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.

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

LANZA DE CRISTOFORIS, MASSIMO;PRECISO, LUCA
2001

Abstract

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.
2001
International Conference on Complex Analysis and Related Topics
International Conference on Complex Analysis and Related Topics
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/2471550
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