Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910336427229184 |
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| author | Giannitsi, Christina Miheisi, Nazar Mousavi, Hamed |
| author_facet | Giannitsi, Christina Miheisi, Nazar Mousavi, Hamed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem Giannitsi, Christina Miheisi, Nazar Mousavi, Hamed Classical Analysis and ODEs Functional Analysis 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. |
| title | Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2402.12457 |