I read "Eigenvalues in Riemannian Geometry (I.Chavel)". At p.188 in this book, I saw the sentence "Thus $v(x,y)$ is expressed as integrals, over supp $h$ and {supp $h$}\V, respectively, against $C^{\infty}$ kernels."
($M$ is a Riemannian manifold, $\Omega$ is a regular domain in $M$.$v$ is a solution of the heat equation on $M \times (\alpha , \beta)$, $h$ is in $C^{\infty}_{c} $($\Omega$) such that $h=1$ on some $V \subseteq \Omega$.
And $v$ forms
$v(x,t)$=$\int_{\Omega} v(y,t_1)h(y)q_{\Omega} (x,y,t-t_1) dV(y)$
+$\int_{t_1}^t d{\tau}$ $\int_{\Omega-V}$ $v(y,\tau)${$2<(grad h)(y),(grad_{y} q_{\Omega}$)(x,y,t_{\tau})>$+ (\Delta h)(y)q_{\Omega}(x,y,t-\tau)dV(y)$,
where $q_{\Omega}$ is Dirichlet heat kernel.
I can't understand last part "integrals,…,agaist $C^{\infty} $ kernels".
It is okay to just teach me meaning "integaral against".
I would appriciate your help.