Fourier convolution operators with symbols equivalent to zero at infinity on Banach function spaces

TL;DR

This paper investigates Fourier convolution operators with symbols that vanish at infinity on certain Banach function spaces, showing that their limit operators are all zero under specific boundedness conditions.

Contribution

It establishes that all limit operators of these convolution operators are zero when the Hardy-Littlewood maximal operator is bounded on the space and its associate.

Findings

Limit operators of the studied convolution operators are zero.

The result applies to Banach spaces where the Hardy-Littlewood maximal operator is bounded.

Provides insight into the asymptotic behavior of convolution operators with zero-at-infinity symbols.

Abstract

We study Fourier convolution operators $W^0(a)$ with symbols equivalent to zero at infinity on a separable Banach function space $X(\mathbb{R})$ such that the Hardy-Littlewood maximal operator is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. We show that the limit operators of $W^0(a)$ are all equal to zero.