Extremals for linear functionals on $H^p$ are outer when $0

Abstract

Let $L$ be a nontrivial bounded linear functional on the Hardy space $H^p$ of the unit disc. We prove that if $0<p<1$ and if $f$ in the unit sphere of $H^p$ satisfies $L(f)=\|L\|$, then $f$ is an outer function. The proof is short and relies on a construction due to de Leeuw and Rudin. Our result complements work of Duren, Romberg, and Shields on the failure of the Hahn--Banach theorem for $H^p$. When applied to the special case $L_k(f) = f^{(k)}(0)/k!$ for $k=1,2,3,\ldots$, it resolves two conjectures of the first-named author and Saksman and one conjecture of Suffridge.