Necessary and sufficient conditions for the bounds of the commutator of a Littlewood-Paley operator with fractional differentiation

For b is an element of L-loc(R-n) and 0 < alpha < 1, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely g Omega,alpha;b(f)(x)=(integral 0 infinity|1/t integral(|x-y|<= t)Omega(x-y)|x-y|n+alpha-1(b(x)-b(y))f(y)dy|(2)dt/t)(1/2). Here, we obtain the necessary and sufficient conditions for the function b to guarantee that g Omega,alpha;b is a bounded operator on L2(R-n). More precisely, if Omega is an element of L(log+L)(1/2)(Sn-1) and b is an element of I alpha(BMO), then g Omega,alpha;b is bounded on L-2(R-n). Conversely, if g Omega,alpha;b is bounded on L-2(R-n), then b is an element of Lip alpha(R-n) for 0 <alpha < 1.