2024/03/19 by Michael Greenblatt, Greenblatt, Michael
Mathematics · Physics and Astronomy · #41A60 #42A38 #42B99 #Advanced Differential Geometry Research #Classical Analysis and ODEs (math.CA) #Cosmology and Gravitation Theories #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2403.12751
openalex publication_date 2024/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on (x1 ∂x1 f(x) ,..., xn ∂xnf(x)), where f(x) is the phase function. We prove a theorem which provides estimates that are uniform with respect to linear perturbations of the phase and investigate some consequences. When the phase function is quasi-homogeneous the theorem gives estimates for the associated surface measure Fourier transforms that are generally not too far off from being sharp. In addition, the theorem provides a new proof, up to endpoints, that the well-known oscillatory integral estimates of Varchenko [V] when the Newton polyhedron of the phase function is nondegenerate extend to corresponding bounds for surface measure Fourier transforms when the index is less than (1)/(2). A sharp version of this was originally proven in [G2].