Complex interpolation between $L^2$ and $H_0^1$

Question

Let $\Omega \subset \mathbb{R}^n$ be any smooth bounded domain and consider the complex interpolation space $X_\theta=[L^2(\Omega), H_0^1(\Omega)]_\theta$, $0<\theta<1$. I want to show the equivalence of norms of $X_\theta$ and $H^\theta(\Omega)$. I get the first one but I'm stuck on the other, namely, $$\|f\|_{X_\theta} \le C_\theta \|f\|_{H^\theta(\Omega)}.$$ If someone could give a proof or any reference for this result, it would be really helpful.