Genuinely sharp estimates for heat kernels on some planar cones being Weyl chambers (Q6592814)

The author derives sharp heat kernel estimate by using purely analytic tools on planar cones with apertures \(\pi/3\) and \(\pi/4\). Precisely, for a planar cone \(C_\alpha\) with aperture \(\alpha\), \(\alpha=\pi/m\), \(m=3\) or \(m=4\), the Dirichlet heat kernel satisfies\N\begin{align*}\Np_t^{C_\alpha}(x,y)\sim \left(1\wedge \frac{\delta_v(x)\delta_v(y)}{t}\right)^{m-2}\left(1\wedge \frac{\delta_{r_1}(x)\delta_{r_1}(y)}{t}\right) \left(1\wedge \frac{\delta_{r_2}(x)\delta_{r_2}(y)}{t}\right) (4\pi t)^{-1}\exp\left(-\frac{|x-y|^2}{4t}\right),\N\end{align*}\Nwhere \(v\) denotes the vertex of the cone \(C_\alpha\), \(r_1\) and \(r_2\) denote the two boundary rays emanating from \(v\), \(\delta_v(x)\), \(\delta_{r_1}(x)\), \(\delta_{r_2}(x)\) denote the Euclidean distance from \(x\in C_\alpha\) to \(v\), \(r_1\), \(r_2\), respectively.