Let be a Schrödinger operator on , , where is a fixed non-negative potential and belongs to the reverse Hölder class for some ; that is, there exists a positive constant such that for every ball . Moreover, the measure satisfies the doubling condition: there exists a positive constant such that for all and . Let be the semigroup of linear operators generated by and be their kernels, that is, for all ,
A weighted Hardy-type space related to is defined by
where for all and is a Muckenhoupt weight. A function is called an atom for the weighted Hardy space associated to a ball for some and , if , , if , then and, if and , then , where for and . The atomic norm of is defined by , where the infimum is taken over all decompositions , and are atoms. The authors establish the following atomic characterization for : Assume that is a non-negative potential and , then the norms and are equivalent. The authors also characterize by the Riesz transforms for : if is a non-negative potential, then there exists a positive constant such that
The authors also prove that the dual space of is , where means that and there exists a positive constant such that for all with and . A positive measure on is called an Carleson measure if
The authors prove that, if , is a non-negative potential in for some and a weight in , then is an Carleson measure, where
and, conversely, if and is a Carleson measure, then . Finally, the authors establish the boundedness of the Hardy-Littlewood maximal operator and the semigroup maximal operator on .