Abstract
| A function of bounded mean oscillation (BMO) is a function whose average
oscillation is bounded. Vanishing mean oscillation (VMO) functions oscillate
like BMO functions, but less so at small scales. Recently, an active research
area in harmonic analysis is the relationship between BMO and VMO functions and
the commutator with a singular integral operator.
I will present our work on characterisations of the function spaces
BMO($\mathbb{R}$) and VMO($\mathbb{R}$) in terms of the boundedness and
compactness on $L^p(\mathbb{R})$ of commutators~$[b,C_A](f) = bC_A(f) - C_A(bf)$
with Cauchy integral~$C_A$. Specifically, we show that for all~$p \in
(1,\infty)$, a locally integrable function~$b$ is in BMO($\mathbb{R}$) if and
only if~$[b,C_A]$ is bounded on $L^p(\mathbb{R})$. We also show that a function
$b \in \text{BMO}(\mathbb{R})$ is in VMO($\mathbb{R}$) if and only if~$[b,C_A]$
is compact on~$L^p(\mathbb{R})$.
This project is a part of my PhD, and is joint work with Ji Li, Lesley Ward
and Brett Wick. |