I am studying Hardy space on the upper half plane $\mathbb{H}$ recently.
Wiki says that there is a isometic isomorphism between Hardy space on the upper half plane and Hardy space on the unit disk $M$:$H^2(\mathbb{H}) \rightarrow H^2(\mathbb{D})$ by the following map: $$(Mf)(z)=\frac{\sqrt{\pi}}{1-z}f(i\frac{1+z}{1-z})$$
By change of variables, I can check that $M$ is an isometric and isomorphism between $L^2(\mathbb{H})$ and $L^2(\mathbb{D})$.But I can not prove $M$ is an isometric isomorphism between Hardy spaces.
Any help would be appreciated!