On estimates of the fundamental solution of an elliptic equation with a small parameter

The behavior of a fundamental solution $\Gamma(x,y;\varepsilon)$ of the elliptic equation
$$ P\biggl(x,-i\varepsilon\,\frac\partial{\partial x}\biggr)u=0 $$
is studied for small $\varepsilon>0$ and fixed $x,y\in\mathbf R^n$. The main result is
$$ \varlimsup_{\varepsilon\to+0}\varepsilon\ln|\Gamma(x,y;\varepsilon)|\leqslant-\rho_P(x,y), $$
where $\rho_P(x,y)$ is the distance between the points $x$ and $y$ in a Finsler metric connected with the function $P(x,\xi)$.
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