Who pays for the compliance with Airworthiness Directives (AD)?
It depends on the agreements between the operator/owner of the aircraft on one side and the manufacturer of the aircraft and its suppliers on the other side. 

Often there will be some form of compensation involved if an AD is issued when that AD is addressing non-compliance with regulations / standard in effect when the aircraft was produced. Normally there is some time limited warranty covering the costs of ADs at least partially.