Singularity of reproducing kernel for elliptic operator

Let $(M,g)$ be a smooth compact Riemannian manifold and dimension $2$, $\Gamma$ a smooth vector bundle over $M$, and suppose $L: W^{k,2}(\Gamma)\to W^{k-2,2}(\Gamma)$ is a second order strongly elliptic operator ($k$ large) that, in coordinates, takes the form $L=\Delta + A$, where $\Delta$ is the vector-valued Laplacian, and $A(x,\partial f^\alpha/\partial x_i)$ is some first order operator with smooth coefficients that only depends on the first derivatives of $f$. Fixing $p\in M$ and $U\in \Gamma_p$, it is known that one can find a reproducing kernel for the functional on $W^{k,2}(\Gamma)$ given by $V\mapsto \langle V(p), U \rangle$, namely a section $G$ such that $$\langle V(p), U \rangle = \int_M \langle LV, G\rangle dv_g$$ $G$ should be smooth, but is singular at the point $p$. Is there any way to find an asymptotic for $G$ at the point $p$? By analogy with the case of the Laplacian, I would expect $G$ is of the form $$G=c\log |z| U + H(z)$$ where $c$ is a constant and $H$ is smooth. I feel there should be a standard technique for this.