Fourier transforms of powers of well-behaved 2D real analytic functions

TL;DR

This paper extends previous work on Fourier transforms of 2D real-analytic functions by identifying a class of well-behaved functions and providing explicit estimates based on the Newton polygon, with improved results for a subclass.

Contribution

It introduces a class of well-behaved functions for which Fourier transform estimates can be explicitly described, and refines these estimates for a specific subclass.

Findings

Explicit Fourier transform estimates in terms of the Newton polygon

Identification of a class of well-behaved functions

More precise estimates for a subclass of functions

Abstract

This paper is a companion paper to [G4], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [G4] are stated in a rather general form. In this paper, we expand on the results of [G4] and show that there is a class of "well-behaved" functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.