A note on singular oscillatory integrals with certain rational phases

Chenyan Wang; Huoxiong Wu

doi:10.5802/crmath.418

Harmonic analysis

A note on singular oscillatory integrals with certain rational phases

Chenyan Wang ¹; Huoxiong Wu ¹

¹ School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 363-370.

Abstract

Let $Ω$ be homogeneous of degree zero with mean value zero, $P$ and $Q$ real polynomials on $ℝ^{n}$ with $Q (0) = 0$ and $Ω \in B_{q}^{0, 0} (S^{n - 1})$ for some $q > 1 .$ This note extends and improves a classical result of Stein and Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to the following general form

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

where $B$ depend only on ${∥ Ω ∥}_{B_{q}^{0, 0} (S^{n - 1})}$ , $n$ and the degrees of $P$ and $Q$ , but not on their coefficients.

Received: 2022-02-23
Accepted: 2022-09-01
Published online: 2023-01-12

DOI: 10.5802/crmath.418

Classification: 42B15, 42B20, 42A50, 42A45

Author's affiliations:

Chenyan Wang ¹; Huoxiong Wu ¹

¹ School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2023__361_G1_363_0,
     author = {Chenyan Wang and Huoxiong Wu},
     title = {A note on singular oscillatory integrals with certain rational phases},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {363--370},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.418},
     language = {en},
}

TY  - JOUR
AU  - Chenyan Wang
AU  - Huoxiong Wu
TI  - A note on singular oscillatory integrals with certain rational phases
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 363
EP  - 370
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.418
LA  - en
ID  - CRMATH_2023__361_G1_363_0
ER  -

%0 Journal Article
%A Chenyan Wang
%A Huoxiong Wu
%T A note on singular oscillatory integrals with certain rational phases
%J Comptes Rendus. Mathématique
%D 2023
%P 363-370
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.418
%G en
%F CRMATH_2023__361_G1_363_0

Chenyan Wang; Huoxiong Wu. A note on singular oscillatory integrals with certain rational phases. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 363-370. doi : 10.5802/crmath.418. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.418/

References
Cited by

[1] Hussain M. Al-Qassem; Leslie C. Cheng; Ayako Fukui; Yibiao Pan Weighted Estimates for Oscillatory Singular Integrals, J. Funct. Spaces Appl., Volume 2013 (2013), 543748, 3 pages | MR | Zbl

[2] Anthony Carbery; Michael Christ; James Wright Multidimensional van der Corput and sublevel set estimates, J. Am. Math. Soc., Volume 12 (1999) no. 4, pp. 981-1015 | DOI | MR | Zbl

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

[4] Masahiro Keitoku; Enji Sato Block spaces on the unit sphere in $ℝ^{n}$ , Proc. Am. Math. Soc., Volume 119 (1993) no. 2, pp. 453-455 | MR | Zbl

[5] S. Lu; Mitchell H. Taibleson; G. Weiss Spaces Generated by Blocks, Publishing House of Beijing Normal University, 1989

[6] Michael Papadimitrakis; Ioannis R. Parissis Singular oscillatory integrals on $ℝ^{n}$ , Math. Z., Volume 266 (2010) no. 1, pp. 169-179 | DOI | Zbl

[7] Ioannis R. Parissis A sharp bound for the Stein–Wainger oscillatory integral, Proc. Am. Math. Soc., Volume 136 (2008) no. 3, pp. 963-972 | DOI | MR | Zbl

[8] Elias M. Stein Oscillation integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis (Beijing, 1984) (Annals of Mathematics Studies), Volume 112, Princeton University Press, 1986, pp. 307-355 | Zbl

[9] Elias M. Stein; Stephen Wainger The estimation of an integral arising in multiplier transformations, Stud. Math., Volume 35 (1970), pp. 101-104 | DOI | MR | Zbl

[10] Xiaofeng Ye; Xiangrong Zhu A note on certain block spaces on the unit sphere, Acta Math. Sin., Engl. Ser., Volume 22 (2006) no. 6, pp. 1843-1846 | MR | Zbl

Cited by Sources:

Comments - Policy