Symmetrization of a family of Cauchy-Like kernels: Global instability.

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].