Oscillatory integral operators with low-order degeneracies. (English) Zbl 1033.35164

Greenleaf, Allan; Seeger, Andreas

Duke Math. J. 112, No. 3, 397-420 (2002).

This paper provides sharp estimates for the norm of oscillatory integrals of the form \[ T_\lambda f(x)= \int e^{i\lambda\Phi(x,z)}\sigma(x, z)\,dz, \] as bounded operators on \(L^2\), where \(\Phi\) is a \(C^\infty\) real-valued function defined on \(\Omega_L\times \Omega_R\), \(\Omega_L\) and \(\Omega_L\) being open sets in \(\mathbb{R}^d\), \(\sigma\in C^\infty_0\) \((\Omega_L\times\Omega_R)\) and \(\lambda\) is a positive parameter. The interest of the authors is in describing how the norm \(\| T_\lambda\|_{L^2\to L^2}\) decays as \(\lambda\to \infty\), under special assumptions on \(\Phi\). In general, this decay is determined by the properties of the canonical relation \[ {\mathcal C}= \{(x,\Phi_x, z-\Phi_z): (x,z)\in \Omega_L\times \Omega_R\} \] as a subset of the product of the cotangent spaces \(T^*\Omega_L\times T^*\Omega_R\). More specifically, it is determined by the properties of the projections \(\pi_L:{\mathcal C}\to T^*\Omega_L\) and \(\pi_R:{\mathcal C}\to T^*\Omega_R\) defined as \[ \pi_L: (x,z)\to (x,\Phi_x(x,z)),\quad \pi_R:(x,z)\to (x,-\Phi_z(x,z)). \] Here, \(\Phi_x\) and \(\Phi_z\) denote the partial gradients. When \(\text{det\,}\Phi_{xz}(x,z)\) is not zero, that is to say when \({\mathcal C}\) is locally the graph of a canonical transformation then \(\| T_\lambda\|_{L^2\to L^2}= O(\lambda^{-d/2})\) [L. Hörmander, Acta Math. 127, 79–183 (1971; Zbl 0212.46601) and Ark Mat. 11, 1–11 (1973; Zbl 0254.42010)]. If the projections have singularities, the decay in \(\lambda\) is less.
The authors provide a careful review of known results for various types of singularities. They state and prove \(L^2\)-estimates under mixed conditions on the singularities of the projections. In particular, if the only singularities of one of the projections are the so-called Whitney folds, Whitney cups or swallowtails, \[ \| T_\lambda\|_{L^2\to L^2}= O(\lambda^{-(d- 1)/2- 1/8})\quad\text{for }\lambda\geq 1. \] If the only singularities of both projections are Whitney folds or Whitney cups, \[ \| T_\lambda\|_{L^2\to L^2}= O(\lambda^{-(d- 1)/2- 1/4})\quad\text{for }\lambda\geq 1. \] The authors also state and prove \(L^2\)-estimates under other mixed conditions on the projections. As a corollary to these results, they obtain sharp \(L^2\)-Sobolev estimates for Fourier integral operators. The proofs of these results are based on some delicate decompositions and estimates.
The paper ends with a comprehensive list of references.

Reviewer: Josefina Alvarez (Las Cruces)

MSC:

35S30	Fourier integral operators applied to PDEs
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47G10	Integral operators
58J40	Pseudodifferential and Fourier integral operators on manifolds

Keywords:

sharp estimates for the norm of oscillatory integrals; singularities; Whitney folds; Whitney cups; swallowtails; \(L^2\)-Sobolev estimates for Fourier integral operators

Citations:

Zbl 0212.46601; Zbl 0254.42010

References:

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