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Because, if this is possible, then it might help us to prove scattering of solutions of (\[Eqn:NlkgWdat\]) with rough data, by analogy with the scattering theory for data in the energy space. This paper gives a positive answer to this question, at least for $s$ close enough to one.\
Of course, one cannot use the energy conservation law as because it can be infinite on $H^{s} \times H^{s-1}$, $s<1$. Instead we work with a mollified energy (defined in (\[Eqn:DefMolNrj\])) that is finite in these rough spaces (see Proposition \[Prop:EstInitMolNrj\]) and that is expected to be almost conserved [^4]: this is the $I$-method, introduced first in [@almckstt] to study global existence of rough solutions of semilinear Schrödinger equations and inspired by the *Fourier truncation method*, designed in [@bourgbook]. Since we basically introduced this multiplier $I$ to make the energy finite, we aim at proving decay estimates involving not only $u$ but also $I$. It was proved in [@triroyradnlkg] that, under the additional assumption of radial symmetry, we can control pretty easily $\| Iu \|_{L_{t}^{p+2} L_{x}^{p+2} (\mathbb{R})}$ by combining the “Almost Morawetz-Strauss estimate” (\[Eqn:AlmMorStrEst\]) with a pointwise decay estimate around the origin (a radial Sobolev inequality). Unfortunately, there is no useful 