I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on $L^{2}(S)$ is called smoothing operator if there is a smooth kernel $k(p,q)$ on $M\times M$, with values $k(p,q)\in \text{Hom}(S_{q},S_{p})$, such that $$As(p)=\int_{M}k(p,q)s(q).\text{vol}(q).$$ Formally, $k$ is a smooth section of $\pi_{1}^{*}S\otimes \pi_{2}^{*}S^{*}$, where $\pi_1$ and $\pi_2$ are the canonical projections of $M\times M$ to $M$."
Here $S\rightarrow M$ is the Clifford bundle.
What I do not understand is the integrand. What does $k(p,q)s(q).\text{vol}(q)$ mean? Is there any canonical action of section of $S\rightarrow M$ on vector fields $TM \rightarrow M$? Or is there any canonical action of of sections $S\rightarrow M$ on forms, so that the expression $k(p,q)s(q).\text{vol}(q)$ makes sense? The expression $q\mapsto k(p,q)s(q).\text{vol}(q)$ should be a $n$-form on $M$, but what means the "dot" operation?
Ben