, H_{t} v
{\rangle}_{L^2}$ by self-adjointness of $H_t$ on $L^2$, while ${\langle}P u , H_{t} v
{\rangle}_{L^2} = {\langle}u , P H_{t} v
{\rangle}_{L^2}$. Indeed, the adjoint domain of $\mathcal{D}_{\min}$ is $\mathcal{D}_{\max}$, so for any $f \in \mathcal{D}_{\min}, g \in
\mathcal{D}_{\max}$, ${\langle}P f, g {\rangle}_{L^2} = {\langle}f, Pg {\rangle}_{L^2}$. But, then since $PH_t = - {\partial}_t H_t$ we see that ${\langle}H_{t} P u , v
{\rangle}_{L^2} = - {\langle}u , {\partial}_t H_t v {\rangle}_{L^2}$. But ${\partial}_t H_t$ is self-adjoint on $L^2$ so ${\langle}u , {\partial}_t H_t v
{\rangle}_{L^2} = {\langle}{\partial}_t H_t u , v {\rangle}_{L^2} = - {\langle}P H_t
u , v {\rangle}_{L^2}$, and thus ${\langle}H_{t} P u , v
{\rangle}_{L^2} = {\langle}P H_{t} u , v
{\rangle}_{L^2}$ for all $u \in \mathcal{D}_{\max}, v \in L^2$, i.e. holds.

The central vehicle for the construction of the heat kernel is the construction of a manifold with corners ${M^2_{\operatorname{heat}}}$ via iterated radial blow up of the natural domain of the heat kernel, namely the space $M \times M \times [0, \infty)_t$; thus the interiors of these two spaces are diffeomorphic, and the blowup process furnishes a ‘blowdown’ map $$\label{eq:singularmodelblowdown}
\beta \colon {M^2_{\operatorname{heat}}}{\longrightarrow}M \times M \times [0, \infty)_{t},$$ which encodes deeper information about 