论文标题
在维度弱型$（1,1）$上
On the dimensional weak-type $(1,1)$ bound for Riesz transforms
论文作者
论文摘要
令$ r_j $表示$ j^{\ text {th}} $ riesz在$ \ mathbb {r}^n $上转换。我们证明存在一个绝对常数$ c> 0 $，这样\ begin {align*} | \ {| r_jf |>λ\} | \ leq c \ left（\ frac {1}λ\ | f \ | _ {l^1（\ Mathbb {r}^n）}+\sup_ν $ f \ in l^1（\ Mathbb {r}^n）$，其中以上上述为$ $ν= \ sum_ {k = 1}^na_kδ_{c_kΔ_{ \ Mathbb {r}^+$ with $ \ sum_ {k = 1}^n a_k \ leq 16 \ | f \ | _ {l^1（\ Mathbb {r}^n）} $。这表明，要确定弱型$（1,1）$不平等的尺寸估计值，它足以研究适用于DIRAC质量有限线性组合的Riesz变换的相应弱型不平等。我们利用这一事实为最著名的维度上限提供了新的证明，而我们的还原结果也适用于Calderón-Zygmund运营商的更一般类别。
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 $λ>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $ν=\sum_{k=1}^Na_kδ_{c_k}$ 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 Calderón-Zygmund operators.
