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

In this paper, our object of investigation is the commutators of the Stein's square functions asssoicated with the Bochner-Riesz means of order lambda lambda defined by G(b,m)(lambda)f(x)=(integral(infinity)(0)|integral(n)(R)(b(x)-b(y))K-m(t)lambda(x-y)f(y)dy|(2)dt/t)(1/2), where K-t(lambda)<^>(xi)=|xi|(2)/t(2)(1-|xi|(2)/t(2))+(lambda-1) and b is an element of BMO(R-n). We show that G(b,m)(lambda) is a compact operator from L-p(w) to L-p(w) for 1<pn+1/2 whenever b is an element of CMO(R-n), where CMO(R-n) is the closure of C-c(infinity)(R-n) in the BMO(R-n) topology.