Estimates of the fundamental solution of a parabolic equation

In this paper the behavior of the fundamental solution of the parabolic equation
$$ \frac{\partial u}{\partial t}+P\biggl(\mathbf x,\frac1i\frac\partial{\partial\mathbf x}\biggr)u=0,\qquad\mathbf x\in\mathbf R^n,\quad t>0, $$
as $t\to+0$ uniformly with respect to $\mathbf x$ is investigated. The basic result is of the form
$$ \varlimsup_{t\to+0}t^\frac1{2m-1}\ln|G(\mathbf x,\mathbf y,t)|\leqslant[\rho_P(\mathbf x,\mathbf y)]^\frac{2m}{2m-1}\cdot\sin\frac\pi{2(2m-1)}, $$
where $\rho_P(\mathbf x,\mathbf y)$ is the distance between $\mathbf x$ and $\mathbf y$ in a Finsler metric defined by the polynomial $P$.
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