Characterizations of $H^1_{\Delta_N}(\mathbb{R}^n)$ and $\rm BMO_{\Delta_N}(\mathbb{R}^n)$ via Weak Factorizations and Commutators

This paper provides a deeper study of the Hardy and $\rm BMO$ spaces associated to the Neumann Laplacian $\Delta_N$. For the Hardy space $H^1_{\Delta_N}(\mathbb{R}^n)$ (which is a proper subspace of the classical Hardy space $H^1(\mathbb{R}^n)$) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space ${\rm BMO}_{\Delta_N}(\mathbb{R}^n)$ (which contains the classical $\rm BMO(\mathbb{R}^n)$) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on $L^\infty(\mathbb{R}^n)$ functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for $H^1(\mathbb{R}^n)$ and $\rm BMO(\mathbb{R}^n)$.