Bounds on the growth of high Sobolev norms of solutions to the fractional nonlinear Schrödinger equation on ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$

We prove polynomial bounds on the growth of high Sobolev norms of solutions to the defocusing fractional nonlinear Schrödinger equation with cubic and Hartree-type nonlinearities in two and three dimensions. Our result is based on the upside-down I-method and the method of higher modified energies. The analysis of the fractional case in higher dimensions is possible due to a higher-dimensional analogue of a resonance inequality, which allows us to control the nonresonant frequency contributions.