Compactness characterization of commutators for Littlewood-Paley operators
Let \(L\) denote one of the Littlewood-Paley operators, such as the Littlewood-Paley \(g\)-function, the Lusin area integral and the Littlewood-Paley \(g^*_{\lambda}\)-function. The authors prove that the commutator \([b,L]\) is a compact operator in \(L^p(R^n)\) if and only if \(b\in\) VMO\((\mathbb{R}^n)\). The Littlewood-Paley operators discussed in this paper are nonlinear, which requires different techniques from traditional methods (such as in [\textit{A. Uchiyama}, Tohoku Math. J., II. Ser. 30, 163--171 (1978; Zbl 0384.47023)]). Their work is thoughtful and innovative.
scientific article; zbMATH DE number 5598067