  In this paper we establish local estimates for the first passage time of a
subordinator under the assumption that it belongs to the Feller class, either
at zero or infinity, having as a particular case the subordinators which are in
the domain of attraction of a stable distribution, either at zero or infinity.
To derive these results we first obtain uniform local estimates for the one
dimensional distribution of such a subordinator, which sharpen those obtained
by Jain and Pruitt in 1987. In the particular case of a subordinator in the
domain of attraction of a stable distribution the results are the analogue of
the results obtained by the authors for non-monotone L\'evy processes. For
subordinators an approach different to that used for non-monotone L\'evy
processes is necessary because the excursion techniques are not available and
also because typically in the non-monotone case the tail distribution of the
first passage time has polynomial decrease, while in the subordinator case it
is exponential.
