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

For b∈Lloc1(Rn), we define the Calderón type commutator for the Littlewood–Paley operator gg function by gΩ,1;b(f)(x)=(∫0∞|1t2∫|x−y|≤tΩ(x−y)|x−y|n−1(b(x)−b(y))f(y)dy|2dtt)1/2, which is a new kind of commutator of Littlewood–Paley operator gg function. In this paper the authors obtain the necessary and sufficient conditions on the function bb to guarantee that gΩ,1;bgΩ,1;b is a bounded operator on L2(Rn)L2(Rn). Namely, if Ω∈L(log+L)1/2(Sn−1)Ω∈L(log+L)1/2(Sn−1) and b∈Lip(Rn)b∈Lip(Rn), then gΩ,1;bgΩ,1;b is bounded on L2(Rn)L2(Rn). And the converse is: if gΩ,1;bgΩ,1;b is bounded on L2(Rn)L2(Rn), then b∈Lip(Rn)b∈Lip(Rn).