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We study growth of higher Sobolev norms of solutions to the one-dimensionalperiodic nonlinear Schrodinger equation (NLS). By a combination of the normalform reduction and the upside-down I-method, we establish \|u(t)\|_{H^s}\lesssim (1+|t|)^{\alpha (s-1)+} with \alpha = 1 for a general powernonlinearity. In the quintic case, we obtain the above estimate with \alpha =1/2 via the space-time estimate due to Bourgain [4], [5]. In the cubic case, weconcretely compute the terms arising in the first few steps of the normal formreduction and prove the above estimate with \alpha = 4/9. These results improvethe previously known results (except for the quintic case.) In Appendix, wealso show how Bourgain's idea in [4] on the normal form reduction for thequintic nonlinearity can be applied to other powers.

A remark on normal forms and the "upside-down" I-method for periodic NLS: growth of higher Sobolev norms