In fact, I don't actually need an equivalence like we get from Arzelà-Ascoli theorem. Just sufficient conditions to get compacity would be enough

Oh, nice references! This will help me a lot, thank you! Definetely looks like it is an open problem. I need to dig more and try to find at least one side of the result: good (for me) sufficient conditions to get compacity. Should be easier than an equivalence.

@DeaneYang in a certain way it is! An analogue of equicontinuity in the sense that, together with a bound, implies compacity on the Holder norm.

I tried to rewrite the question to be more clear, but I don't know how to express that in a better way since I'm not looking to solve an exercise or something like that. I'm looking for a general result on how to show compactness on Hölder norm (in a similar fashion as Arzelà-Ascoli for supremum norm)