Extremal problems in BMO and VMO involving the Garsia norm
Konstantin M. Dyakonov
TL;DR
The paper investigates extremal problems for the Garsia norm $\|\cdot\|_G$ on ${\rm BMO}$ and ${\rm VMO}$, linking norm attainment to inner-outer factorization and the geometry of unit balls. It proves that every nonnegative $\eta\in L^ ty$ arises as the modulus of a ${\rm G}$-extremal function from a small subalgebra (e.g., $H^ ty+C$), and it gives sharp criteria for when outer and inner factors yield ${\rm G}$-extremality, including an Axler-type refinement. It shows that ${\rm BMO}_{\rm na}$-unit functions with $\|f\|_G=1$ are extreme points of ball$({\rm BMO})$, and that unit-norm ${\rm VMO}$ functions are exactly the extreme points of ball$({\rm VMO})$, yielding a simple geometric picture for inner functions. The work also provides a geometric characterization of inner functions and ends with an explicit construction of a $G$-extremal outer function in $QA$, illustrating the extremal phenomena in concrete terms.
Abstract
Given an $L^2$ function $f$ on the unit circle $\mathbb T$, we put $$Φ_f(z):=\mathcal P(|f|^2)(z)-|\mathcal Pf(z)|^2,\qquad z\in\mathbb D,$$ 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 $\sup_{z\in\mathbb D}Φ_f(z)^{1/2}$, and the space ${\rm BMO}$ is formed by the functions $f\in L^2$ with $\|f\|_G<\infty$. If $\|f\|^2_G=Φ_f(z_0)$ 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.