Second-order Riesz transforms associated with magnetic Schrödinger operators

Let A=−(1i∇−a)2+V be a magnetic Schrödinger operator on RnRn, where a∈Lloc2(Rn)n and 0≤V∈Lloc1(Rn). We show that Lp(Rn)Lp(Rn) boundedness of LjLkA−1LjLkA−1 and VA−1VA−1 for some interval of p   automatically implies boundedness of the same operators and their commutators on Lwp(Rn) for certain Muckenhoupt weights w, and on the Musielak–Orlicz Hardy type spaces.