Convolution kernels of 2D Fourier multipliers based on real analytic functions

TL;DR

This paper establishes sharp decay estimates for convolution kernels linked to 2D Fourier multipliers derived from real-analytic functions, enhancing understanding of their directional decay properties.

Contribution

It provides new sharp decay estimates for convolution kernels associated with a broad class of 2D Fourier multipliers based on real-analytic functions.

Findings

Established sharp decay rates for convolution kernels

Derived directional decay estimates for specific directions

Applied results to a general class of multipliers

Abstract

In this paper, estimates are proven for convolution kernels associated to multipliers from a reasonably general class of compactly supported two-dimensional functions constructed out of real-analytic functions. These estimates are both for overall decay rate and decay rate in specific directions. The estimates are sharp for a certain range of exponents appearing in the theorems.