Where can I find the following result?
Let $M$ be a compact Riemannian $C^{\infty}$-manifold, $\Delta$ be the Laplace operator, and $n$ be the dimension of $M$. Then, $(n/2+1)$-th power of the Green's operator of $\Delta$ is an integral operator defined by a continuous symmetric kernel.
I was told to check de Rham's book Differential Manifolds. I did and I am sure it is where Green's form is discussed, but I do not really see where exactly. There should be books with more contemporary wording? I do not really see where $(n/2+1)$ comes from.
Let $\mathcal G_s$ be the Green kernel of $(-\Delta_g)^s$, $s>0$. For non-integer $s$, this can be defined on compact manifolds via the spectral measure of $\Delta_g$, and on non-compact manifolds via Bessel-type kernels, see e.g. [2] below.
Then $\mathcal G_s$ has a continuous kernel $G_s$ provided $s>n/2$, satisfying $$G_s(x,y)=\sum_{j=1}^\infty\lambda_j^{-s}\phi_j(x)\phi_j(y) $$ where $(\phi_j)_{j\geq 0}$ is a CONS of eigenfunctions for $-\Delta_g$ of eigenvalues $\lambda_j$ counted with multiplicity. Continuity holds since the series converges uniformly. Symmetry is immediate.
This is (originally?) shown in [1]. A more accessible proof can be found in [2, Thm. 7.2.18].
Note that, if $M$ is closed (and connected!), $-\Delta_g$ is invertible only on $L^2$-functions with $0$-mean, which is why the series starts from $j=1$.
[1] S. Minakshisundaram, Å. Pleijel, Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds, Canadian J. Math. 1949
[2] P. Buser, Geometry and spectra of compact Riemann surfaces, 1992