Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem

We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb{Z}[i]$ action $T$, the limit $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{D(N)}\sum _{\mathcal{N}(n)\le N}d(n)f(T^{n}x)$$ converges for every $f\in L^p$, where $\mathcal{N}(n)=n\bar{n}$, and $D(N)={\sum }_{\mathcal{N}(n)\le N}d(n)$, and $1<p\le \mathrm{\infty }$. To do so we study the averages $$A_{N}f(x)=\frac{1}{D(N)}\sum _{\mathcal{N}(n)\le N}d(n)f(x-n),$$ and obtain improving and weighted maximal inequalities for our operator, in the process.