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

Daniel Spector*, Cody B. Stockdale

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Article number2050072
JournalCommunications in Contemporary Mathematics
Issue number7
Publication statusPublished - 2021 Nov 1
Externally publishedYes


  • Riesz transforms
  • dimensional dependence
  • weak-type estimates

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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