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

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}$^n)$,wheretheabovesupremumistakenovermeasuresoftheform$\nu=\sum_{k=1}^Na_k\delta_{c_k}$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.