  The local regularity of functional time series is studied under
$L^p-m-$appro\-ximability assumptions. The sample paths are observed with error
at possibly random design points. Non-asymptotic concentration bounds of the
regularity estimators are derived. As an application, we build nonparametric
mean and autocovariance functions estimators that adapt to the regularity and
the design, which can be sparse or dense. We also derive the asymptotic
normality of the mean estimator, which allows honest inference for irregular
mean functions. Simulations and a real data application illustrate the
performance of the new estimators.
