On the compactness of commutators of Hardy operators

Shaoguang Shi, Zunwei Fu and Shanzhen Lu

Vol. 307 (2020), No. 1, 239–256

DOI: 10.2140/pjm.2020.307.239

Abstract

We focus on the need for the compactness characterizations of the commutators of Hardy operators. More precisely, we prove that the commutators of Hardy operators, including the fractional Hardy operator, are compact operators on $L^p\left({ℝ}^n\right)$$\left(1<p<\infty \right)$ spaces if and only if the symbol functions of the commutators belong to $CVMO\left({ℝ}^n\right)$ spaces (the central $BMO\left({ℝ}^n\right)$ closure of $C_c^{\infty }\left({ℝ}^n\right)$).

Keywords

Hardy operator, commutator, compactness, BMO

Mathematical Subject Classification 2010

Primary: 42B25, 47B47, 47G10

Milestones

Received: 24 July 2018

Revised: 20 December 2019

Accepted: 21 March 2020

Published: 8 August 2020