17 USC S 512(f) Misrepresentations.
Any person who knowingly materially misrepresents under this section—
(1) that material or activity is infringing, or
(2) that material or activity was removed or disabled by mistake or misidentification,
shall be liable for any damages, including costs and attorneys’ fees, incurred by the alleged infringer, by any copyright owner or copyright owner’s authorized licensee, or by a service provider, who is injured by such misrepresentation, as the result of the service provider relying upon such misrepresentation in removing or disabling access to the material or activity claimed to be infringing, or in replacing the removed material or ceasing to disable access to it.
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Actual damages isn't much of deterrence, though, considering the typical scenario. Proving actual damages is costly and the ratio of typical damage amounts (nil to moderate) to litigation costs makes it an unattractive option.
The level of abuse at which a court may penalize a party independent of a statutory provision is quite high, so that's not a meaningful deterrent, either, though the situation is so bad there have been some high-profile cases.
For real deterrence you'd need statutory damages, similar to statutory damages for copyright infringement. But Fair Use is such a muddied area of law that litigation costs might still make it unattractive to pursue statutory damages for most people.
It's all kind of moot in the context of Youtube because most copyright disputes happen through Youtube's own internal policies and mechanisms, which are mostly controlled by the terms of Youtube's private contracts.
The best solution is twofold: 1) reform copyright by circumscribing copyrightable subject matter and clarifying fair use, and 2) make self-hosting of content easier, more in line with the late 1990s utopic vision of everybody as their own publisher--as opposed to the current situation where Youtube, Facebook, etc are the new publishers and individuals remain confined to content creation.