The Fourier transform is an extremizer of a class of bounded operators
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913669650055168 |
|---|---|
| author | Saucedo, Miquel Tikhonov, Sergey |
| author_facet | Saucedo, Miquel Tikhonov, Sergey |
| contents | We show that, for a natural class of rearrangement admissible spaces $X$ and $Y$, the Fourier operator is bounded between $X$ and $Y$ if and only if any operator of joint strong type $(1,\infty; 2,2)$ is also bounded between $X$ and $Y$.
By using this result, we fully characterize the weighted Fourier inequalities of the form $$\qquad\qquad \lVert\widehat{f}u \rVert_q \leq C \lVert fv\rVert_p,\quad 1\leq p\leq \infty,\,0<q\leq \infty,$$ for radially monotone weights $(u,v)$.
This answers a long-standing problem posed by Benedetto-Heinig, Jurkat-Sampson, and Muckenhoupt. In the case of $p\le q$, such a characterization has been known since the 1980s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fourier transform is an extremizer of a class of bounded operators Saucedo, Miquel Tikhonov, Sergey Classical Analysis and ODEs Functional Analysis 42B10, 42B35 (Primary) 46E30 (Secondary) We show that, for a natural class of rearrangement admissible spaces $X$ and $Y$, the Fourier operator is bounded between $X$ and $Y$ if and only if any operator of joint strong type $(1,\infty; 2,2)$ is also bounded between $X$ and $Y$. By using this result, we fully characterize the weighted Fourier inequalities of the form $$\qquad\qquad \lVert\widehat{f}u \rVert_q \leq C \lVert fv\rVert_p,\quad 1\leq p\leq \infty,\,0<q\leq \infty,$$ for radially monotone weights $(u,v)$. This answers a long-standing problem posed by Benedetto-Heinig, Jurkat-Sampson, and Muckenhoupt. In the case of $p\le q$, such a characterization has been known since the 1980s. |
| title | The Fourier transform is an extremizer of a class of bounded operators |
| topic | Classical Analysis and ODEs Functional Analysis 42B10, 42B35 (Primary) 46E30 (Secondary) |
| url | https://arxiv.org/abs/2501.17505 |