Algebra of convolution type operators with continuous data on Banach function spaces

TL;DR

This paper establishes that certain Banach function spaces with bounded Hardy-Littlewood maximal operators possess unconditional wavelet bases and characterizes their compact operators as generated by multiplication and Fourier convolution operators.

Contribution

It proves the existence of unconditional wavelet bases under boundedness conditions and describes the structure of compact operators in these spaces.

Findings

Banach function spaces with bounded Hardy-Littlewood maximal operator have unconditional wavelet bases.

The ideal of compact operators is contained in the algebra generated by multiplication and Fourier convolution operators.

The results connect maximal operator boundedness with basis existence and operator algebra structure.

Abstract

We show that if the Hardy-Littlewood maximal operator is bounded on a reflexive Banach function space $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$, then the space $X(\mathbb{R})$ has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in $X(\mathbb{R})$, we prove that the ideal of compact operators $\mathcal{K}(X(\mathbb{R}))$ on the space $X(\mathbb{R})$ is contained in the Banach algebra generated by all operators of multiplication $aI$ by functions $a\in C(\dot{\mathbb{R}})$, where $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$, and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$, the Fourier multiplier analogue of $C(\dot{\mathbb{R}})$.