On the boundedness of a family of oscillatory singular integrals
Let $\Omega \in H^1(\mathbb{S}^{n-1})$ with mean value zero, $P$ and $Q$ be polynomials in $n$ variables with real coefficients and $Q(0)=0$. We prove that \[ \Biggl |\mbox {p.v.}\int _{\mathbb{R}^n}e^{i(P(x)+1/Q(x))}\frac{\Omega (x/|x|)}{|x|^n}\mathrm{d}x\Biggr | \le A \Vert \Omega \Vert _{H^1(\mat...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.523/ |
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| Summary: | Let $\Omega \in H^1(\mathbb{S}^{n-1})$ with mean value zero, $P$ and $Q$ be polynomials in $n$ variables with real coefficients and $Q(0)=0$. We prove that
\[ \Biggl |\mbox {p.v.}\int _{\mathbb{R}^n}e^{i(P(x)+1/Q(x))}\frac{\Omega (x/|x|)}{|x|^n}\mathrm{d}x\Biggr | \le A \Vert \Omega \Vert _{H^1(\mathbb{S}^{n-1})} \]
where $A$ may depend on $n$, $\deg (P)$ and $\deg (Q)$, but not otherwise on the coefficients of $P$ and $Q$.The above result answers an open question posed in [13]. Additional boundedness results of similar nature are also obtained. |
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| ISSN: | 1778-3569 |