Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on Rn

McIntosh, David I.

Canadian journal of mathematics, Tome 47 (1995) no. 4, pp. 852-876

Résumé

Let R+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of Rn. A function φ: Rn → R+ is called a weight function if ʃRn φ dλ = 1. Let (X, F, μ) be a non-atomic, finite measure space, let ƒ: X → R+, and suppose { Tν}ν∊Rn is an ergodic, aperiodic Rn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: Rn → Rn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk -1 / dλ are L∞ (Rn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(Rn) function with bounded support.

Détail

DOI : 10.4153/CJM-1995-044-0

Mots-clés : 28D99, 60F99, maximal ergodic theorems

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