Marcus Pasquariello will give a talk at 09/17/2024

Many of the fundamental spaces in harmonic analysis, such as $BMO(\mathbb{R}^n)$, are defined in terms of averages. Is it possible to find equivalent definitions using medians instead of averages? In these two talks, we will discuss the median values $M_f(s, Q)$, and prove a charaterization of $BMO$ in terms of them. Namely, we show that $f \in BMO$ if and only if $\sup_Q[M_f(t, Q) - M_f(s, Q)] < \infty$ for some (or all) $0 < s < t < 1$.

We present two applications of this characterization. Firstly, we determine exactly which homeomorphisms $g: \mathbb{R} \to \mathbb{R}$ satisfy $g \circ f \in BMO$ for all $f \in BMO$. Secondly, we give a geometric characterization of the sets $E$ whose corresponding log-distance function $\log \text{dist}(\cdot, E)$ is in $BMO$.