On the Fourier decay of multiplicative convolutions
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911784449867776 |
|---|---|
| author | Orponen, Tuomas de Saxcé, Nicolas Shmerkin, Pablo |
| author_facet | Orponen, Tuomas de Saxcé, Nicolas Shmerkin, Pablo |
| contents | We prove the following. Let $μ_{1},\ldots,μ_{n}$ be Borel probability measures on $[-1,1]$ such that $μ_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative convolution of the measures $μ_{1},\ldots,μ_{n}$ has power Fourier decay: there exists a constant $τ= τ(s_{1},\ldots,s_{n}) > 0$ such that
\[
\left| \int e^{-2πi ξ\cdot x_{1}\cdots x_{n}} \, dμ_{1}(x_{1}) \cdots \, dμ_{n}(x_{n}) \right| \leq |ξ|^{-τ}
\]
for sufficiently large $|ξ|$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_{j}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03068 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Fourier decay of multiplicative convolutions Orponen, Tuomas de Saxcé, Nicolas Shmerkin, Pablo Classical Analysis and ODEs 42A38, 28A20, 11L07 We prove the following. Let $μ_{1},\ldots,μ_{n}$ be Borel probability measures on $[-1,1]$ such that $μ_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative convolution of the measures $μ_{1},\ldots,μ_{n}$ has power Fourier decay: there exists a constant $τ= τ(s_{1},\ldots,s_{n}) > 0$ such that \[ \left| \int e^{-2πi ξ\cdot x_{1}\cdots x_{n}} \, dμ_{1}(x_{1}) \cdots \, dμ_{n}(x_{n}) \right| \leq |ξ|^{-τ} \] for sufficiently large $|ξ|$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_{j}$. |
| title | On the Fourier decay of multiplicative convolutions |
| topic | Classical Analysis and ODEs 42A38, 28A20, 11L07 |
| url | https://arxiv.org/abs/2309.03068 |