2014/11/17 by Philip T. Gressman, Gressman, Philip T.
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Classical Analysis and ODEs (math.CA) #Cosmology and Gravitation Theories #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1411.4680
openalex publication_date 2014/11/17 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We consider the question of when it is possible to force a degenerate scalar\noscillatory integral to decay as fast as a nondegenerate one by restricting the\nsupport to the region where the Hessian determinant of the phase is bounded\nbelow. We show in two dimensions that the desired outcome is not always\npossible, but does occur for a broad class of phases which may be described in\nterms of the Newton polygon. The estimates obtained are uniform with respect to\nlinear perturbation of the phase and uniform in the cutoff value of the Hessian\ndeterminant. In the course of the proof, we investigate a\ngeometrically-invariant approach to making uniform estimates of qualitatively\nnondegenerate oscillatory integrals. The approach illuminates a previously\nunknown, fundamental relationship between the asymptotics of oscillatory\nintegrals and the Schr "odinger equation.\n