  We study nonlinear stability of pulled fronts in scalar parabolic equations
on the real line of arbitrary order, under conceptual assumptions on existence
and spectral stability of fronts. In this general setting, we establish sharp
algebraic decay rates and temporal asymptotics of perturbations to the front.
Some of these results are known for the specific example of the Fisher-KPP
equation, and our results can thus be viewed as establishing universality of
some aspects of this simple model. We also give a precise description of how
the spatial localization of perturbations to the front affects the temporal
decay rate, across the full range of localizations for which asymptotic
stability holds. Technically, our approach is based on a detailed study of the
resolvent operator for the linearized problem, through which we obtain sharp
linear time decay estimates that allow for a direct nonlinear analysis.
