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Dec 26, 2020 at 8:16 comment added Martin Väth For the history, one probably has to distinguish the $C^2$ and $C^1$ case: The $C^2$ case is completely analogous to the finite-dimensional case. But the $C^1$ case does not follow by approximation as in finite dimensions but requires a rather different approach. My guess is that it was indeed one of the 4 mentioned authors who did the $C^1$ case first.
Dec 25, 2020 at 23:23 comment added Thomas Rot The modulo two case goes back to Cacciopoli probably. Smale also defined it.
Dec 25, 2020 at 8:02 history edited Martin Väth CC BY-SA 4.0
Added more remarks and references
Dec 25, 2020 at 7:46 history answered Martin Väth CC BY-SA 4.0