#### Vol. 307, No. 1, 2020

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On the compactness of commutators of Hardy operators

### Shaoguang Shi, Zunwei Fu and Shanzhen Lu

Vol. 307 (2020), No. 1, 239–256
##### 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 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