Bergman Space Properties of Fractional Derivatives of the Cauchy Transform of a Certain Self-Similar Measure

Let μ be a self-similar measure with compact support K. The Hausdorff dimension of K is α. The Cauchy transform of μ is denoted by F(z). For 0<β<1, we define the function F[β], which compares with the fractional derivative of F of order β. Let Φ(z)=F(1/z),|z|<1. In this paper, we prove that Φ[β] belongs to Ap for 0<p<1/(β+1), and (Φ′)[β] belongs to Ap for 1≤p<1/β≤1/(2−α), where Ap is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem.