Characterizations of HΔN1(Rn) and BMOΔN(Rn) via weak factorizations and commutators

This paper provides a deeper study of the Hardy and BMO spaces associated to the Neumann Laplacian ΔN. For the Hardy space HΔN1(Rn) (which is a proper subspace of the classical Hardy space H1(Rn)) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space BMOΔN(Rn) (which contains the classical BMO(Rn)) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on L∞(Rn) 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 H1(Rn) and BMO(Rn).