Commutators of convolution type operators on some Banach function spaces

Abstract

We study the boundedness of Fourier convolution operators $W^0(b)$ and the compactness of commutators of $W^0(b)$ with multiplication operators $aI$ on some Banach function spaces $X(\mathbb{R})$ for certain classes of piecewise quasicontinuous functions $a\in PQC$ and piecewise slowly oscillating Fourier multipliers $b\in PSO_{X,1}^\diamond$. We suppose that $X(\mathbb{R})$ is a separable rearrangement-invariant space with nontrivial Boyd indices or a reflexive variable Lebesgue space, in which the Hardy--Littlewood maximal operator is bounded. Our results complement those of Isaac De La Cruz--Rodríguez, Yuri Karlovich, and Iván Loreto Hernández obtained for Lebesgue spaces with Muckenhoupt weights.