A note on weak factorization of Meyer-type Hardy space via Cauchy integral operator

This paper provides a weak factorization for the Meyer-type Hardy space $H^1_b(\mathbb{R})$, and characterizations of its dual ${\rm BMO}_b(\mathbb{R})$ and its predual ${\rm VMO}_b(\mathbb{R})$ via boundedness and compactness of a suitable commutator with the Cauchy integral $\mathscr{C}_{\Gamma}$, respectively. Here $b(x)=1+iA'(x)$ where $A'\in L^{\infty}(\mathbb{R})$, and the Cauchy integral $\mathscr{C}_{\Gamma}$ is associated to the Lipschitz curve $\Gamma=\{x+iA(x)\, : \, x\in \mathbb{R}\}$.