A Class of Oscillatory Singular Integrals with Rough Kernels and Fewnomials Phases

Abstract

This paper is concerned with the oscillatory singular integral operator \(T_Q\) defined by

$$\begin{aligned} T_Qf(x)=\mathrm{p.v.}\int _{{\mathbb {R}^n}}f(x-y)\frac{\Omega (y)}{|y|^n}e^{iQ(|y|)}dy, \end{aligned}$$

where \(Q(t)=\sum _{1\le i\le m}a_it^{\alpha _i}\) is a real-valued polynomial on \(\mathbb {R}\), \(\Omega \) is a homogenous function of degree zero on \(\mathbb {R}^n\) with mean value zero on the unit sphere \(S^{n-1}\). Under the assumption of that \(\Omega \in H^1(S^{n-1})\), the authors show that \(T_Q\) is bounded on the weighted Lebesgue spaces \(L^p(\omega )\) for \(1<p<\infty \) and \(\omega \in \tilde{A}_{p}^{I}(\mathbb {R}_+)\) with the uniform bound only depending on m, the number of monomials in polynomial Q, not on the degree of Q as in the previous results. This result is new even in the case \(\omega \equiv 1\), which can also be regarded as an improvement and generalization of the result obtained by Guo in [New York J. Math. 23 (2017), 1733-1738].