no. G10
Analyse harmonique
On the boundedness of a family of oscillatory singular integrals
Al-Qassem, Hussain1  ; Cheng, Leslie2 ; Pan, Yibiao3
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, Tome 361 (2023) no. G10, pp. 1673-1681

Let Ω∈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. ∫ ℝ n e i(P(x)+1/Q(x)) Ω(x/|x|) |x| n d x≤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 :
Accepté le :
Publié le :
DOI : 10.5802/crmath.523
Classification : 42B20, 42B30, 42B35
Keywords: oscillatory integrals, singular integrals, Calderón–Zygmund kernels, Hardy spaces

Al-Qassem, Hussain 1 ; Cheng, Leslie 2 ; Pan, Yibiao 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.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2023__361_G10_1673_0,
     author = {Al-Qassem, Hussain and Cheng, Leslie and Pan, Yibiao},
     title = {On the boundedness of a family of oscillatory singular integrals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1673--1681},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     number = {G10},
     doi = {10.5802/crmath.523},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.523/}
}
TY  - JOUR
AU  - Al-Qassem, Hussain
AU  - Cheng, Leslie
AU  - Pan, Yibiao
TI  - On the boundedness of a family of oscillatory singular integrals
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1673
EP  - 1681
VL  - 361
IS  - G10
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.523/
DO  - 10.5802/crmath.523
LA  - en
ID  - CRMATH_2023__361_G10_1673_0
ER  - 
%0 Journal Article
%A Al-Qassem, Hussain
%A Cheng, Leslie
%A Pan, Yibiao
%T On the boundedness of a family of oscillatory singular integrals
%J Comptes Rendus. Mathématique
%D 2023
%P 1673-1681
%V 361
%N G10
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.523/
%R 10.5802/crmath.523
%G en
%F CRMATH_2023__361_G10_1673_0
Al-Qassem, Hussain; Cheng, Leslie; Pan, Yibiao. On the boundedness of a family of oscillatory singular integrals. Comptes Rendus. Mathématique, Tome 361 (2023) no. G10, pp. 1673-1681. doi: 10.5802/crmath.523

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :