TY - JOUR

T1 - On the dimensional weak-type (1, 1) bound for Riesz transforms

AU - Spector, Daniel

AU - Stockdale, Cody B.

N1 - Publisher Copyright:
© 2021 The Author(s).

PY - 2021/11/1

Y1 - 2021/11/1

N2 - 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.

AB - 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.

KW - Riesz transforms

KW - dimensional dependence

KW - weak-type estimates

UR - http://www.scopus.com/inward/record.url?scp=85097548712&partnerID=8YFLogxK

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U2 - 10.1142/S0219199720500728

DO - 10.1142/S0219199720500728

M3 - Article

AN - SCOPUS:85097548712

VL - 23

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 7

M1 - 2050072

ER -