On weighted compactness of commutators of square function and semi-group maximal function associated to Schrodinger operators

Let Delta be the laplacian operator on R-n and V be a nonnegative potential satisfying an appropriate reverse Holder inequality. The Littlewood-Paley square function g associated with the Schrodinger operator L = -Delta + V is defined by: g(f)(x) = (integral(infinity)(0) vertical bar d/dte(-tL)(f)(x)vertical bar(2)tdt)(1/2). In this paper, we show that the commutators of g are compact operators on L-p(w) for 1 < p < infinity if b is an element of CMO theta(rho ) and w is an element of A(p)(rho,theta) , where CMO theta(rho)(R-n) denotes the closure of C-c(infinity) (R-n) in the BMO theta(rho) topology and A(p)(rho,theta) is a weighted class which is more larger than Muckenhoupt A(p) weight class. An extra weight condition in a previous weighted compactness result is removed for the commutators of the semi-group maximal function defined by T*(f)(x) = sup(t>0) vertical bar e(-tL)f(x)vertical bar.