Two weight Commutators in the Dirichlet and Neumann Laplacian settings
  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 $\mathbb{R}^n_+$.
