If the abuser has been convicted of a domestic violence misdemeanor or felony, can s/he keep or buy a gun?
No. Under federal law, if the abuser has been convicted of a felony or a domestic violence misdemeanor, s/he cannot have or buy a gun.* If you're not sure if the abuser has been convicted of a domestic violence misdemeanor, see What crimes are considered domestic violence misdemeanors?
In addition, Utah state law has many additional reasons** that a person cannot own a firearm, which you can read about in I am a victim of domestic violence and the abuser has a gun. Is that legal?
* 18 USC 922(g)(9)
** U.C.A. § 76-10-503