### Abstract

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ℓ^{1}-sum in the sparse operator is replaced by an ℓ^{r}-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A_{2}-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

Original language | English |
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Pages (from-to) | 1-40 |

Number of pages | 40 |

Journal | Journal of Geometric Analysis |

DOIs | |

Publication status | E-pub ahead of print - 2020 |

### Keywords

- Mihlin multiplier theorem
- Muckenhoupt weight
- Rademacher maximal operator
- Singular integral operator
- Space of homogeneous type
- Sparse domination