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

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