Function space properties of the Cauchy transform on the Sierpinski gasket

Abstract

Let $ S_j(z) = \varepsilon_j +(z-\varepsilon_j)/2 $ be an iterated function system, where $ \varepsilon_j = e^{2j\pi i/3} $ for $ j = 0, 1, 2 $. Then, there exists a uniform self-similar measure $ \mu $ supported on a compact set $ K $, which is the attractor of $ \{S_j\}_{j = 0}^2 $. The Hausdorff dimension of the attractor $ K $ is $ \alpha = \log 3/\log 2 $. Let $ F(z) = \int_{K}(z-\omega)^{-1}d\mu(\omega) $ be the Cauchy transform of $ \mu $. In this paper, we consider the Hardy space and the multiplier property of $ F $. We prove that $ F' $ belongs to $ H^p $ for $ 0 < p < 1/(2-\alpha) $ and that $ F $ is a multiplier of some class of function space.

Keywords

• cauchy transform
• sierpinski gasket
• self-similar measure
• hardy space
• multiplier