On a Carleson-Radon Transform (the non-resonant setting)
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912103214874624 |
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| author | Hsu, Martin Lie, Victor |
| author_facet | Hsu, Martin Lie, Victor |
| contents | Given a curve $\vecγ=(t^{α_1}, t^{α_2}, t^{α_3})$ with $\vecα=(α_1,α_2,α_3)\in \mathbb{R}_{+}^3$, we define the Carleson-Radon transform along $\vecγ$ by the formula $$ C_{[\vecα]}f(x,y):=\sup_{a\in \mathbb{R}}\left|p.v.\,\int_{\mathbb{R}} f (x-t^{α_1},y-t^{α_2})\,e^{i\,a\,t^{α_3}}\,\frac{dt}{t}\right|\,.$$ We show that in the \emph{non-resonant} case, that is, when the coordinates of $\vecα$ are pairwise disjoint, our operator $ C_{[\vecα]}$ is $L^p$ bounded for any $1<p<\infty$. Our proof relies on the (Rank I) LGC-methodology introduced in arXiv:1902.03807 and employs three key elements:
1) a partition of the time-frequency plane with a linearizing effect on both the argument of the input function and on the phase of the kernel;
2) a sparse-uniform dichotomy analysis of the Gabor coefficients associated with the input/output function;
3) a level set analysis of the time-frequency correlation set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Carleson-Radon Transform (the non-resonant setting) Hsu, Martin Lie, Victor Classical Analysis and ODEs 42B05, 42B08, 42B20, 42A16, 42A20 Given a curve $\vecγ=(t^{α_1}, t^{α_2}, t^{α_3})$ with $\vecα=(α_1,α_2,α_3)\in \mathbb{R}_{+}^3$, we define the Carleson-Radon transform along $\vecγ$ by the formula $$ C_{[\vecα]}f(x,y):=\sup_{a\in \mathbb{R}}\left|p.v.\,\int_{\mathbb{R}} f (x-t^{α_1},y-t^{α_2})\,e^{i\,a\,t^{α_3}}\,\frac{dt}{t}\right|\,.$$ We show that in the \emph{non-resonant} case, that is, when the coordinates of $\vecα$ are pairwise disjoint, our operator $ C_{[\vecα]}$ is $L^p$ bounded for any $1<p<\infty$. Our proof relies on the (Rank I) LGC-methodology introduced in arXiv:1902.03807 and employs three key elements: 1) a partition of the time-frequency plane with a linearizing effect on both the argument of the input function and on the phase of the kernel; 2) a sparse-uniform dichotomy analysis of the Gabor coefficients associated with the input/output function; 3) a level set analysis of the time-frequency correlation set. |
| title | On a Carleson-Radon Transform (the non-resonant setting) |
| topic | Classical Analysis and ODEs 42B05, 42B08, 42B20, 42A16, 42A20 |
| url | https://arxiv.org/abs/2411.01660 |