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

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 $1\le p<\infty$. Moreover, if $X=\mathbb{C}$, then this complementation preserves the positivity of multipliers.