Geodesic
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

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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 
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     title = {On the boundedness of a family of oscillatory singular integrals},
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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

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