Some remarks about Cauchy integrals

A basic theme in the wonderful books and surveys of Stein, Weiss, and Zygmund is that Hilbert transforms, Poisson kernels, heat kernels, and related objects are quite interesting and fundamental. I certainly like this point of view. There is a variety of ways in which things can be interesting or fundamental, of course. In the last several years there have been striking developments connected to Cauchy integrals, and in this regard I would like to mention the names of Pertti Mattila, Mark Melnikov, and Joan Verdera in particular. I think many of us are familiar with the remarkable new ideas involving Menger curvature, and indeed a lot of work using this has been done by a lot of people, and continues to be done. Let us also recall some matters related to symmetric measures. Let μ be a nonnegative Borel measure on the complex plane C, which is finite on bounded sets. Following Mattila, μ is said to be symmetric if for each point a in the support of μ and each positive real number r we have that the integral of z − a over the open ball with center a and radius r with respect to the measure dμ(z) is equal to 0. One might think of this as a kind of flatness condition related to the existence of principal values of Cauchy integrals. If μ is equal to a constant times 1-dimensional Lebesgue measure on a line, then μ is a symmetric measure. For that matter, 2-dimensional Lebesgue measure on C is symmetric, and there are other possibilities. Mattila discusses this, and shows that a symmetric measure which satisfies some additional conditions is equal to a constant times 1-dimensional Lebesgue measure on a line. Mattila uses this to show that existence almost everywhere of principle values of a measure implies rectifiability properties of the measure. Mattila