\(L^2\) estimates for oscillatory singular integral operators with analytic phrases (Q2386520)

Let \(S\) and \(\chi\) be two \(C^\infty\) real functions on \({\mathbb R}\times{\mathbb R}\), and \(\chi\) be compactly supported in a small neighborhood of the origin in \({\mathbb R}\times{\mathbb R}\). In this paper, the authors consider the following oscillatory singular integral operator that for \(\lambda\in\mathbb R\) and \(x\in\mathbb R\), \[ T_\lambda f(x)=\int_{\mathbb R}e^{i\lambda S(x,y)}K(x-y)\chi(x,y)f(y)\,dy, \] where \(K\) is a \(C^2\) function on \(\mathbb R\setminus\{0\}\) and there exist \(\mu\in (0,1)\) and \(A\geq 0\) such that for all \(z\in\mathbb R\) and \(i=0,\,1,\,2\), \[ \bigg| \frac {d^i}{dz^i}K(z)\bigg| \leq A| z| ^{-\mu-i}. \] They prove that if \(S\) is real analytic on \({\mathbb R}\times{\mathbb R}\) and \(\delta\) is its Newton decay, then \(T_\lambda\) is bounded on \(L^2(\mathbb R)\) with the operator norm no more than \(C| \lambda| ^{-\delta(1-\mu)/2}\) as \(\lambda\to\infty\).