  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 $\Gamma=\{x+iA(x)\, : \, x\in \mathbb{R}\}$.
