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In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian ΔN+ on the upper half space R+ n and the reflection Neumann Laplacian ΔN on Rn with respect to the weights associated to ΔN+ and ΔN respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to ΔN is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt–Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via the area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on Rn and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt Ap weights with p∈(1,2] while the previously known related results cover only p∈(1,n+1/n]. We also point out that this two weight commutator theorem might not be true in the setting of general operators L, and in particular we show that it is not true when L is the Dirichlet Laplacian ΔD+ on R+ n.
- Dirichlet and Neumann Laplacians
- Riesz transform
- Two weight