to x_0^{k(m)} u$ in a neighborhood of $\mathop x\limits^ \circ ^*$.
Faculty of Science, The University of Tokyoの論文
1983-12-10