Comptes Rendus
Harmonic analysis
On the boundedness of a family of oscillatory singular integrals
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1673-1681.

Let ΩH 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. n e i(P(x)+1/Q(x)) Ω(x/|x|) |x| n d xAΩ 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.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.523
Classification: 42B20, 42B30, 42B35
Keywords: oscillatory integrals, singular integrals, Calderón–Zygmund kernels, Hardy spaces

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.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

[1] Lennart Carleson On convergence and growth of partial sums of Fourier series, Acta Math., Volume 116 (1966), pp. 135-157 | DOI | MR | Zbl

[2] Ronald R. Coifman; Guido Weiss Extensions of Hardy spaces and their use in analysis, Bull. Am. Math. Soc., Volume 83 (1977), pp. 569-645 | DOI | MR | Zbl

[3] Leonardo Colzani Hardy spaces on spheres, Ph. D. Thesis, Washington University, St. Louis (1982)

[4] Charles Fefferman Inequalities for strongly singular convolution operators, Acta Math., Volume 124 (1970), pp. 9-36 | DOI | MR | Zbl

[5] Magali Folch-Gabayet; James Wright An estimate for a family of oscillatory integrals, Stud. Math., Volume 154 (2003) no. 1, pp. 89-97 | DOI | Zbl

[6] Martin Golubitsky; Victor Guillemin Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Springer, 1973 | DOI

[7] Loukas Grafakos Classical and Modern Fourier Analysis, Pearson/Prentice Hall, 2004

[8] Duong Phong; Elias M. Stein Hilbert integrals, singular integrals, and Radon transforms I, Acta Math., Volume 157 (1986), pp. 99-157 | DOI | MR

[9] Fulvio Ricci; Elias M. Stein Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal., Volume 73 (1987), pp. 179-194 | DOI | MR | Zbl

[10] Elias M. Stein Beijing Lectures in Harmonic Analysis, Annals of Mathematics Studies, 112, Princeton University Press, 1986

[11] Elias M. Stein Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993

[12] Elias M. Stein; Stephen Wainger Problems in harmonic analysis related to curvature, Bull. Am. Math. Soc., Volume 84 (1978), pp. 1239-1295 | DOI | MR | Zbl

[13] Chenyan Wang; Huoxiong Wu A note on singular oscillatory integrals with certain rational phases, C. R. Math. Acad. Sci. Paris, Volume 361 (2023), pp. 363-370 | MR | Zbl

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