Hardy's Theorem for the $(k,\frac{2}{n})-$Fourier Transform
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914489860882432 |
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| author | Jilani, Hanen Negzaoui, Selma |
| author_facet | Jilani, Hanen Negzaoui, Selma |
| contents | By comparing a function and its $(k, \frac{2}{n})-$Fourier transform to a Gaussian analogue, $e^{-na|x|^\frac{2}{n}}$, we establish a Hardy-type uncertainty principle using Phragmén-Lindlöf lemma. Furthermore, we investigate the heat equation in this context, deriving a dynamical version of Hardy's theorem that illustrates the temporal evolution of the uncertainty principle. We also extend our results to $L^p-L^q$ versions, proving Miyachi-type and Cowling-Price-type theorems for the $(k,\frac{2}{n})$-Fourier transform. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_01094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hardy's Theorem for the $(k,\frac{2}{n})-$Fourier Transform Jilani, Hanen Negzaoui, Selma Classical Analysis and ODEs 42A38, 44A15, 33C10, 35A22 By comparing a function and its $(k, \frac{2}{n})-$Fourier transform to a Gaussian analogue, $e^{-na|x|^\frac{2}{n}}$, we establish a Hardy-type uncertainty principle using Phragmén-Lindlöf lemma. Furthermore, we investigate the heat equation in this context, deriving a dynamical version of Hardy's theorem that illustrates the temporal evolution of the uncertainty principle. We also extend our results to $L^p-L^q$ versions, proving Miyachi-type and Cowling-Price-type theorems for the $(k,\frac{2}{n})$-Fourier transform. |
| title | Hardy's Theorem for the $(k,\frac{2}{n})-$Fourier Transform |
| topic | Classical Analysis and ODEs 42A38, 44A15, 33C10, 35A22 |
| url | https://arxiv.org/abs/2503.01094 |