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 Delta(N). For the Hardy space H-Delta N(1) (R-n) (which is a proper subspace of the classical Hardy space H-1 (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 BMO Delta N (R-n) (which contains the classical BMO(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-infinity(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 (R-n) and BMO(R-n). (C) 2017 Elsevier Inc. All rights reserved.