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

## TL;DR

This paper establishes a weak factorization for Meyer-type Hardy spaces and characterizes their duals and preduals via boundedness and compactness of commutators with the Cauchy integral operator associated to a Lipschitz curve.

## Contribution

It introduces a weak factorization for $H^1_b(
eal)$ and characterizes ${
m BMO}_b(
eal)$ and ${
m VMO}_b(
eal)$ via commutator boundedness and compactness.

## Key findings

- Weak factorization for Meyer-type Hardy space $H^1_b(
eal)$.
- Characterizations of ${
m BMO}_b(
eal)$ and ${
m VMO}_b(
eal)$ via commutator properties.
- Connection between the Cauchy integral operator and function space dualities.

## Abstract

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