Noncompactness of Fourier Convolution Operators on Banach Function Spaces

TL;DR

This paper proves that Fourier convolution operators on certain Banach function spaces are only compact if their symbol is zero, highlighting the noncompactness of nontrivial operators, especially on weighted Lebesgue spaces.

Contribution

It establishes a necessary and sufficient condition for the compactness of Fourier convolution operators on Banach function spaces, showing they are trivial unless zero.

Findings

Fourier convolution operators are noncompact unless their symbol is zero.

On weighted Lebesgue spaces with Muckenhoupt weights, nontrivial Fourier convolution operators are never compact.

The result links the properties of the symbol to the operator's compactness on Banach spaces.

Abstract

Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. Suppose $a$ is a Fourier multiplier on the space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.