\(L^p\) decay estimates for weighted oscillatory integral operators on \(\mathbb R\). (English) Zbl 1097.45007

Pramanik, Malabika; Yang, Chan Woo

Rev. Mat. Iberoam. 21, No. 3, 1071-1095 (2005).

If \(f\) and \(g\) are real-analytic functions on a neighborhood of the origin in \(\mathbb R^2\) with \(f(0,0)=g(0,0)=0\), \(\chi\) is a smooth function of compact support in this neighborhood of \((0,0)\), the decay rate in \(\lambda\) of \(\| T_{\lambda}\| _{L^p}\) as \(\lambda\to\infty\) is studied, where \[ T_{\lambda}\phi(x)=\int_{\mathbb R}e^{i\lambda f(x, y)}| g(x, y)| ^{\epsilon/2}\chi(x,y)\phi(y)\, dy, \] is the oscillatory integral operator with the phase function \(f\) and weight \(g\). In this paper a region \(\mathcal A\subset[0,1]\times \mathbb R\) is defined. For the necessity, it is shown that if \(T_\lambda\) is bounded in \(L^p(\mathbb R)\) with \(\| T_\lambda\| _{L^p}\leq O(\lambda^{-\alpha})\), then \((1/p,\alpha)\in\mathcal A\). Moreover, if \((1/2,\alpha)\in\mathcal A\), then \(\| T_\lambda\| _{L^2}\leq O(\lambda^ {-\alpha})\), and if \((1/p,\alpha)\in\text{int}(\mathcal A)\), then \(\| T_{\lambda}\| _ {L^p}\leq O(\lambda^{-\alpha})\).

Reviewer: Ching-Hua Chang (Hualien)

MSC:

45M05	Asymptotics of solutions to integral equations
45P05	Integral operators

Keywords:

oscillatory integral operators; decay rate; weight; asymptotics

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