Cauchy Transforms On A Class Of Fractals And Problems Related To Them

Let K be the 3-level Sierpinski gasket with vertices 1, e^2πi/3,e^4πi/3;it has Hausdorff dimension α = 1 + log2/log3. Let μ be the normalized a-Hausdorff measure on K, i.e. μ(K) = 1. In the first part of this thesis, we consider mainly the Cauchy transform of the measure μ, F(z) = ∫_K(z - w)^-1dμ(w), and some auxiliary functions related to such transform. Firstly, we study the properties of these auxiliary functions. Then making use of these properties, we get some analytic and fractal properties and a property of F(z) on function spaces .In the second part of this thesis, let K be the Sierpinski gasket with vertices 1, e^2πi/3,e^4πi/3. It has Hausdorff dimension α = log2/log3. Let μ, be the normalized a-Hausdorff measure on K. We consider an auxiliary function related to the Cauchy transform of the measure μ,, and get that it preserves sign on negative real axis. The property play a role in the study of the transform.