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On the dimensional weak-type (1,1) bound for Riesz transforms
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-11-30 , DOI: 10.1142/s0219199720500728 Daniel Spector 1 , Cody B. Stockdale 2
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-11-30 , DOI: 10.1142/s0219199720500728 Daniel Spector 1 , Cody B. Stockdale 2
Affiliation
Let R j denote the j th Riesz transform on ℝ n . We prove that there exists an absolute constant C > 0 such that
| { | R j f | > λ } | ≤ C 1 λ ∥ f ∥ L 1 ( ℝ n ) + sup ν | { | R j ν | > λ } |
for any λ > 0 and f ∈ L 1 ( ℝ n ) , where the above supremum is taken over measures of the form ν = ∑ k = 1 N a k δ c k for N ∈ ℕ , c k ∈ ℝ n , and a k ∈ ℝ + with ∑ k = 1 N a k ≤ 1 6 ∥ f ∥ L 1 ( ℝ n ) . This shows that to establish dimensional estimates for the weak-type ( 1 , 1 ) 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.
中文翻译:
关于 Riesz 变换的维弱类型 (1,1) 界
让R j 表示j th Riesz 变换ℝ n . 我们证明存在一个绝对常数C > 0 这样
| { | R j F | > λ } | ≤ C 1 λ ∥ F ∥ 大号 1 ( ℝ n ) + 支持 ν | { | R j ν | > λ } |
对于任何λ > 0 和F ∈ 大号 1 ( ℝ n ) ,其中上述上限值被采取的措施形式ν = ∑ ķ = 1 ñ 一种 ķ δ C ķ 为了ñ ∈ ℕ ,C ķ ∈ ℝ n , 和一种 ķ ∈ ℝ + 和∑ ķ = 1 ñ 一种 ķ ≤ 1 6 ∥ F ∥ 大号 1 ( ℝ n ) . 这表明建立弱类型的维度估计( 1 , 1 ) Riesz 变换的不等式只要研究应用于狄拉克质量的有限线性组合的 Riesz 变换的相应弱型不等式就足够了。我们使用这个事实给出了最知名的维数上限的新证明,而我们的归约结果也适用于更一般的 Calderón-Zygmund 算子类。
更新日期:2020-11-30
中文翻译:
关于 Riesz 变换的维弱类型 (1,1) 界
让