In Cook's famous paper on $\mathsf{NC}$, he cites the following result:
PROPOSITION 4.7 (Cook and Ruzzo, 1983). $\mathsf{AC}^k$ consists of those problems solvable by uniform unbounded fan-in circuit families in $O(\log^k n)$ depth and $n^{O(1)}$ size.
where $\mathsf{AC}^k$ here is defined for $k \ge 1$ as the class of problems solvable by an ATM in $O(\log n)$ space and $O(\log^k n)$ alternation depth. (Of course, the definition of $\mathsf{AC}^k$ usually goes the other way around, but given the equivalence it is all just a matter of presentation.)
Note the above is the logspace-uniform version of $\mathsf{AC}$. It was later acknowledged that for $\mathsf{AC}^0$ the more restrictive dlogtime-uniformity is preferable. There is also an interesting result regarding the (in)equality of dlogtime- and logspace-uniformity of $\mathsf{AC}^0$. We also know that dlogtime-uniform $\mathsf{AC}^0$ is characterizable as the class of problems solvable by ATMs in $O(\log n)$ time (instead of space above) and $O(1)$ alternation depth.
Given that dlogtime- and logspace-uniformity most likely does make a difference for $\mathsf{AC}^0$ I am interested whether the Cook and Ruzzo result can be extended to the case of $k = 0$ and logspace-uniform $\mathsf{AC}^0$. Unfortunately, in Cook's paper, "Cook and Ruzzo, 1983" is only listed as "unpublished theorem," so I was unable to check whether the proof also works for $k=0$ (and whether Cook and Ruzzo had simply neglected it for some reason).
