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 \(L^2\) boundedness. In particular, a famous proof of Melnikov-Verdera (A geometric proof of the \(L^2\) boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7:325–331, 1995) relies upon an iconic symmetrization identity of Melnikov (Analytic capacity: a discrete approach and the curvature of measures. Mat Sb 186(6):57–76, 1995) 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. 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 Melnikov and Verdera (A geometric proof of the \(L^2\) boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7:325–331, 1995) 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 (Lanzani and Pramanik, Symmetrization of a Cauchy-like kernel on curves. J Funct Anal (Preprint). arXiv:2001.09375) 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 and Prat, Some Calderòn–Zygmund kernels and their relation to Wolff capacity. Math Z 282:435–460, 2016) and Chunaev, A new family of singular integral operators whose \(L^2\) boundedness implies rectifiability. J Geom Anal 27:2725–2757, 2017).