The heat kernel of a weighted Maass-Laplacian with real weights

We derive closed formula for the heat kernel $K_{\mathbb{H}}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_{\mathbb{H}}$ is strictly monotone decreasing function. We also derive some important asymptotic formulae for the heat kernel $K_{\mathbb{H}}$.