Title Characterisation of BMO($\mathbb{R}$) and VMO($\mathbb{R}$) via commutators with Cauchy integrals Trang Nguyen (University of South Australia) Trang Thi Thien Nguyen Analysis and Partial Differential Equations 15:00:00 2017-09-25  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.