论文标题
类似考奇（Cauchy）家族的对称性：全球不稳定
Symmetrization of a family of Cauchy-Like kernels: Global instability
论文作者
论文摘要
Cauchy转换在谐波和复杂分析中的基本作用导致了许多不同的证明其$ l^2 $界限。特别是，梅尔尼科夫 - 弗雷拉（Melnikov-Verdera）的著名证明[18]依赖于Melnikov [17]的标志性对称身份[17]将通用的Cauchy kernel $ K_0 $与Menger Curvature联系起来。 $ k_0 $的真实和虚构部分也具有类似的身份。在奇异积分运算符和几何度量理论的研究中，这种连接具有极高的生产力。 \ vskip0.1in在本文中，给定任何函数$ h：\ mathbb c \ rightarrow \ mathbb r $，我们考虑了一个不均匀的变体$ k_h $ of $ k_0 $，这是受复杂函数理论的启发。虽然很容易看出具有集成内核$ k_h $的操作员对所有$ h $都符合$ l^2 $结合，但对于$ k_h $的每个真实和想象中的每个零件的对称身份都表现出界限和积极性的鲁棒性，而在[18]中以及在[18]以及许多后续作品中都是至关重要的。确实，我们在这里表明，对于$ \ Mathbb c $上的任何连续$ h $，$ \ {k_h \} _ H $的唯一成员的对称性具有正确的属性为$ k_0 $！这种全球不稳定在曲线的限制设置中对我们先前对对称身份的调查[12]，其中$ \ {k_h \} _ H $的子家庭与此处考虑的全局对应物的行为截然不同。我们的证明方法与Chousionis-Prat [5]和Chunaev [6]的最新工作中的技术有一些重叠。
The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its $L^2$ boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon an iconic symmetrization identity of Melnikov [17] linking the universal Cauchy kernel $K_0$ to Menger curvature. Analogous identities hold for the real and the imaginary parts of $K_0$ as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. \vskip0.1in In this article, given any function $h: \mathbb C \rightarrow \mathbb R$, we consider an inhomogeneous variant $K_h$ of $K_0$ which is inspired by complex function theory. While an operator with integration kernel $K_h$ is easily seen to be $L^2$-bounded for all $h$, the symmetrization identities for each of the real and imaginary parts of $K_h$ show a striking lack of robustness in terms of boundedness and positivity, two properties that were critical in [18] and in subsequent works by many authors. Indeed here we show that for any continuous $h$ on $\mathbb C$, the only member of $\{K_h\}_h$ whose symmetrization has the right properties is $K_0$! This global instability complements our previous investigation [12] of symmetrization identities in the restricted setting of a curve, where a sub-family of $\{K_h\}_h$ displays very different behaviour than its global counterparts considered here. Our methods of proof have some overlap with techniques in recent work of Chousionis-Prat [5] and Chunaev [6].
