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...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-11-01
|
| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.523/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206220815859712 |
|---|---|
| author | Al-Qassem, Hussain Cheng, Leslie Pan, Yibiao |
| author_facet | Al-Qassem, Hussain Cheng, Leslie Pan, Yibiao |
| author_sort | Al-Qassem, Hussain |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-3642aa2954fb4d289370073bfb3ba4a9 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3642aa2954fb4d289370073bfb3ba4a92025-02-07T11:11:47ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G101673168110.5802/crmath.52310.5802/crmath.523On the boundedness of a family of oscillatory singular integralsAl-Qassem, Hussain0Cheng, Leslie1Pan, Yibiao2Mathematics Program, Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, 2713, Doha, QatarDepartment of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, U.S.A.Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.523/oscillatory integralssingular integralsCalderón–Zygmund kernelsHardy spaces |
| spellingShingle | Al-Qassem, Hussain Cheng, Leslie Pan, Yibiao On the boundedness of a family of oscillatory singular integrals Comptes Rendus. Mathématique oscillatory integrals singular integrals Calderón–Zygmund kernels Hardy spaces |
| title | On the boundedness of a family of oscillatory singular integrals |
| title_full | On the boundedness of a family of oscillatory singular integrals |
| title_fullStr | On the boundedness of a family of oscillatory singular integrals |
| title_full_unstemmed | On the boundedness of a family of oscillatory singular integrals |
| title_short | On the boundedness of a family of oscillatory singular integrals |
| title_sort | on the boundedness of a family of oscillatory singular integrals |
| topic | oscillatory integrals singular integrals Calderón–Zygmund kernels Hardy spaces |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.523/ |
| work_keys_str_mv | AT alqassemhussain ontheboundednessofafamilyofoscillatorysingularintegrals AT chengleslie ontheboundednessofafamilyofoscillatorysingularintegrals AT panyibiao ontheboundednessofafamilyofoscillatorysingularintegrals |