Speaker: Ji Li, Macquarie University

Title: Weak factorization of Hardy spaces and characterization of BMO spaces in the Bessel setting

Abstract: It is well-known that the classical Hardy space $H^p$, $0 “A function $f$ is in $H^1( \mathbb{D})$ if and only if there exist $g,h \in H^2(\mathbb{D})$ with $f= g\cdot h$ and $\|f\|_{H^1(\mathbb{D})}=\|g\|_{H^2(\mathbb{D})}\|h\|_{H^2(\mathbb{D})}$.”

This factorization plays an important role in studying function theory and operator theory connected to the spaces $H^1(\mathbb{D})$, $H^2(\mathbb{D})$ and the space $BMOA(\mathbb{D})$ (analytic BMO). The analogue of the Riesz factorization theorem, sometimes referred to as strong factorization, is not true for real-variable Hardy space $H^1(\mathbb{R}^n)$. Nevertheless, Coifman, Rochberg and Weiss provided a suitable replacement that works in studying function theory and operator theory of $H^1(\mathbb{R}^n)$, the weak factorization via a bilinear form related to the Riesz transform (Hilbert transform in dimension 1).

We study the analogue of the result of Coifman, Rochberg and Weiss for the Hardy spaces associated with differential operators $L$ developed in recently ten years. Then we further provide a characterization of BMO spaces associated with $L$ in terms of the commutators related to the Riesz transform $\nabla L^{-1/2}$. Examples of such operators $L$ include the Neumann Laplacian and the Bessel operators.

This is joint work with Xuan Duong, Brett D. Wick and Dongyong Yang.