Estimates for strongly singular operators along curves
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918483208437760 |
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| author | Folch-Gabayet, Magali Sáenz, Ricardo A. |
| author_facet | Folch-Gabayet, Magali Sáenz, Ricardo A. |
| contents | For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2πi γ(t)}}{ψ(t)} dt, \] where $k\ge1$ and $ψ$ and $γ$ are functions defined near the origin such that $ψ(t)\to 0$ and $|γ(t)|\to\infty$ as $t\to 0$. We give sufficient regularity and growth conditions on $ψ$ and $γ$ for its multiplier to be a bounded function, and thus for the operator to be bounded on $L^2(\mathbb R^2)$. We consider an extension to $L^p(\mathbb R^2)$, for certain $p's$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimates for strongly singular operators along curves Folch-Gabayet, Magali Sáenz, Ricardo A. Classical Analysis and ODEs 42B20, 44A15 For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2πi γ(t)}}{ψ(t)} dt, \] where $k\ge1$ and $ψ$ and $γ$ are functions defined near the origin such that $ψ(t)\to 0$ and $|γ(t)|\to\infty$ as $t\to 0$. We give sufficient regularity and growth conditions on $ψ$ and $γ$ for its multiplier to be a bounded function, and thus for the operator to be bounded on $L^2(\mathbb R^2)$. We consider an extension to $L^p(\mathbb R^2)$, for certain $p's$. |
| title | Estimates for strongly singular operators along curves |
| topic | Classical Analysis and ODEs 42B20, 44A15 |
| url | https://arxiv.org/abs/2412.07703 |