Extremal problems in BMO and VMO involving the Garsia norm

Abstract

Given an $L^2$ function $f$ on the unit circle $\mathbb T$, we put $$\Phi_f(z):=\mathcal P(|f|^2)(z)-|\mathcal Pf(z)|^2,\qquad z\in\mathbb D,$$ where $\mathbb D$ is the open unit disk and $\mathcal P$ is the Poisson integral operator. The Garsia norm $\|f\|_G$ is then defined as $\sup_{z\in\mathbb D}\Phi_f(z)^{1/2}$, and the space ${\rm BMO}$ is formed by the functions $f\in L^2$ with $\|f\|_G<\infty$. If $\|f\|^2_G=\Phi_f(z_0)$ for some point $z_0\in\mathbb D$, then $f$ is said to be a norm-attaining ${\rm BMO}$ function, written as $f\in{\rm BMO}_{\rm na}$. Note that ${\rm BMO}_{\rm na}$ contains ${\rm VMO}$, the space of functions with vanishing mean oscillation. We study, first, the functions $f$ in $L^\infty$ (as well as in $L^\infty\cap{\rm BMO}_{\rm na}$) with the property that $\|f\|_G=\|f\|_\infty$. The analytic case, where $L^\infty$ gets replaced by $H^\infty$, is discussed in more detail. Secondly, we prove that every function $f\in{\rm BMO}_{\rm na}$ with $\|f\|_G=1$ is an extreme point of ${\rm ball}\,({\rm BMO})$, the unit ball of ${\rm BMO}$ with respect to the Garsia norm. This implies that the extreme points of ${\rm ball}\,({\rm VMO})$ are precisely the unit-norm ${\rm VMO}$ functions. As another consequence, we arrive at an amusing "geometric" characterization of inner functions.