Fourier decay and L^p Sobolev smoothing for weighted hypersurface measures in \mathbb R³

2026/06/30 by Michael Greenblatt
Mathematics · #math.CA #msc:42B20

paper · pdf

26 pages. v2: Various small corrections and improvements to the exposition. Comment: this paper builds on the author's unpublished preprint arXiv:1409.4059v3 and some parts of this paper are taken from this unpublished work

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We consider local hypersurface measures in \mathbb R³ whose density is allowed to have a weight function constructed from real analytic functions in a broad sense. We prove L^p Sobolev smoothing theorems for convolutions with such surface measures and Fourier transform decay rate results for these measures, generalizing and subsuming earlier results for smooth densities. Our theorems are sharp in an appropriate sense and can be described in terms of relatively simple properties of the surfaces and weight functions.

Citations

• Smooth and singular maximal averages over 2D hypersurfaces and associated Radon transforms
• Convolution kernels of 2D Fourier multipliers based on real analytic\n functions
• Uniform bounds for Fourier transforms of surface measures in R³ with nonsmooth density
• Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude
• Oscillatory Integral Decay, Sublevel Set Growth, and the Newton Polyhedron
• Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena
• Decay of the Fourier transform of surfaces with vanishing curvature
• Oscillating integrals and Newton polyhedra
• The Newton polyhedron and oscillatory integral operators
• Fourier transforms of irregular mixed homogeneous hypersurface measures