$L^2$-Quasi-compact and hyperbounded Markov operators

A Markov operator $P$ on a probability space $(S,Σ,μ)$, with $μ$ invariant, is called {\it hyperbounded} if for some $1 \le p<q \le \infty$ 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 $L^r(S,μ)$, $1<r< \infty$. 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).

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