On the boundedness of a family of oscillatory singular integrals

Abstract

Let $\Omega \in H^1(\mathbb{S}^{n-1})$ with mean value zero, $P$ and $Q$ be polynomials in $n$ variables with real coefficients and $Q(0)=0$. We prove that \[ \Biggl |\mbox {p.v.}\int _{\mathbb{R}^n}e^{i(P(x)+1/Q(x))}\frac{\Omega (x/|x|)}{|x|^n}\mathrm{d}x\Biggr | \le A \Vert \Omega \Vert _{H^1(\mathbb{S}^{n-1})} \] 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.

Keywords

• oscillatory integrals
• singular integrals
• Calderón–Zygmund kernels
• Hardy spaces