I am currently reading the book Heat Kernels and Dirac Operators and I don't quite understand the definition of the heat kernel, definition 2.15:
Given a Laplacian $H$, a family $p=(p_t)_{t\geq 0}$ of kernels is called a heat kernel if it "satisfies the heat equation $(\partial_t+H_x)p_t(x,y)$" (and some other requirements). But what exactly does it mean that $p$ solves the heat equation? Even if we fix $(t,y)$, then the function $p(t,{}\cdot{},y)$ is not an element of $\Gamma(M,E)$ and hence we can not apply $H$ to it. So how does the heat equation need to be understood in this case?
Edit: Here are two guesses of mine - is any of it correct?
- One idea would be that $(\partial_t+H_x)p=0$ is a shorthand for the following statement: \begin{equation}\tag{1} \forall y\in M:\forall v\in E_y:(\partial_t+H_x)p({}\cdot{},y)v=0 \end{equation}
- Another idea would be to require \begin{equation}\tag{2} \forall\phi\in\Gamma(M,E):(\partial_t+H_x)P\phi=0, \end{equation} where $$(P\phi)(t,x)=(P_t\phi)(x)=\int_M p(t,x,y)\phi(y)|dy|$$ and $y\mapsto|dy|$ is the Riemannian density.