Let $\mathbb D= \{z\in\mathbb{C}: |z|<1\}$. Let $Hol(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows.
$$H^p=\{f\in Hol(\mathbb {D}):\sup_{r<1}\int_{0}^{2\pi} |f(re^{i\theta}|^pd\theta<\infty\}\;\;\;(0<p<\infty),\\ \|f\|_p^p= \sup_{r<1}\int_{0}^{2\pi} |f(re^{i\theta}|^pd\theta ,$$
$$H^\infty=\{f\in Hol(\mathbb {D}):\sup_{z\in\mathbb D}|f(z)|<\infty\}, \|f\|_\infty = \sup_{z\in\mathbb D}|f(z)| .$$ I wanted to know if the above defined hardy spaces have a multiplication that is submultiplicative with the defined norms? Or if there is a text/ reference with the details? This post here suggests that point wise multiplication is submultiplicative.
Only $H^\infty$ is a Banach space under pointwise multiplication. The trouble with the other $H^p$'s is that, for $f\in H^p$, you only have that $f^2\in H^p$ if $f\in H^{2p}$ and there are clearly counter examples to this.