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 $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(T^n x)$$ converges for every $f\in L^p$, where $\mathscr{N} (n) = n \bar{n}$, and $D(N) = \sum _{\mathscr{N} (n) \leq N} d(n)$, and $1<p\leq \infty$. To do so we study the averages $$A_N f (x) = \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(x-n) ,$$ and obtain improving and weighted maximal inequalities for our operator, in the process.