On Weighted Compactness of Commutators of Stein’s Square Functions Associated with Bochner-Riesz means

Abstract

In this paper, our object of investigation is the commutators of the Stein’s square functions asssoicated with the Bochner-Riesz means of order \({\uplambda }\) defined by

$$\begin{aligned} G_{b,m}^{\uplambda }f(x)=\Big (\int _0^\infty \Big |\int _{{\mathbb {R}}^n}(b(x)-b(y))^mK_t^{\uplambda }(x-y)f(y)dy \Big |^2\frac{dt}{t}\Big )^{\frac{1}{2}}, \end{aligned}$$

where \(\widehat{K_t^{\uplambda }}({\upxi })=\frac{|{\upxi }|^2}{t^2}\Big (1-\frac{|{\upxi }|^2}{t^2}\Big )_+^{{\uplambda }-1}\) and \(b\in \mathrm BMO(\mathbb {R}^n)\). We show that \(G_{b,m}^{\uplambda }\) is a compact operator from \(L^p(w)\) to \(L^p(w)\) for \(1<p<\infty \) and \({\uplambda }>\frac{n+1}{2}\) whenever \(b\in \mathrm CMO({\mathbb {R}^n})\), where \(\textrm{CMO}(\mathbb {R}^n)\) is the closure of \(\mathcal {C}_c^\infty (\mathbb {R}^n)\) in the \(\textrm{BMO}(\mathbb {R}^n)\) topology.