A note on singular oscillatory integrals with certain rational phases
Let $\Omega $ be homogeneous of degree zero with mean value zero, $P$ and $Q$ real polynomials on $\mathbb{R}^n$ with $Q(0)=0$ and $\Omega \in B_q^{0,0}(S^{n-1})$ for some $q>1.$ This note extends and improves a classical result of Stein and Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.418/ |
| _version_ | 1797651511146184704 |
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| author | Wang, Chenyan Wu, Huoxiong |
| author_facet | Wang, Chenyan Wu, Huoxiong |
| author_sort | Wang, Chenyan |
| collection | DOAJ |
| description | Let $\Omega $ be homogeneous of degree zero with mean value zero, $P$ and $Q$ real polynomials on $\mathbb{R}^n$ with $Q(0)=0$ and $\Omega \in B_q^{0,0}(S^{n-1})$ for some $q>1.$ This note extends and improves a classical result of Stein and Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to the following general form
\[ \left|\text{p.~v.}\int _{\mathbb{R}^n}e^{i(P(x)+1/Q(x))}\frac{\Omega (x/|x|)}{|x|^n}dx\right|\le B, \]
where $B$ depend only on $\Vert \Omega \Vert _{B_q^{0,0}(S^{n-1})}$, $n$ and the degrees of $P$ and $Q$, but not on their coefficients. |
| first_indexed | 2024-03-11T16:16:50Z |
| format | Article |
| id | doaj.art-fca3bdd3bba4473388db3284f1314c40 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:50Z |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-fca3bdd3bba4473388db3284f1314c402023-10-24T14:20:20ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G136337010.5802/crmath.41810.5802/crmath.418A note on singular oscillatory integrals with certain rational phasesWang, Chenyan0Wu, Huoxiong1School of Mathematical Sciences, Xiamen University, Xiamen 361005, ChinaSchool of Mathematical Sciences, Xiamen University, Xiamen 361005, ChinaLet $\Omega $ be homogeneous of degree zero with mean value zero, $P$ and $Q$ real polynomials on $\mathbb{R}^n$ with $Q(0)=0$ and $\Omega \in B_q^{0,0}(S^{n-1})$ for some $q>1.$ This note extends and improves a classical result of Stein and Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to the following general form \[ \left|\text{p.~v.}\int _{\mathbb{R}^n}e^{i(P(x)+1/Q(x))}\frac{\Omega (x/|x|)}{|x|^n}dx\right|\le B, \] where $B$ depend only on $\Vert \Omega \Vert _{B_q^{0,0}(S^{n-1})}$, $n$ and the degrees of $P$ and $Q$, but not on their coefficients.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.418/ |
| spellingShingle | Wang, Chenyan Wu, Huoxiong A note on singular oscillatory integrals with certain rational phases Comptes Rendus. Mathématique |
| title | A note on singular oscillatory integrals with certain rational phases |
| title_full | A note on singular oscillatory integrals with certain rational phases |
| title_fullStr | A note on singular oscillatory integrals with certain rational phases |
| title_full_unstemmed | A note on singular oscillatory integrals with certain rational phases |
| title_short | A note on singular oscillatory integrals with certain rational phases |
| title_sort | note on singular oscillatory integrals with certain rational phases |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.418/ |
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