Let $\mathbb A^2 \left (\mathbb D^2 \right )$ be the Bergman space consisting of square integrable holomorphic functions on $\mathbb D^2$ and $\mathbb P : L^2 \left (\mathbb D^2 \right ) \longrightarrow \mathbb A^2 \left (\mathbb D^2 \right )$ be the orthogonal projection. Then $\mathbb P$ maps $L^q \left (\mathbb D^2 \right )$ boundedly onto $\mathbb A^q \left (\mathbb D^2 \right )$ for all $q \geq 2,$ where $\mathbb A^q \left (\mathbb D^2 \right )$ is the Bergman space consisting of $L^q$-integrable holomorphic functions on $\mathbb D^2.$
How can it be shown? Any suggestion in this regard would be greatly appreciated.
Thanks for your kind attention.
