Maximality theorems for Fréchet algebras

Abstract:

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 $p \geq 1$.