Let , , be a non-negative locally integrable function that belongs to a reverse Hölder class for some , that is, there exists such that the reverse Hölder inequality
holds for every ball in . The critical radius function is defined as follows:
Let be a Schrödinger operator. And let be the associated Riesz transform vector and be its adjoint operator.
The space is the set of locally integrable functions satisfying
for all and , where and . Denote . Moreover, the authors introduce classes of weights that are given in terms of the critical radius function and they are larger than the classical weights.
For , the authors consider the commutator
where or .
Their main results are to obtain weighted , , and weak estimates for the commutator .