Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves
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| Format: | Preprint |
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2024
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| _version_ | 1866912965233475584 |
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| author | Daskalakis, Leonidas Fragkos, Anastasios |
| author_facet | Daskalakis, Leonidas Fragkos, Anastasios |
| contents | For $c\in(1,2)$ we consider the following operators \[ \mathcal{C}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad \mathcal{C}^{\mathsf{sgn}}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\mathsf{sign(n)} \lfloor |n|^{c} \rfloor}}{n}\bigg| \text{,}
\] and prove that both extend boundedly on $\ell^p(\mathbb{Z})$, $p\in(1,\infty)$. The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages
\[
A_Nf(x)=\frac{1}{N}\sum_{n=1}^Nf(T^nS^{\lfloor n^c\rfloor}x)\text{,}
\]
where $T,S\colon X\to X$ are commuting measure-preserving transformations on a $σ$-finite measure space $(X,μ)$, and $f\in L_μ^p(X)$, $p\in(1,\infty)$. The point of departure for both proofs is the study of exponential sums with phases $ξ_2 \lfloor |n^c|\rfloor+ ξ_1n$ through the use of a simple variant of the circle method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves Daskalakis, Leonidas Fragkos, Anastasios Dynamical Systems Classical Analysis and ODEs 42B25, 42B20, 42A45, 37A46 For $c\in(1,2)$ we consider the following operators \[ \mathcal{C}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad \mathcal{C}^{\mathsf{sgn}}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\mathsf{sign(n)} \lfloor |n|^{c} \rfloor}}{n}\bigg| \text{,} \] and prove that both extend boundedly on $\ell^p(\mathbb{Z})$, $p\in(1,\infty)$. The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages \[ A_Nf(x)=\frac{1}{N}\sum_{n=1}^Nf(T^nS^{\lfloor n^c\rfloor}x)\text{,} \] where $T,S\colon X\to X$ are commuting measure-preserving transformations on a $σ$-finite measure space $(X,μ)$, and $f\in L_μ^p(X)$, $p\in(1,\infty)$. The point of departure for both proofs is the study of exponential sums with phases $ξ_2 \lfloor |n^c|\rfloor+ ξ_1n$ through the use of a simple variant of the circle method. |
| title | Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves |
| topic | Dynamical Systems Classical Analysis and ODEs 42B25, 42B20, 42A45, 37A46 |
| url | https://arxiv.org/abs/2412.15766 |