  In the presence of quasiperiodic potentials, the celebrated Kitaev chain
presents an intriguing phase diagram with ergodic, localized and and
multifractal states. In this work, we generalize these results by studying the
localization properties of the Aubry-Andr\'e-Harper model in the presence of
long-range hopping and superconducting pairing amplitudes. These amplitudes
decay with power-law exponents $\xi$ and $\alpha$ respectively. To this end, we
review and compare a toolbox of global and local characterization methods in
order to investigate different types of transitions between ergodic, localized
and multifractal states. We report energy-dependent transitions from ergodic to
multifractal states for pairing terms with $\alpha<1$ and energy-dependent
transitions from ergodic to localized states with an intermediate multifractal
region for $\alpha>1$. The size of the intermediate multifractal region depends
not only on the value of the superconducting pairing term $\Delta$, but also on
the energy band. The transitions are not described by a mobility edge, but
instead we report hybridization of bands with different types of localization
properties. This leads to coexisting multifractal regimes where fractal
dimensions follow different distributions.
