Hardy inequality and fractional Leibnitz rule for perturbed Hamiltonians on the line

We consider the following perturbed Hamiltonian $\mathcal{H}= -\partial_x^2 + V(x)$ on the real line. The potential $V(x)$ is a real - valued function of short range type. We study the equivalence of classical homogeneous Sobolev type spaces $\dot{H}^s_p$, $p \in (1,\infty)$ and the corresponding perturbed homogeneous Sobolev spaces associated with the perturbed Hamiltonian. It is shown that the assumption zero is not a resonance guarantees that the perturbed and unperturbed homogeneous Sobolev norms of order $s = \gamma - 1 \in [0,1/p)$ are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Sobolev spaces of order $s \in [0,1/p)$ invariant.