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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. 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