In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian $\Delta_{ N_+}$ on the upper half space $\mathbb{ R}^n_+$ and the reflection Neumann Laplacian $\Delta_N$ on $\mathbb{ R}^n$ with respect to the weights associated to $\Delta_{ N_+}$ and $\Delta_{ N}$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to $\Delta_{ N}$ 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 area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on $\mathbb{ R}^n$ and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt $A^p$ weights with $p\in (1,2]$ while the previously known related results cover only $p\in (1,{ n+1\over n}]$. 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 $\Delta_{ D_+}$ on $\mathbb{ R}^n_+$.
Categories: math.AP math.CA