Complementation of the subspace of radial multipliers in the space of Fourier multipliers on $\mathbb{R}^n$

TL;DR

This paper proves that the subspace of radial Fourier multipliers is contractively complemented within the space of all Fourier multipliers on certain function spaces, with positivity preserved in the scalar case.

Contribution

It establishes the contractive complementation of radial multipliers in Fourier multiplier spaces on Bochner spaces, extending understanding of their structure and positivity properties.

Findings

Radial multipliers form a contractively complemented subspace

Positivity of multipliers is preserved when the Banach space is scalars

Results apply to Fourier multipliers on L^p spaces with Banach space values

Abstract

In this short note, we prove that the subspace of radial multipliers is contractively complemented in the space of Fourier multipliers on the Bochner space $\mathrm{L}^p(\mathbb{R}^n,X)$ where $X$ is a Banach space and where $1 \leq p <\infty$. Moreover, if $X = \mathbb{C}$, then this complementation preserves the positivity of multipliers.