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

Abstract

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 ${\dot{H}}_p^s$, $p\in (1,\infty )$ 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.

References in corpus (1)

• A remark on an endpoint Kato-Ponce inequality