我们研究了由实数集$\u211d$上的概率测度$μ_n$给出的卷积算子序列的几乎处处收敛性。若该算子序列在某一特定函数类$\u2131$上构成近似恒等元，还需满足哪些额外条件才能使得对所有$f \u2208 \u2131$，都有$μ_n \u2217 f \to f$几乎处处成立？我们聚焦于单个概率测度$μ$的压缩序列$C_{t_n}μ$这一特殊情形，其中$t_n \to 0$，从而该算子序列是一个近似恒等元。
英文摘要：
We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures $μ_n$ on $\mathbb R$. If this sequence of operators constitutes an approximate identity on a particular class of functions $\mathcal F$, under what additional conditions do we have $μ_n \ast f \to f$ a.e. for all $f \in \mathcal F$? We focus on the particular case of a sequence of contractions $C_{t_n}μ$ of a single probability measure $μ$, with $t_n \to 0$, so that that the sequence of operators is an approximate identity.
We show that effective $2\ell$-multiple correlations imply quantitative $\ell$-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.
We prove a.e. convergence of continuous-time quadratic averages with respect to two commuting $\mathbb{R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb{R}^2$-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.
We prove that for any holomorphic map, and any bounded orbit which does not accumulate to a singular set or to an attracting cycle, its lower Lyapunov exponent is non-negative. The same result holds for unbounded orbits, for maps with a bounded singular set. Furthermore, the orbit may accumulate to infinity or to a singular set, as long as it is slow enough.
We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform con…
It is shown that the cubic nonconventional ergodic averages of any order with a bounded aperiodic multiplicative function or von Mangoldt weights converge almost surely.
Motivated by Bourgain's work on pointwise ergodic theorems, and the work of Stein and Stein-Wainger on maximally modulated singular integrals without linear terms, we prove that the maximally monomially modulated discrete Hilbert transform, \[ \mathcal{C}_df(x) := \sup_λ\left| \sum_{m \neq 0} f(x-m) \frac{e^{2πi λm^d}}{m} \right| \] is bounded on all $\ell^p, \ 2 - \frac{1}{d^2 + 1} < p < \infty$, for any $d \geq 2$. We also establish almost everywhere pointwise convergence of the modulated ergodic Hilbert transforms (as $λ\to 0$) \[ \sum_{m \neq 0} T^m f(x) \cdot \frac{e^{2πi λm^d}}{m} \] for…
It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of $L^\infty$, which we generically call global obse…
