Ergodic averages with prime divisor weights in L-1

Buczolich, Zoltán (2019) Ergodic averages with prime divisor weights in L-1. ERGODIC THEORY AND DYNAMICAL SYSTEMS, 39 (4). pp. 889-897. ISSN 0143-3857

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Abstract

We show that omega (n) and Omega (n), the number of distinct prime factors of n and the number of distinct prime factors of n counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in L-1. That is, if g denotes one of these functions and S-g, K = Sigma(n <= K )g(n) then for every ergodic dynamical system (X, A, mu, tau) and every f is an element of L-1(X), lim(K -> infinity)1/S-g, K Sigma(K)(n=1)g(n)f(tau(n)x) = integral(x) f d mu for mu almost every x is an element of X. This answers a question raised by Cuny and Weber, who showed this result for L-p, p > 1.

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