Boundedness and Compactness of commutator of Hardy-Littlewood maximal operator

TL;DR

This paper investigates the boundedness and compactness properties of the commutator of the bilinear Hardy-Littlewood maximal operator within homogeneous Triebel-Lizorkin spaces, revealing new results even in the linear case.

Contribution

It establishes the boundedness and compactness of the commutator operator, providing new insights in both bilinear and linear settings.

Findings

The commutator is bounded on certain function spaces.

The commutator acts as a compact operator on product Lebesgue spaces.

Results are novel even for the linear Hardy-Littlewood maximal operator.

Abstract

We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.