Ergodic averages with prime divisor weights in $L^1$

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 $S_{g,K}={\sum }_{n\le K}g(n)$ then for every ergodic dynamical system $(X,\mathcal{A},\mu ,\tau )$ and every $f\in L^1(X)$ $$\underset{K\to \infty }{\lim }\frac{1}{S_{g,K}}\sum _{n=1}^{K}g(n)f({\tau }^{n}x)={\int }_{X}fd\mu \text{ for μ a.e. }x\in X.$$ This answers a question raised by C. Cuny and M. Weber who showed this result for $L^p$, $p>1$.