If you own up to breaking the glass, you'll be forgiven.
If you own up to something wrong that you have done, you admit that you did it. The headmaster is waiting for someone to own up Last year my husband owned up to a secret affair with his secretary. = admit
[ 'On ] (adjective.) before 12th century. Middle English owen, from Old English Agen; akin to Old High German eigan own, Old Norse eiginn, Old English Agan to possess; more at OWE.