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/ |
| Summary: | 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. |
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| ISSN: | 1778-3569 |