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Variation of Calderón–Zygmund operators with matrix weight
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-10-24 , DOI: 10.1142/s0219199720500625 Xuan Thinh Duong 1 , Ji Li 1 , Dongyong Yang 2
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-10-24 , DOI: 10.1142/s0219199720500625 Xuan Thinh Duong 1 , Ji Li 1 , Dongyong Yang 2
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Let p ∈ ( 1 , ∞ ) , ρ ∈ ( 2 , ∞ ) and W be a matrix A p weight. In this paper, we introduce a version of variation 𝒱 ρ ( 𝒯 n , ∗ ) for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the L p ( W ) -boundedness of 𝒱 ρ ( 𝒯 n , ∗ ) with norm
∥ 𝒱 ρ ( 𝒯 n , ∗ ) ∥ L p ( W ) → L p ( W ) ≤ C [ W ] A p 1 + 1 p − 1 − 1 p
by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón–Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón–Zygmund operator.
中文翻译:
Calderón–Zygmund 算子与矩阵权重的变化
让p ∈ ( 1 , ∞ ) ,ρ ∈ ( 2 , ∞ ) 和W 是一个矩阵一种 p 重量。在本文中,我们介绍了一个版本的变体𝒱 ρ ( 𝒯 n , * ) 对于具有满足 Dini 条件的连续模的矩阵 Calderón–Zygmund 算子。然后我们得到大号 p ( W ) -有界性𝒱 ρ ( 𝒯 n , * ) 有规范
∥ 𝒱 ρ ( 𝒯 n , * ) ∥ 大号 p ( W ) → 大号 p ( W ) ≤ C [ W ] 一种 p 1 + 1 p - 1 - 1 p
首先证明标量 Calderón-Zygmund 算子的变分的稀疏支配,然后提供矩阵 Calderón-Zygmund 算子的变分的凸体稀疏支配。这里的关键步骤是关于标量 Calderón-Zygmund 算子的局部最大截断算子的弱类型估计。
更新日期:2020-10-24
中文翻译:
Calderón–Zygmund 算子与矩阵权重的变化
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