Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908279317200896 |
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| author | Krause, Ben |
| author_facet | Krause, Ben |
| contents | We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our methods, we are able to quickly extend a result of Bourgain, namely the pointwise convergence of ergodic averages of integer parts of real-variables polynomials, to a broader class of functions, previously considered in a wide range of contexts by Boshernitzan-Jones-Wierdl. Namely, the following averages converge almost everywhere \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor P(n) \rfloor} f, \; \; \; f \in L^p(X,μ), \ P \in \mathbb{R}[\cdot], \] for any $σ$-finite measure space equipped with a measure-preserving transformation, $T:X \to X$, whenever $1 < p \leq \infty$ if $P$ is linear, and $4/3 < p \leq \infty$ otherwise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory Krause, Ben Classical Analysis and ODEs Dynamical Systems We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our methods, we are able to quickly extend a result of Bourgain, namely the pointwise convergence of ergodic averages of integer parts of real-variables polynomials, to a broader class of functions, previously considered in a wide range of contexts by Boshernitzan-Jones-Wierdl. Namely, the following averages converge almost everywhere \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor P(n) \rfloor} f, \; \; \; f \in L^p(X,μ), \ P \in \mathbb{R}[\cdot], \] for any $σ$-finite measure space equipped with a measure-preserving transformation, $T:X \to X$, whenever $1 < p \leq \infty$ if $P$ is linear, and $4/3 < p \leq \infty$ otherwise. |
| title | Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory |
| topic | Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2502.12887 |