Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential onL 2 ( ( 0 , ∞ ) , x αd x ) , namely L f ( x ) = − f ′ ′( x ) − α x f ′ ( x ) + V ( x ) f ( x ) . We assume that α > 0 and V ∈ L l o c 1 ( ( 0 , ∞ ) , x αd x )is a nonnegative function. By definition, a function f ∈ L 1 ( ( 0 , ∞ ) , x αd x )belongs to the Hardy spaceH 1 ( L )ifsup t > 0e − t L f ∈ L 1 ( ( 0 , ∞ ) , x αd x ) . Under certain assumptions on V we characterize the spaceH 1 ( L )in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to L for α ∈ ( 0 , 1 ) with no additional assumptions on the potential V .