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|>λ\}|\leq C\left(\frac{1}λ\|f\|_{L^1(\mathbb{R}^n)}+\sup_ν |\{|R_jν|>λ\}|\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 $\nu ={\sum }_{k=1}^N{a}_k{\delta }_{c_k}$ for $N\in \mathbb{N}$, $c_k\in {\mathbb{R}}^n$, and $a_k\in {\mathbb{R}}^{+}$ with ${\sum }_{k=1}^N{a}_k\le 16\Vert f{\Vert }_{L^1({\mathbb{R}}^n)}$. 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 Calderón-Zygmund operators.

17 pages