# On the dimensional weak-type (1, 1) bound for Riesz transforms

Daniel Spector*, Cody B. Stockdale

*此作品的通信作者

## 摘要

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.

原文 英語 2050072 Communications in Contemporary Mathematics 23 7 https://doi.org/10.1142/S0219199720500728 已發佈 - 2021 11月 1 是

• 數學(全部)
• 應用數學