## 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 |
---|---|

Pages (from-to) | 9366-9405 |

Number of pages | 40 |

Journal | Journal of Geometric Analysis |

Volume | 31 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

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

## Fingerprint

Dive into the research topics of 'On Pointwise ℓ^{r}-Sparse Domination in a Space of Homogeneous Type'. Together they form a unique fingerprint.