Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on L2((0,∞),xαdx), namely Lf(x)=−f′′(x)−αxf′(x)+V(x)f(x).We assume that α>0 and V∈Lloc1((0,∞),xαdx) is a nonnegative function. By definition, a function f∈L1((0,∞),xαdx) belongs to the Hardy space H1(L) if supt>0e−tLf∈L1((0,∞),xαdx).Under certain assumptions on V we characterize the space H1(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.