On the boundedness of a family of oscillatory singular integrals

Hussain Al-Qassem; Leslie Cheng; Yibiao Pan

doi:10.5802/crmath.523

Analyse harmonique

On the boundedness of a family of oscillatory singular integrals

Hussain Al-Qassem ¹ ; Leslie Cheng ² ; Yibiao Pan ³

¹ Mathematics Program, Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, 2713, Doha, Qatar
² Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, U.S.A.
³ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1673-1681.

Résumé

Let $Ω \in H^{1} (𝕊^{n - 1})$ with mean value zero, $P$ and $Q$ be polynomials in $n$ variables with real coefficients and $Q (0) = 0$ . We prove that

|p . v . \int_{ℝ^{n}} e^{i (P (x) + 1 / Q (x))} \frac{Ω (x / | x |)}{{| x |}^{n}} d x| \leq A {∥ Ω ∥}_{H^{1} (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.

Reçu le : 2023-01-21
Accepté le : 2023-06-09
Publié le : 2023-11-17

DOI : 10.5802/crmath.523

Classification : 42B20, 42B30, 42B35
Mots clés : oscillatory integrals, singular integrals, Calderón–Zygmund kernels, Hardy spaces

Affiliations des auteurs :

Hussain Al-Qassem ¹ ; Leslie Cheng ² ; Yibiao Pan ³

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the boundedness of a family of oscillatory singular integrals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1673--1681},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.523},
     language = {en},
}

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Hussain Al-Qassem; Leslie Cheng; Yibiao Pan. On the boundedness of a family of oscillatory singular integrals. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1673-1681. doi : 10.5802/crmath.523. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.523/

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