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

TL;DR

This paper proves that prime divisor counting functions serve as effective weights in the pointwise ergodic theorem for functions in L^1, extending previous results known for p>1 to the case p=1.

Contribution

It establishes the validity of prime divisor weights in the L^1 setting for the ergodic theorem, answering a question posed by Cuny and Weber.

Findings

Prime divisor functions are good weights in L^1 ergodic averages.

The result extends previous p>1 findings to p=1.

Almost everywhere convergence is achieved for these weights.

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