L²-Quasi-compact and hyperbounded Markov operators

A Markov operator P on a probability space (S, Σ μ) with μ invariant, is called hyperbounded if for some 1 ≤ p≤ q ≤ ∞ it maps (continuously) L^p into L^q. We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all L^r(S, μ), 1 < r < ∞. We prove, using a method similar to Foguel’s, that a hyperbounded Markov operator has periodic behavior similar to that of Harris recurrent operators, and for the ergodic case obtain conditions for aperiodicity. Given a probability ν on the unit circle, we prove that if the convolution operator P_νf:= ν ⋇ f is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that P_ν is not hyperbounded, and there is ν with all powers singular such that P_ν is hyperbounded. As an application, we prove that if P_ν is hyperbounded, then for any sequence (n_k) of distinct positive integers with bounded gaps, (n_kx) is uniformly distributed mod 1 for ν almost every x (even when ν is singular).