Analyse harmonique

A note on singular oscillatory integrals with certain rational phases

Wang, Chenyan¹ ; Wu, Huoxiong¹ 

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

Comptes Rendus. Mathématique, Tome 361 (2023) no. G1, pp. 363-370

Résumé

Let $\Omega$ be homogeneous of degree zero with mean value zero, $P$ and $Q$ real polynomials on ${ℝ}^n$ with $Q\left(0\right)=0$ and $\Omega \in B_q^{0,0}\left(S^{n-1}\right)$ 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

$$\left|\text{p.}\text{v.}{\int }_{{ℝ}^n}{e}^{i\left(P\right(x)+1/Q(x\left)\right)}\frac{\Omega (x/|x\left|\right)}{{\left|x\right|}^n}dx\right|\le B,$$

where $B$ depend only on ${\parallel \Omega \parallel }_{B_q^{0,0}\left(S^{n-1}\right)}$, $n$ and the degrees of $P$ and $Q$, but not on their coefficients.

Détail

Reçu le : 2022-02-23
Accepté le : 2022-09-01
Publié le : 2023-01-12

DOI : 10.5802/crmath.418

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits