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 A p for 0 < p < 1 / (β + 1) , and (Φ ′) [ β ] belongs to A p for 1 ≤ p < 1 / β ≤ 1 / (2 − α) , where A p is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem. [ABSTRACT FROM AUTHOR]