The Cheeger-Gromov compactness theorem says the following. Let us fix $n\in \mathbb{N}$ and positive constants $K,D,v$. Let $\{(M_i^n,g_i)\}$ be a sequence of closed infinitely smooth $n$-dimensional Riemannian manifolds with $|Sec(M_i)|\leq K$, diameter at most $D$ and volume at least $v$. Then, after a choice of a subsequence, there exist a closed smooth $n$-dimensional manifold $M$ with a Riemannian metric $g$ of class $C^{1,\alpha}$ for any $\alpha\in(0,1)$, and diffeomorphisms $\phi_i\colon M\to M_i$ such that $\phi_i^*(g_i)$ converges to $g$ in $C^{1,\alpha}$ for any $\alpha\in (0,1)$.

**Question 1.** What is a reference for this result in this particular form?

**Question 2.** I have an impression that I have heard that there are some refinements of the above result saying that the diffeomorphisms $\phi_i$ can be chosen in such a way that the pull-back under them of the Riemann curvature tensor of $g_i$ converges to some tensor in a Sobolev space. Is that correct? What is the precise statement? A reference?