I am not a lawyer but my understanding is that it’s more than that: see https://www.law.cornell.edu/uscode/text/17/512 - infringement covers not just that you have copyright but also that the user doesn’t have a valid right to use it, which includes fair use.
Diebold lost a case where someone published internal emails and they used the DMCA to try to get them taken offline, and a judge agreed that this was fair use:
What I'm referring to is that perjury seems to be limited to cases where the claim to represent the copyright holder is unauthorized. One can file any number of DMCA claims on otherwise false grounds (such as fair use, etc), and if the court then chooses to sue upon DMCA appeal, they can cost the user far more than they should be able to, even if the user wins, as long as the person filing the DMCA is authorized by the copyright holder to do so.
I would like to see such perjury charges be expanded to situations where it is clear the copyright holder's representative has acted in bad faith regarding whether infringement even happened, or whether it was justified infringement (fair use).
Diebold lost a case where someone published internal emails and they used the DMCA to try to get them taken offline, and a judge agreed that this was fair use:
https://www.eff.org/cases/online-policy-group-v-diebold