Could someone please point me to an English reference for the construction of the heat kernel on a compact manifold using the method of convolution, as described in the book of Berger, Gauduchon and Mazet (which is in French)?
This book is already quite detailed, but I am having trouble deriving, for example, Lemme E.III.7 (the $C^l$ statement in particular), so it would be nice to see another treatment of this.
Cheers.
One possible reference would be Steven Rosenberg's "The Laplacian on a Riemannian Manifold", chapter 3, which develops the same method. For non-compact manifolds admitting an action of certain groups with compact quotient, you could see Donnelly's 1979 paper. The first part of this paper essentially adapts the original argument to the cocompact setting with very few changes in technique. See https://projecteuclid.org/euclid.ijm/1256048110.
Another reference for the original construction could be http://www.math.mcgill.ca/toth/spectral%20geometry.pdf.