Maximality theorems for Fréchet algebras

The algebra of unbounded holomorphic functions $\bigcap_{p \geq 1}H^p(\partial \mathbb{D} )$ that is contained in the algebra $\bigcap_{p \geq 1}L^p(\partial \mathbb{D} )$ is studied. For $f$ in $\bigcap_{p \geq 1}L^p(\partial \mathbb{D} )$ but not in $\bigcap_{p \geq 1}H^p(\partial \mathbb{D} )$, we show that the algebra generated by $\bigcap_{p \geq 1}H^p(\partial \mathbb{D} )$ and $f$ is dense in $L^p(\partial \mathbb{D} )$ for all $p \geq 1$.