Symmetrization of a family of Cauchy-Like kernels: Global instability
Abstract
The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its L2 boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon an iconic symmetrization identity of Melnikov [17] linking the universal Cauchy kernel K0 to Menger curvature. Analogous identities hold for the real and the imaginary parts of K0 as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. 0.1in In this article, given any function h: C → R, we consider an inhomogeneous variant Kh of K0 which is inspired by complex function theory. While an operator with integration kernel Kh is easily seen to be L2-bounded for all h, the symmetrization identities for each of the real and imaginary parts of Kh 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 C, the only member of \Kh\h whose symmetrization has the right properties is K0! This global instability complements our previous investigation [12] of symmetrization identities in the restricted setting of a curve, where a sub-family of \Kh\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].