On homogeneous Besov spaces for $1D$ Hamiltonians without zero resonance

We consider 1-D Laplace operator with short range potential V(x), such that $$(1+|x|)^\gamma V(x) \in L^1(R), \ \ \gamma > 1.$$ We study the equivalence of classical homogeneous Besov type spaces $\dot{B}^s_p(R)$, $p \in (1,\infty)$ and the corresponding perturbed homogeneous Besov spaces associated with the perturbed Hamiltonian $\mathcal{H}= -\partial_x^2 + V(x)$ on the real line. It is shown that the assumptions $1/p < \gamma -1$ and zero is not a resonance guarantee that the perturbed and unperturbed homogeneous Besov norms of order $s \in [0,1/p)$ are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Besov spaces of order $s \in [0,1/p)$ invariant.