Title: Connections Between Continuous and Dyadic Function Spaces in Harmonic Analysis
Speaker: Lesley Ward (Univ. South Australia)
Abstract: The function spaces of harmonic analysis come in both continuous and dyadic flavours. These function spaces include the functions of bounded mean oscillation (BMO), the functions of vanishing mean oscillation (VMO), the Hardy space ${H}^{1}$, and the classes of ${A}_{p}$ weights and reverse-Holder weights. We know of two types of connections between the continuous and dyadic versions of such a space. First, averaging procedures take us from the dyadic to the continuous version. Second, the continuous version can be written as an intersection (for BMO, VMO, ${A}_{p}$ and reverse-Holder weights), or a sum (for ${H}^{1}$), of finitely many dyadic versions. We give an overview of recent work on these dyadic structure theorems, both when the underlying space is Euclidean ${ℝ}^{n}$ and also in the setting of spaces of homogeneous type $\left(X,d,\mu \right)$ in the sense of Coifman and Weiss. We consider both the one-parameter and product situations. Our results build on earlier work by Garnett, Jones, and Treil. This is joint work with P. Chen, A. Kairema, J. Li, J. Pipher, M.C. Pereyra, and X. Xiao.