This applies not only to conditions inflicted by the weapon that bears the weapon upgrade itself, but also conditions of the same type inflicted by skills. For example: A Necromancer wielding a barbed weapon can lengthen the foes' bleeding caused by the popular Signet of Agony + Plague Sending combo. Likewise, an Elementalist wielding a heavy weapon can lengthen the weakness caused by Enervating Charge.
Indirect condition application is not always effected by a Lengthens Prefix. See below for examples: