heat kernel on closed manifolds - error in Chavel's book?

Question

first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.

In Isaac Chavel's book Eigenvalues in Riemannian Geometry, Chapter VI, pages 151-154, the heat kernel for compact manifolds is constructed.

I am hoping for someone that is familiar with the consctrucion of the heat kernel in Chavel's book.

On page 154, the final formula for the heat kernel $p$ on the closed Riemannian manifold $M$ reads

$p=H_k+((L_xH_k)\ast F)\hspace{25em}(A)$

where $H_k$ is a parametrix for the heat operator $L:=\Delta-\partial_t$ on $M$ and $F=\sum_{l=1}^\infty (L_xH_k)^{\ast l}$.

Shouldn't the correct formula be

$p=H_k+ (H_k\ast F)\hspace{26em}(B)$

?

Formula $(B)$ would also correspond to the ansatz he made in equation $(42)$ on page 153. I read up other books and they all seem to use $(B)$. Additionally, $(A)$ doesn't make much sense for me. However, this still bugs me and I wanted to ask if I make an obvious mistake here, e.g. are $(A)$ and $(B)$ actually the same?

In summary, I want to know if $(A)$ is a typo or intended and would really appreciate any help.

Thanks in advance.

Answer 1

Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that $$L(H_k * F) = (LH_k)*F - F,$$ so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then $$ L p = LH_k + (L H_k)*F - L F = LH_k + \sum_{l=2}^\infty (LH_k)^{*l} - \sum_{l=1}^\infty (LH_k)^{*l} = 0,$$ where all sums converge and differentiation under the sum is allowed by all the estimates on $LH_k$. Also, if you look at formula (42), you see that the definition $p= H_k + H_k * F$ is indeed the one he meant to make.