Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness
Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness
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DOI：
10.1512/iumj.2021.70.8573
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发表时间：
2021
影响因子：
1.1
通讯作者：
X. Duong;M. Lacey;Ji Li;B. Wick;Qingyan Wu
中科院分区：
文献类型：
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作者：
X. Duong;M. Lacey;Ji Li;B. Wick;Qingyan Wu
本文研究了C中有界强伪凸区域D上Cauchy型积分C的交换子，其中边界bD满足Lanzani-Stein最近的结果中的最小正则性条件C.我们指出，在这种情况下，柯西型积分C是本质部分C（它是Calderón-Zygmund算子）和余项R（它不再是Calderón-Zygmund算子）之和。证明了交换子[B，C]在L（bD）（1 < p < ∞）上有界的充要条件是B在bD上的BMO空间中.此外，交换子[B，C]在L（bD）（1 < p < ∞）上是紧的当且仅当B在bD上的VMO空间中.我们的方法也可应用于C中有界强C-线性凸区域D中Cauchy-Leray积分的交换子，其中边界bD满足最小正则性C。这样一个柯西-勒雷积分是一个卡尔德龙-齐格蒙德算子，正如兰扎尼-斯坦最近的结果所证明的。我们还指出，我们的方法提供了Cauchy-SzegIgn算子在C中具有光滑边界的有界强伪凸域D上的交换子的有界性和紧性的另一个证明（Krantz-Li首次建立）.
In this paper we study the commutator of Cauchy type integrals C on a bounded strongly pseudoconvex domain D in C with boundary bD satisfying the minimum regularity condition C as in the recent result of Lanzani–Stein. We point out that in this setting the Cauchy type integrals C is the sum of the essential part C which is a Calderón–Zygmund operator and a remainder R which is no longer a Calderón–Zygmund operator. We show that the commutator [b,C] is bounded on L(bD) (1 < p < ∞) if and only if b is in the BMO space on bD. Moreover, the commutator [b, C] is compact on L(bD) (1 < p < ∞) if and only if b is in the VMO space on bD. Our method can also be applied to the commutator of Cauchy–Leray integral in a bounded, strongly C-linearly convex domain D in C with the boundary bD satisfying the minimum regularity C. Such a Cauchy–Leray integral is a Calderón–Zygmund operator as proved in the recent result of Lanzani–Stein. We also point out that our method provides another proof of the boundedness and compactness of commutator of Cauchy–Szegő operator on a bounded strongly pseudoconvex domain D in C with smooth boundary (first established by Krantz–Li).