Not sure I understand. Could you provide the precise statement that you want a reference for?
– Deane YangJan 12 '11 at 23:26

There are various versions of Arzela-Ascoli, but typically one starts with a sequence of functions that has a uniform bound on $C^{k,\alpha}$ norm, an one want to extract a subsequence converging to in $C^{m,\beta}$ norm where $m+\beta<k+\alpha$, and one also gets information about regularity of the limit. There are also some subtleties when $\alpha = 0$. I found that I do not have a sufficiently firm grip on these matters.
– Igor BelegradekJan 12 '11 at 23:49

Maybe a book that explains Schauder estimates for elliptic PDE's? Gilbarg and Trudinger? Or the book by L. C. Evans?
– Deane YangJan 13 '11 at 1:53

Deane, I don't have the books handy, and won't get to them before Friday (due to snowstorm) but what I see in google.books while searching them isn't promising. Anyway, I agree that these should be the first books to look at, thanks.
– Igor BelegradekJan 13 '11 at 2:10

The standard version of Arzela-Ascoli theorem says an equicontinuous family of functions is compact in the $C^0$ topology. I believe that it is straightforward to show that a uniform $C^\alpha$ bound implies equicontinuity. I also believe it is reasonably straightforward extend this argument to the general case. But I haven't worked out the details.
– Deane YangJan 13 '11 at 4:25