Quantitative Estimates for Square Functions with New Class of Weights

Let L be a non-negative self-adjoint operator on L2(X) where X is a metric space with a doubling measure. Assume that the kernels of the semigroup generated by − L satisfy a suitable upper bound related to a critical function ρ but these kernels are not assumed to satisfy any regularity conditions on spacial variables. In this paper, we prove the quantitative weighted estimates for some square functions associated to L which include the vertical square function, the conical square function and the g-functions. The novelty of our results is that the square functions associated to L might have rough kernels, hence do not belong to the Calderon-Zygmund class, and the class of weights is larger than the class of Muckenhoupt weights. Our results have applications in various settings of Schrodinger operators such as magnetic Schrodinger operators on the Euclidean space $\mathbb {R}^n$ and Schrodinger operators on doubling manifolds.