993.
In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian
on the upper half space
and the reflection Neumann Laplacian
on
with respect to the weights associated to
and
respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to
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
and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt
weights with
while the previously known related results cover only
. 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
on
.
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