  We consider the transfer operators of non-uniformly expanding maps for
potentials of various regularity, and show that a specific property of
potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay
of the transfer operator of the same speed than entailed by the constant
potential. The method relies neither on Markov partitions nor on inducing, but
on functional analysis and duality, through the simplest principles of optimal
transportation. As an application, we notably show that for any map of the
circle which is expanding outside an arbitrarily flat neutral point, the set of
H{\"o}lder potentials exhibiting a spectral gap is dense in the uniform
topology. The method applies in a variety of situation, including
Pomeau-Manneville maps with regular enough potentials, or uniformly expanding
maps of low regularity with their natural potential; we also recover in a
united fashion variants of several previous results.
