This is the same question I asked in stackexchange: http://math.stackexchange.com/questions/519152/asymptotic-of-the-heat-kernel

I read Proposition 3.23(page 101) in Rosenberg's book "The Laplacian on a Riemannian Manifold" and not quite clear how to get the estimate $(4\pi t)^{n/2}|Q_k * H_k|\leq C \cdot t^{k+1}$ on a compact manifold. Since $|Q_k|\leq C \cdot t^{k-(n/2)}$ and $H_k = \eta\cdot e^{-r^2/4t}(4\pi t)^{-n/2}(u_0 + t u_1 + \cdots + t^k u_k)$, it amounts to show the following estimate $$ (4\pi t)^{n/2}\left|\int_0^t C \tau^{k-n/2}\int_M H_k(t-\tau,x,y)dy\right|\leq C_1 t^{k+1} $$ for some constant when $t\in [0, T]$ and $T$ is small. Since $M$ is compact the terms without $t$ are uniformly bounded and it reduces to the following one(maybe with different constants) $$ t^{n/2} \int_0^t \tau^{k - n/2}(t-\tau)^{-n/2} d\tau \leq C t^{k+1}. $$ It seems that the above inequality does not hold. I would appreciate if someone can point out where I am wrong.