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

Abstract

Let Rj denote the jth Riesz transform on Rn. We prove that there exists an absolute constant C > 0 such that |{|Rjf| > λ}|≤ C 1 λ×f×L1(Rn) +supν|{|Rjν| > λ}| for any λ > 0 and f L1(Rn), where the above supremum is taken over measures of the form ν =×k=1Na kdck for N N, ck Rn, and ak R+ with ×k=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
DOIs
Publication statusAccepted/In press - 2020
Externally publishedYes

Keywords

  • dimensional dependence
  • Riesz transforms
  • weak-type estimates

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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