Weighted estimates for square functions associated with operators

Let L be a non-negative self-adjoint operator on L2(Rn). Suppose that the kernels of the analytic semigroup e-tL satisfy the upper bound related to a critical function rho but without any assumptions of smooth conditions on spacial variables. In this paper, we consider the weighted inequalities for square functions associated with L, which include the vertical square functions, the conical square functions and the Littlewood-Paley g-functions. A new bump condition related to the critical function is given for the two-weighted boundedness of square functions associated with L. Besides, we also prove the weighted inequalities for square functions associated with L on weighted variable Lebesgue spaces with new classes of weights considered in [5]. As applications, our results can be applied to magnetic Schrodinger operator, Laguerre operators.