Cesàro averages for Goldbach representations with summands in arithmetic progressions

Let [Formula: see text] be the von Mangoldt function, let [Formula: see text] be an integer and let [Formula: see text] be the counting function for the Goldbach numbers with summands in arithmetic progression modulo a common integer [Formula: see text]. We prove an asymptotic formula for the weighted average, with Cesàro weight of order [Formula: see text], with [Formula: see text], of this function. Our result is uniform in a suitable range for [Formula: see text].

Symmetric Spaces of Measurable Functions: Old and New Advances

The article is an extensive review in the theory of symmetric spaces of measurable functions. It contains a number of new (recent) and old (known) results in this field. For the most of the results, we give their proofs or exact references, where they can be found. The symmetric spaces under consideration are Banach (or quasi-Banach) latices of measurable functions equipped with symmetric (rearrangement invariant) norm (or quasinorm). We consider symmetric spaces E = E(Ω, Fμ, μ) ⊂ L0 (Ω, Fμ, μ) on general measure spaces (Ω, Fμ, μ), where the measures μ are assumed to be finite or infinite σ-finite and nonatomic, while there are no assumptions that (Ω, Fμ, μ) is separable or Lebesgue space. In the first section of the review, we describe main classes and basic properties of symmetric spaces, consider minimal, maximal, and associate spaces, the properties (A), (B), and (C), and Fatou’s property. The list of specific symmetric spaces we use includes Orlicz LΦ(Ω, Fμ, μ), Lorentz ΛW (Ω, Fμ, μ), Marcinkiewicz MV (Ω, Fμ, μ), and Orlicz-Lorentz LW,Φ (Ω, Fμ, μ) spaces, and, in particular, the spaces Lp (w), Mp(w), Lp,q, and L∞(U ). In the second section, we deal with the dilation (Boyd) indexes of symmetric spaces and some applications of classical Hardy-Littlewood operator H. One of the main problems here is: when H acts as a bounded operator on a given symmetric space E(Ω, Fμ, μ)? A spacial attention is paid to symmetric spaces, which have Hardy-Littlewood property (HLP) or weak Hardy-Littlewood property (WHLP). In the third section, we consider some interpolation theorems for the pair of spaces (L1 , L∞) including the classical Calderon-Mityagin theorem. As an application of general theory, we prove in the last section of review Ergodic Theorems for Cesaro averages of positive contractions in symmetric spaces. Studying various types of convergence, we are interested in Dominant Ergodic Theorem (DET ), Individual (Pointwise) Ergodic Theorem (IET ), Order Ergodic Theorem (OET ), and also Mean (Statistical) Ergodic Theorem (MET ).

Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n ∑ k = 0 n − 1 λ − n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.

Uniform Convergence of Cesaro Averages for Uniquely Ergodic $C^*$-Dynamical Systems

Consider a uniquely ergodic $C^*$-dynamical system ba\-sed on a unital $*$-endomorphism $\Phi$ of a $C^*$-algebra. We prove the uniform convergence of Cesaro averages $\frac1{n}\sum_{k=0}^{n-1}\lambda^{-n}\Phi(a)$ for all values $\lambda$ in the unit circle which are not eigenvalues corresponding to "measurable non continuous" eigenfunctions. This result generalises the analogous one in commutative ergodic theory presented in [19], which turns out to be a combination of the Wiener-Wintner Theorem (cf. [22]) and the uniformly convergent ergodic theorem of Krylov and Bogolioubov (cf. [15]).

Ergodic measures of Markov semigroups with the e-property

We study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property, an equicontinuity condition. We introduce a weak concentrating condition around a compact set K and show that this condition has several implications on the set of ergodic measures, one of them being the existence of a Borel subset K0 of K with a bijective map from K0 to the ergodic measures, by sending a point in K0 to the weak limit of the Cesàro averages of the Dirac measure on this point. We also give sufficient conditions for the set of ergodic measures to be countable and finite. Finally, we give a quite general condition under which the Cesàro averages of any measure converge to an invariant measure.

A pointwise ergodic theorem for Fuchsian groups

We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesàro averages of spherical averages in a Fuchsian group.