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

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Let Rj denote the jth Riesz transform on ℝn. We prove that there exists an absolute constant C>0 such that [Formula: see text] for any λ>0 and f∈L¹(ℝⁿ), where the above supremum is taken over measures of the form [Formula: see text]. 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.

source:https://www.worldscientific.com/doi/10.1142/S0219199720500728

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