Extremal problems in BMO and VMO involving the Garsia norm

Abstract

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