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

Abstract

Let [Formula: see text] denote the [Formula: see text] Riesz transform on [Formula: see text]. We prove that there exists an absolute constant [Formula: see text] such that [Formula: see text] for any [Formula: see text] and [Formula: see text], where the above supremum is taken over measures of the form [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] with [Formula: see text]. This shows that to establish dimensional estimates for the weak-type [Formula: see text] 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.