Necessary and sufficient conditions for the quantitative weighted bounds of the Calderón type commutator for the Littlewood–Paley operator

In this paper, we study the necessary and sufficient conditions for the quantitative weighted bounds of the Calderón type commutator for the Littlewood–Paley operator. Let \(g_{\Omega ,1;b}\) be the Calderón type commutator for the Littlewood–Paley operator where \(\Omega \) is homogeneous of degree zero and satisfies the cancellation condition on the unit sphere, and \(b\in Lip(\mathbb {R}^n)\). More precisely, for the sufficiency, we use a new operator \(\widetilde{G}_{\Omega ,m;b}^j\). Through the Calderón–Zygmund decomposition and the grand maximal operator \(\mathcal {M}_{\widetilde{G}_{\Omega ,m;b}^j}\) of weak type (1,1), we establish a sparse domination of \(\widetilde{G}_{\Omega ,m;b}^j\). And then applying the interpolation theorem with change of measures and the relationship between the operators \(g_{\Omega ,1;b}\) and \(\widetilde{G}_{\Omega ,m;b}^j\), we get the weighted bounds of the Calderón type commutators for the Littlewood–Paley operator \(g_{\Omega ,1;b}\). In addition, for the necessity, through the local mean oscillation, we obtain Lip-type characterizations of \(Lip(\mathbb {R}^n)\) via the weighted bounds of the Calderón type commutators for the Littlewood–Paley operator.