First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of \emph{uniform dual ergodicity} for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides \emph{second (and higher) order asymptotics}. In the process, we derive a \emph{higher order Tauberian theorem} for scalar power series, which to our knowledge, has not previously been covered.