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In this paper we study the cubic fractional nonlinear Schrodinger equation(NLS) on the torus and on the real line. Combining the normal form and therestricted norm methods we prove that the nonlinear part of the solution issmoother than the initial data. Our method applies to both focusing anddefocusing nonlinearities. In the case of full dispersion (NLS) and on thetorus, the gain is a full derivative, while on the real line we get aderivative smoothing with an $\epsilon$ loss. Our result lowers the regularityrequirement of a recent theorem of Kappeler et al. on the periodic defocusingcubic NLS, and extends it to the focusing case and to the real line. We alsoobtain estimates on the higher order Sobolev norms of the global smoothsolutions in the defocusing case.

Smoothing for the fractional Schrodinger equation on the torus and the real line