论文标题
$ l^2 $ -quasi-compact和庞大的马尔可夫操作员
$L^2$-Quasi-compact and hyperbounded Markov operators
论文作者
论文摘要
Markov操作员$ p $在概率空间$（s，σ，μ）$中，带有$μ$不变的$，如果有一些$ 1 \ le p <q \ le \ le \ le \ le \ le \ le infty $ it Map（连续）$ l^p $ in $ l^q $。 我们从最近的Glück结果中推断出，庞大的$ p $是准紧凑的，因此均匀地呈现了，在所有$ l^r（s，μ）$中，$ 1 <r <\ r <\ infty $。我们证明，使用类似于福格尔（Foguel）的方法，庞大的马尔可夫操作员的周期性行为与哈里斯（Harris）经常性操作员的行为相似，而对于千古案例，则可以获得大术的条件。 给定单位圆的概率$ν$，我们证明，如果卷积运算符$p_νf：=ν*f $是体重的，则$ν$是无原来的。我们表明，$ν$绝对连续，因此$p_ν$没有过多，并且所有势力都有$ν$，因此$p_ν$是夸张的。作为一个应用程序，我们证明，如果$p_ν$超重，那么对于任何序列$（n_k）$的独特正整数 有限的差距，$（n_kx）$是均匀分布的mod 1，对于$ν$，几乎每个$ x $（即使$ν$都是单数）。
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).
