Is $H^\infty(\mathbb{C_+}) \subset H^2(\mathbb{C_+})$?

Question

I know that $H^\infty(\mathcal{D}(0,1)) \subset H^2(\mathcal{D}(0,1))$, and that there is a mapping $$f: \mathcal{D}(0,1)\to \mathbb{C}_+ : z\mapsto s=\frac{1-z}{1+z}.$$

However in $\mathbb{C_+}$ I have an "infinite domain of integration" somehow, so I don;t know if the result can be extended.

Answer 1

The constant function $1$ belongs to $H^{\infty}(\mathbb C_{+})$ but not to $H^{2}(\mathbb C_{+})$.