  We give conditions under which nonuniformly expanding maps exhibit lower
bounds of polynomial type for the decay of correlations and for a large class
of observables. We show that if the Lasota-Yorke type inequality for the
transfer operator of a first return map are satisfied in a Banach space B, and
the absolutely continuous invariant measure obtained is weak mixing, in terms
of aperiodicity, then under some renewal condition, the maps have polynomial
decay of correlations for observables in B. We also provide some general
conditions that give aperiodicity for expanding maps in higher dimensional
spaces. As applications, we obtain lower bounds for piecewise expanding maps
with an indifferent fixed point and for which we also allow non-Markov
structure and unbounded distortion. The observables are functions that have
bounded variation or satisfy quasi-H\"older conditions and have their support
bounded away from the neutral fixed points.
