A functional analytic approach to the study of the dependence of the Cauchy integral upon the contour of integration, and applications

We present a PDE approach to the study of the dependence of the Cauchy integral \[ C[\phi,f](\cdot)\equiv \frac{1}{2\pi i}\,\mathrm{p.v.}\int_{\phi} \frac{f\circ \phi^{(-1)}(\xi)}{\xi-\phi(\cdot)}\, d\xi \] upon the plane oriented simple closed curve $\phi$ and the density function $f$. Both $f$ and $\phi$ are defined on the counterclockwise oriented boundary $\partial {\bf D}$ of the plane unit disk ${\bf D}$. We assume that both $\phi$ and $f$ belong to the Schauder space $C^{m,\alpha}_{*}(\partial{\bf D},{\bf C})$ of the $m$ times continuously differentiable complex-valued functions on $\partial {\bf D}$ whose $m$-th order derivatives are $\alpha$-H\"older continuous, with $\alpha\in ]0,1[$, $m\geq 1$. As it is well-known, under such conditions, $C[\phi,f]$ is also of class $C^{m,\alpha}_{*}(\partial{\bf D},{\bf C})$. The PDE approach yields to a theorem of complex-analytic dependence of $C[\phi,f]$ upon $(\phi,f)$, which can be regarded as an extension to a Schauder space setting of a known result of Coifman and Meyer. Then we show that such result can be employed in the perturbation analysis of the conformal sewing problem and of other problems.