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 Calderón-Zygmund class, and the class of weights is larger than the class of Muckenhoupt weights. Our results have applications in various settings of Schrödinger operators such as magnetic Schrödinger operators on the Euclidean space \(\mathbb {R}^n\) and Schrödinger operators on doubling manifolds.

令L为L ²（X）上的非负自伴随算子，其中X为具有倍增测度的度量空间。假设由-L生成的半群的核满足与临界函数ρ有关的合适上限，但是不假定这些核满足空间变量的任何规则性条件。在本文中，我们证明了一些与L相关的平方函数的定量加权估计，其中包括垂直平方函数，圆锥平方函数和g函数。我们的结果的新颖之处在于，与L相关的平方函数可能具有粗糙的内核，因此不属于Calderón-Zygmund类，并且权重的类别大于Muckenhoupt权重的类别。我们的结果在Schrödinger算子的各种设置中都有应用，例如在欧几里得空间\（\ mathbb {R} ^ n \）上的磁性Schrödinger算子和在双重流形上的Schrödinger算子。