  We obtain polynomial bounds on the growth in time of Sobolev norm of
solutions to the cubic defocusing nonlinear Schrodinger equation on two
dimensional product space. We also give the angular improved bilinear
Strichartz estimates for frequency localized functions, which estimates are
used for enhancement of a smoothing estimates. Such upper bounds for the growth
of Sobolev norms measure the transfer of energy from low to high modes as time
grows on.
