en_tn_lite_do_not_use/rut/03/13.md

if he will perform for you the duty of a kinsman

Boaz is referring to the expectation that the closest male relative of Ruth's dead husband would marry her and help carry on his family name. (See: rc://en/ta/man/jit/figs-explicit)

by the life of Yahweh

"as surely as Yahweh lives." This was a common Hebrew vow.