The Fourier transform in weighted rearrangement invariant spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909251061940224 |
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| author | Mastyło, Mieczysław Sinnamon, Gord |
| author_facet | Mastyło, Mieczysław Sinnamon, Gord |
| contents | It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on $\mathbb R^n$, modified by a weight, then the weight is bounded above and below and the space is equivalent to $L^2$. Also, if it is bounded from $L^p$ to $L^q$, each modified by the same weight, then the weight is bounded above and below and $1\le p=q'\le 2$. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_07047 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Fourier transform in weighted rearrangement invariant spaces Mastyło, Mieczysław Sinnamon, Gord Functional Analysis Analysis of PDEs 35B30 (Primary) 42B15 (Secondary) It is shown that if the Fourier transform is a bounded map on a rearrangement-invariant space of functions on $\mathbb R^n$, modified by a weight, then the weight is bounded above and below and the space is equivalent to $L^2$. Also, if it is bounded from $L^p$ to $L^q$, each modified by the same weight, then the weight is bounded above and below and $1\le p=q'\le 2$. Applications prove the non-boundedness on these spaces of an operator related to the Schrödinger equation. |
| title | The Fourier transform in weighted rearrangement invariant spaces |
| topic | Functional Analysis Analysis of PDEs 35B30 (Primary) 42B15 (Secondary) |
| url | https://arxiv.org/abs/2302.07047 |