## Abstract

Let L be a non-negative self-adjoint operator on L^{2}(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 ℝ^{n} and Schrödinger operators on doubling manifolds.

Original language | English |
---|---|

Number of pages | 25 |

Journal | Potential Analysis |

Early online date | 27 May 2021 |

DOIs | |

Publication status | E-pub ahead of print - 27 May 2021 |

## Keywords

- Critical function
- Quatitative weighted estimate
- Square function
- Heat kernel