## Abstract

Let Rj denote the jth Riesz transform on ℝn. We prove that there exists an absolute constant C > 0 such that (equation presented) for any λ > 0 and f L1(ℝn), where the above supremum is taken over measures of the form ν =ck=1Na kαck for N, ck ℝn, and ak ℝ+ with ck=1Na k ≤ 16∥F∥L1(ℝn). This shows that to establish dimensional estimates for the weak-type (1, 1) inequality for the Riesz transforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón-Zygmund operators.

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

Article number | 2050072 |

Journal | Communications in Contemporary Mathematics |

Volume | 23 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2021 Nov 1 |

Externally published | Yes |

## Keywords

- Riesz transforms
- dimensional dependence
- weak-type estimates

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics