L2-Quasi-compact and hyperbounded Markov operators

IF 0.8 2区 数学 Q2 MATHEMATICS
Guy Cohen, Michael Lin
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引用次数: 0

Abstract

A Markov operator P on a probability space (S, Σ μ) with μ invariant, is called hyperbounded if for some 1 ≤ pq ≤ ∞ it maps (continuously) Lp into Lq.

We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all Lr(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 Pνf:= νf is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that Pν is not hyperbounded, and there is ν with all powers singular such that Pν is hyperbounded. As an application, we prove that if Pν is hyperbounded, then for any sequence (nk) of distinct positive integers with bounded gaps, (nkx) is uniformly distributed mod 1 for ν almost every x (even when ν is singular).

L2-准紧密和超界马尔可夫算子
概率空间(S, Σ μ)上的马尔可夫算子 P 具有 μ 不变性,如果对于某些 1 ≤ p≤ q ≤ ∞,它(连续地)将 Lp 映射到 Lq,则称为超边界算子 P。我们从格吕克(Glück)的一个最新结果推导出,超边界算子 P 在所有 Lr(S, μ), 1 < r < ∞ 中都是准紧凑的,因此是均匀遍历的。我们用类似福格尔的方法证明,超边界马尔可夫算子具有与哈里斯循环算子类似的周期行为,并为遍历情况获得了非周期性的条件。给定单位圆上的概率ν,我们证明,如果卷积算子Pνf:= ν ⋇ f是超边界的,那么ν是无原子的。我们证明存在绝对连续的 ν,使得 Pν 不是超边界的,并且存在所有幂都是奇异的 ν,使得 Pν 是超边界的。作为应用,我们证明如果 Pν 是超界的,那么对于任何具有有界间隙的不同正整数序列 (nk),(nkx) 对于 ν 几乎每个 x 都是均匀分布 mod 1 的(即使 ν 是奇异的)。
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来源期刊
CiteScore
1.70
自引率
10.00%
发文量
90
审稿时长
6 months
期刊介绍: The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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