Fefferman multiplier theorem for Hardy martingales
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908527565471744 |
|---|---|
| author | Rzeszut, Maciej |
| author_facet | Rzeszut, Maciej |
| contents | A well-known theorem due to Fefferman provides a characterization of Fourier multipliers from $H^1(\mathbb{T})$ to $\ell^1$, i.e. sequences $\left(λ_n\right)_{n=0}^\infty$ such that \[\sum_{n=0}^\infty \left|λ_n \widehat{f}(n)\right|\lesssim \|f\|_{L^1(\mathbb{T})},\] where $f(x)=\sum_{n=0}^\infty \widehat{f}(n)e^{inx}$. We extend it to the space $H^1\left(\mathbb{T}^\mathbb{N}\right)$ of Hardy martingales, i.e. the subspace of $L^1$ on the countable product $\mathbb{T}^\mathbb{N}$ consisting of all $f$ such that the differences $Δ_nf=f_{n}-f_{n-1}$ of the martingale wrt the standard filtration generated by $f$ satisfy \[\left(t\mapsto Δ_n f\left(x_1,\ldots,x_{n-1},t\right)\right)\in H^1(\mathbb{T}). \] The key ingredient is a theorem due to P. F. X. Müller stating that the classical Davis-Garsia decomposition \[\mathbb{E} \left(\sum_{n=0}^\infty \left|Δ_n f\right|^2\right)^\frac{1}{2}\simeq \inf_{f=g+h} \mathbb{E}\sum_{n=0}^\infty \left|Δ_n g\right|+ \mathbb{E}\left(\sum_{n=0}^\infty \mathbb{E}\left(\left|Δ_n f\right|^2\mid \mathcal{F}_{n-1}\right)\right)^\frac{1}{2}\] may be done within the space of Hardy martingales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fefferman multiplier theorem for Hardy martingales Rzeszut, Maciej Probability Functional Analysis A well-known theorem due to Fefferman provides a characterization of Fourier multipliers from $H^1(\mathbb{T})$ to $\ell^1$, i.e. sequences $\left(λ_n\right)_{n=0}^\infty$ such that \[\sum_{n=0}^\infty \left|λ_n \widehat{f}(n)\right|\lesssim \|f\|_{L^1(\mathbb{T})},\] where $f(x)=\sum_{n=0}^\infty \widehat{f}(n)e^{inx}$. We extend it to the space $H^1\left(\mathbb{T}^\mathbb{N}\right)$ of Hardy martingales, i.e. the subspace of $L^1$ on the countable product $\mathbb{T}^\mathbb{N}$ consisting of all $f$ such that the differences $Δ_nf=f_{n}-f_{n-1}$ of the martingale wrt the standard filtration generated by $f$ satisfy \[\left(t\mapsto Δ_n f\left(x_1,\ldots,x_{n-1},t\right)\right)\in H^1(\mathbb{T}). \] The key ingredient is a theorem due to P. F. X. Müller stating that the classical Davis-Garsia decomposition \[\mathbb{E} \left(\sum_{n=0}^\infty \left|Δ_n f\right|^2\right)^\frac{1}{2}\simeq \inf_{f=g+h} \mathbb{E}\sum_{n=0}^\infty \left|Δ_n g\right|+ \mathbb{E}\left(\sum_{n=0}^\infty \mathbb{E}\left(\left|Δ_n f\right|^2\mid \mathcal{F}_{n-1}\right)\right)^\frac{1}{2}\] may be done within the space of Hardy martingales. |
| title | Fefferman multiplier theorem for Hardy martingales |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2509.07616 |