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

Abstract

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.

Figures (1)

• Figure 1: Tessallation of the plane by $B_q$

Theorems & Definitions (33)

• Theorem 1.2
• Theorem 1.4
• Theorem 1.6
• Corollary 1.8
• Lemma 2.4
• proof
• Remark 2.18
• Lemma 3.1
• Remark 3.3
• proof