In order for a cause of action against a new defendant to relate back to the date of a claim against another defendant, the plaintiff must establish: (1) that the cause of action arises out of the same conduct, transaction, or occurrence; (2) that the additional party is united in interest with the original party; and (3) that the additional party knew or should have known that, but for the plaintiff's mistake as to the identity of the proper parties, the action would have been brought against the additional party as well.
Practice point: In a medical malpractice action, the defendants are considered united in interest when one is vicariously liable for the acts of the other.
Ferrera v. Jerome Zisfein, NY Slip 00096 (2d Dep't January 9, 2019)
Here is the decision.