An oscillatory integral estimate associated to rational phases (Q1407753)

The authors study \(L^p({\mathbb R}^n)\)-mapping properties (\(1<p<\infty\)) of the singular integral operator \[ H_{\gamma} f(x)=\text{p.v.}\int_{-\infty}^{\infty} f(x-\gamma(t))\,\frac{dt}{t} \] by obtaining the uniform oscillatory integral estimate \[ \left| \text{p.v.}\int_{-\infty}^{\infty} e^{i\varphi(t)}\,\frac{dt}{t}\right| \leq C_{\varphi}\tag{*} \] for a proper function \(\varphi\) defined on \(\mathbb R\) and a constant \(C_{\varphi}>0\) depending on \(\varphi\). Here, \(\gamma(t)=(\varphi_1(t),\cdots,\varphi(t))\) and each \(\varphi_j\) is a proper function defined on \(\mathbb R\). Originally, \textit{E. M. Stein} and \textit{S. Wainger} [Stud. Math. 35, 101--104, (1970; Zbl 0202.12401)] proved the \(L^p({\mathbb R}^n)\)-mapping properties (\(1<p<\infty\)) of \(H_{\gamma}\) when \(\varphi=P\) is a real polynomial of degree \(d\) and \(\gamma(t)=(P_1(t),\cdots,P_n(t))\) for real polynomials \(P_j\), by obtaining a constant \(C_{\varphi}=C_d>0\) depending only on \(d\) but not depending on the coefficients of \(P\) in the estimate (*). In this paper, the authors prove that if \(\varphi=P(t)/Q(t)\) for real polynomials \(P\) and \(Q\), and \(\gamma(t)=(t,P(t)/Q(t))\), then \(H_{\gamma}\) is bounded on \(L^p({\mathbb R}^2), 1<p<\infty\), with bounds independent of the coefficients of \(P\) and \(Q\), by obtaining a constant \(C_{\varphi}=C(\deg(P),\deg(Q))>0\) not depending on the coefficients of \(P\) and \(Q\) in (*).