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

Abstract: Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |{|R_jf|>\lambda}|\leq C\left(\frac{1}{\lambda}|f|{L^1(\mathbb{R}^n)}+\sup{\nu} |{|R_j\nu|>\lambda}|\right) \end{align*} for any $\lambda>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $\nu=\sum_{k=1}^Na_k\delta_{c_k}$ for $N \in \mathbb{N}$, $c_k \in \mathbb{R}^n$, and $a_k \in \mathbb{R}^+$ with $\sum_{k=1}^N a_k \leq 16|f|_{L^1(\mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms 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 Calder\'on-Zygmund operators.