Let $S_j(z)={\epsilon }_j+(z-{\epsilon }_j)/2$ be an iterated function system, where ${\epsilon }_j=e^{2j\pi i/3}$ 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 $F^{\prime }$ belongs to $H^p$ for $0<p<1/(2-\alpha )$ and that $F$ is a multiplier of some class of function space.