The Riemann Boundary Value Problem on Non-rectifiable Arcs and the Cauchy Transform
In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable arcs on the complex plane. We construct this integral as the convolution of the distribution (2πiz)-1 with a certain distribution such that its support is a non-rectifiable arc. These convolutions are called Cauchy transforms. As an application, solvability conditions of the Riemann boundary value problem are derived under very weak conditions on the boundary.
KeywordsNon-rectifiable arc metric dimension Cauchy transform Riemann boundary value problem
Unable to display preview. Download preview PDF.
- 1.L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution theory and Fourier Analysis, Springer Verlag, 1983.Google Scholar
- 4.I. Feder, Fractals, Mir Publishers, Moscow, 1991.Google Scholar
- 5.B.A. Kats, The Cauchy transform of certain distributions with application, Complex Analysis and Operator Theory, to appear.Google Scholar