  In this paper, we first establish the equivalence of three types of error
bounds: uniformized Kurdyka-{\L}ojasiewicz (u-KL) property, uniformized
level-set subdifferential error bound (u-LSEB) and uniformized H\"{o}lder error
bound (u-HEB) for prox-regular functions. Then we study the behavior of the
level-set subdifferential error bound (LSEB) and the local H\"{o}lder error
bound (LHEB) which is expressed respectively by Moreau envelopes, under
suitable assumptions. Finally, in order to illustrate our main results and to
compare them with those of recent references, some examples are also given.
