### Abstract

Let X be a metric space with a doubling measure and let L be a linear operator in *L*^{2} (X) which generates a semigroup e ^{-}^{t}* ^{L}* whose kernels p

_{t}(x, y) , t > 0 , satisfy the Gaussian upper bound. In this article, we prove

*sharp L*

_{w}

^{p}norm inequalities for a number of square functions associated to

*L*including the vertical square functions, the Lusin area integral square functions and the Littlewood–Paley g-functions. We note that our conditions on the heat kernels p

_{t}(x, y) are mild in the sense that the associated kernels of the square functions do not possess enough regularity for those operators to belong to the standard class of Calderón–Zygmund operators.

Original language | English |
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Pages (from-to) | 874-900 |

Number of pages | 27 |

Journal | Journal of Geometric Analysis |

Volume | 30 |

Issue number | 1 |

Early online date | 4 Mar 2019 |

DOIs | |

Publication status | Published - Jan 2020 |

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### Keywords

- Heat kernel
- Square function
- Sharp weighted estimate