Split functions, Fourier transforms and multipliers

TL;DR

This paper investigates how a splitting operator affects the Fourier transform's L^p norm and the operator norm of Fourier multipliers, with results mainly for even integer p and functions with compact support.

Contribution

It provides new insights into the behavior of splitting operators on Fourier transforms and multipliers, especially under specific conditions like even p and compact support.

Findings

Splitting operator S_t impacts Fourier transform norms.

Results are stronger for functions with compact support.

Mainly applicable when p is an even integer.

Abstract

We study the effect of a splitting operator S_t on the L^p norm of the Fourier transform of a function f and on the operator norm of a Fourier multiplier m. Most of our results assume p is an even integer, and are often stronger when f or m has compact support.